On the Korányi spherical maximal function on Heisenberg groups

Abstract

We prove $L^p\rightarrow L^q$ estimates for the local maximal operator associated with dilates of the Kóranyi sphere in Heisenberg groups. These estimates are sharp up to endpoints and imply new bounds on sparse domination for the corresponding global maximal operator. We also prove sharp $L^p\rightarrow L^q$ estimates for spherical means over the Korányi sphere, which can be used to improve the sparse domination bounds in (Ganguly and Thangavelu in J Funct Anal 280(3):108832, 2021) for the associated lacunary maximal operator.

1 Introduction

Let ${{\mathbb {H}}}^n={{\mathbb {R}}}^{2n} \times {{\mathbb {R}}}$ be the Heisenberg group of real Euclidean dimension $2n+1$. We shall use the notation $x=({\underline{x}}, {{\bar{x}}})$, $y=({\underline{y}}, {{\bar{y}}})$, with ${\underline{x}}, {\underline{y}}\in {{\mathbb {R}}}^{2n}$ and ${{\bar{x}}}, {{\bar{y}}}\in {{\mathbb {R}}}$, to denote elements of ${{\mathbb {H}}}^n$. The group law is given by

$$\begin{aligned} x \cdot y =({\underline{x}}+{\underline{y}}, {{\bar{x}}}+{{\bar{y}}}+ \tfrac{1}{2}{\underline{x}}^\intercal J{\underline{y}}), \end{aligned}$$

where

$$\begin{aligned}J:=\begin{pmatrix} 0 &{} &{} I_n\\ -I_n &{} &{} 0 \end{pmatrix} \end{aligned}$$

is the $2n\times 2n$ standard symplectic matrix (and $I_n$ is the $n\times n$ identity matrix). Further, the Korányi norm of an element x is defined to be

$$\begin{aligned} |x|_K:=(|{\underline{x}}|^4+|{{\bar{x}}}|^2)^{\frac{1}{4}}. \end{aligned}$$

This norm is homogeneous of degree one with respect to the natural parabolic dilation structure $\delta _t(({\underline{x}},{{\bar{x}}})):=(t{\underline{x}},t^2 {{\bar{x}}})$ on ${{\mathbb {H}}}^n$. Let $S_K$ be the sphere centred at the origin, of radius one with respect to the Korányi norm. There exists a unique Radon measure $\mu$ on $S_K$ induced by the Haar measure on ${{\mathbb {H}}}^n$. For $t>0$, the averaging operator associated with the $\mu _t$, the t-dilate of this measure, is given by

$$\begin{aligned} {\mathcal {A}}_tf(x):=f*\mu _t(x)=\int _{S_K} f({\underline{x}}-t\underline{\omega },{{\bar{x}}}-t^2{{{\bar{\omega }}}}-\tfrac{t}{2} {\underline{x}}^\intercal J\underline{\omega } ))\,d\mu ((\underline{\omega },{{{\bar{\omega }}}})) \end{aligned}$$

(1.1)

and let ${\mathcal {A}}f(x,t):={\mathcal {A}}_tf(x)$.

Fig. 1

The Korányi sphere centred at the origin in ${{\mathbb {H}}}^1$

In 1981, Cowling [10] considered the global Korányi maximal function

$$\begin{aligned} {\mathfrak {M}} f(x):=\sup _{t>0}|{\mathcal {A}}_t f(x)|. \end{aligned}$$

He showed that ${\mathfrak {M}}$ is bounded on $L^p({{\mathbb {H}}}^n)$ for all $p>\frac{2n+1}{2n}$ and $n\ge 2$. In this paper, we consider the local maximal function

$$\begin{aligned} {\mathcal {M}}f(x):=\sup _{t\in [1,2]}|{\mathcal {A}}_t f(x)|. \end{aligned}$$

In light of the recent $L^p\rightarrow L^q$ estimates for the local maximal operator associated with codimension two spheres in the Heisenberg groups [2, 26] (see Remark 1.3), it is natural to seek similar estimates for ${{\mathcal {M}}}$. Our main theorem contains $L^p\rightarrow L^q$ estimates for ${\mathcal {M}}$ which are sharp up to endpoints (with the off-diagonal estimates being new) (Figs. 1, 2).

Theorem 1.1

Let $n\ge 2$. Let ${{\mathcal {R}}}$ be the closed quadrilateral with corners

$$\begin{aligned} \begin{aligned} Q_1&=(0,0), \qquad Q_2=\left( \tfrac{2n}{2n+1}, \tfrac{2n}{2n+1}\right) ,\\ Q_3&=\left( \tfrac{2n}{2n+1},\tfrac{1}{2n+1}\right) , \quad Q_4=\left( \tfrac{n(2n+1)}{2n^2+2n+2},\tfrac{n}{2n^2+2n+2}\right) . \end{aligned} \end{aligned}$$

Then

• (i) ${\mathcal {M}}:L^p({{\mathbb {H}}}^n)\rightarrow L^q({{\mathbb {H}}}^n)$ is bounded if $(\frac{1}{p}, \frac{1}{q})$ belongs to the interior of ${{\mathcal {R}}}$, or to the open boundary segment $(Q_2,Q_3)$, or to the half open boundary segment $[Q_1,Q_2)$.

• (ii) ${\mathcal {M}}$ does not map $L^p({{\mathbb {H}}}^n)$ to $L^q({{\mathbb {H}}}^n)$ if $(\frac{1}{p}, \frac{1}{q})\notin {{\mathcal {R}}}$.

• (iii) ${\mathcal {M}}$ does not map $L^p({{\mathbb {H}}}^n)$ to $L^p({{\mathbb {H}}}^n)$ for $(\tfrac{1}{p},\tfrac{1}{q})=Q_2$.

Fig. 2

The region ${\mathcal {R}}$ in Theorem 1.1, for $n=2$

Recently, in [11], Ganguly and Thangavelu considered spherical means over the unit Korányi sphere

$$\begin{aligned} {\mathcal {A}}_1f(x)=f*\mu (x). \end{aligned}$$

They proved that ${\mathcal {A}}_1$ maps $L^p({{\mathbb {H}}}^n)$ to $L^q({{\mathbb {H}}}^n)$ provided that $(\tfrac{1}{p}, \tfrac{1}{q})$ belongs to the interior of the triangle with corners (0, 0), (1, 1), $(\tfrac{2n}{2n+1}, \frac{1}{2n+1})$, as well as the line joining the points (0, 0) and (1, 1). Our second theorem is a sharpened version of their result.

Theorem 1.2

Let $n\ge 1$. The inequality

$$\begin{aligned} \Vert f*\mu \Vert _{L^q({{\mathbb {H}}}^n)} \lesssim \Vert f\Vert _{L^p({{\mathbb {H}}}^n) } \end{aligned}$$

holds for all $f\in L^p({{\mathbb {H}}}^n)$ if and only if $(\tfrac{1}{p}, \tfrac{1}{q})$ belongs to the closed triangle with corners (0, 0), (1, 1) and $\left( \frac{2n+1}{2n+2}, \frac{1}{2n+2}\right) .$

As we shall see, Theorem 1.2 can be proved using essentially known results on generalized Radon transforms and oscillatory integral operators with non-vanishing rotational curvature (Hörmander’s classical $L^2$ theory [33, Ch. IX.1]). The fact that the rotational curvature associated to the Kóranyi spherical averaging operator is non-vanishing has been observed by Schmidt in his unpublished Master’s thesis [28]; for the sake of completeness we shall show it in this paper as well. In contrast, the off-diagonal estimates in Theorem 1.1 are completely new and the estimate at $Q_4$, in particular, relies on a new kind of scaling argument (which we shall discuss below).

In [11], the authors’ main objective was to prove sharp $L^p$ estimates for the lacunary Korányi maximal function

$$\begin{aligned} {\mathfrak {M}}^{\text {lac}}f(x):=\sup _{k>0}|{\mathcal {A}}_{2^k} f(x)|. \end{aligned}$$

They showed that for $n\ge 2$, ${\mathfrak {M}}^{\text {lac}}$ is bounded on $L^p({{\mathbb {H}}}^n)$ for all $1<p<\infty$. Closely following Lacey’s approach in the Euclidean case [16], the authors achieved this by establishing a $(p,q')$-sparse domination result for ${\mathfrak {M}}^{\text {lac}}$ which made use of an induction argument relying on the $L^p\rightarrow L^q$ estimates for ${\mathcal {A}}_1$.

Although we are interested in the estimates contained in Theorems 1.1 and 1.2 in their own right, it can be shown using the arguments in [11] (see also [2]) that our results also imply sparse bounds on ${\mathfrak {M}}$ and ${\mathfrak {M}}^{\text {lac}}$ which are sharp up to endpoints (or even sharp in the case of ${\mathfrak {M}}^{\text {lac}}$).

There has been considerable progress in the recent years on the problem of establishing $L^p$ improving properties of localized maximal functions associated with surfaces, both in the Euclidean and Heisenberg settings. A few remarks are in order to shed light on the chief features of our problem which make it different from some related bodies of work.

Remarks 1.3

(i) Non-vanishing Rotational Curvature: A different maximal function on the Heisenberg group, considered in [2, 3, 21,22,23, 26], is associated to averages over codimension two spheres contained in a dilation invariant subspace of ${{\mathbb {H}}}^n$. In contrast, the averaging operator ${\mathcal {A}}_t$ in (1.1) is associated to a surface of codimension one which respects the non-isotropic dilation structure on ${{\mathbb {H}}}^n$. Further, unlike the codimension two case, the oscillatory integral operators associated with ${\mathcal {A}}_t$ possess non-vanishing rotational curvature at fixed time t (this fact is also implicit in [24] and proven directly in [28]). In this sense, ${\mathcal {A}}_t$ could be considered to be a faithful analog of the Euclidean spherical average in the Heisenberg setting.

(ii) Vanishing of Cinematic Curvature: Unlike the Euclidean spherical averaging operator, the rank of the cinematic curvature matrix (7.11) associated with ${\mathcal {A}}$ is zero at the north and south poles of the Korányi sphere. To deal with the issue of flatness (that is, the absence of cinematic curvature in the sense of [30]) at the poles, we need to employ a scaling argument. The first step in such an argument is to dyadically decompose the relevant kernel based on the distance from the poles, followed by rescaling each piece to a region where the curvature does not vanish. This idea was used by Iosevich [14] to establish the $L^p$ boundedness of global maximal operators associated to families of flat, finite type curves in ${{\mathbb {R}}}^2$. It was also used in [18] and more recently [17] to prove $L^p\rightarrow L^q$ estimates for local maximal functions along some finite type curves in ${{\mathbb {R}}}^2$ and hypersurfaces in ${{\mathbb {R}}}^3$. However, unlike [14, 17, 18], our problem is based in a non-Euclidean setting. The Heisenberg group structure calls for a new type of scaling argument.

1.1 Comparison with the Euclidean convolution structure

The Heisenberg group structure plays a crucial role in our analysis. It shows itself in the presence of the bilinear term involving the symplectic matrix J in the defining equation of the Korányi sphere centred at x and of radius t, given by

$$\begin{aligned} |{\underline{x}}-{\underline{y}}|^4+|{{\bar{x}}}-{{\bar{y}}}+\tfrac{1}{2}{\underline{x}}^{\intercal }J{\underline{y}}|^2=t^4. \end{aligned}$$

(1.2)

To illustrate its significance, we contrast our situation with that of the Korányi sphere in ${{\mathbb {R}}}^{2n+1}$ when the translations are Euclidean and not given by the Heisenberg law. The defining equation of such a sphere centred at x and of radius t is given by

$$\begin{aligned} |{\underline{x}}-{\underline{y}}|^4+|{{\bar{x}}}-{{\bar{y}}}|^2=t^4. \end{aligned}$$

(1.3)

Fig. 3

A comparison between the regions of $L^p\rightarrow L^q$ boundedness for the Korányi spherical maximal operator corresponding to Heisenberg convolution (in grey) versus the Euclidean convolution (in blue), for $n=2$

The quadrilateral ${\mathcal {R}}=Q_1Q_2Q_3Q_4$ in Fig. 3 (in grey) depicts the region of almost sharp $L^p\rightarrow L^q$ estimates for the Korányi spherical maximal operator with Heisenberg group structure (given by Theorem 1.1). Since the necessary conditions for this operator (see Sect. 9) also apply to the Korányi spherical maximal operator described with respect to Euclidean convolution, the region corresponding to the sharp $L^p\rightarrow L^q$ estimates for the Euclidean version is contained in ${\mathcal {R}}$. In fact, this containment is proper, as can be seen by means of a standard Knapp example adapted to a homogeneous Euclidean hypersurface of degree four, which yields the sharpness of the edge $P_3'P_4'$ in the interior of ${\mathcal {R}}$. The quadrilateral ${\mathcal {P}}=P_1P_2P_3P_4$ (in blue) depicts the region in which scaling type arguments can be used to obtain a positive result for the Korányi maximal operator in the Euclidean setting. Its coordinates in ${{\mathbb {R}}}^{2n+1}$ are given by

$$\begin{aligned} \begin{aligned}&P_1=(0,0), \qquad P_2=\left( \tfrac{n}{n+1}, \tfrac{n}{n+1}\right) , \\&P_3=\left( \tfrac{n}{n+1},\tfrac{1}{n+1}\right) , \quad P_4=\left( \tfrac{n(2n+1)}{2n^2+3n+2},\tfrac{2n}{2n^2+3n+2}\right) . \end{aligned} \end{aligned}$$

The estimate at $P_2$ is the same as the $L^p$ bound contained in [9, Theorem 3.1] for global maximal operators associated to averages over Euclidean hypersurfaces which are graphs of smooth homogeneous functions with non-vanishing Hessian away from the origin, with the degree of homogeneity being at most twice the dimension of the surface. The estimates corresponding to $P_3$ and $P_4$ can be obtained by following the same arguments as in the proof of Theorem 1.1 (but in a more straightforward manner). Eschewing detailed calculations, we now briefly discuss the reasons behind these differences between the Heisenberg and Euclidean settings.

In the Heisenberg case, the term ${\underline{x}}^{\intercal }J{\underline{y}}$ changes the geometry of the Korányi sphere as it is translated away from the origin and is responsible for the non-vanishing of the rotational curvature, even at the poles. As a result, Hörmander’s classical $L^2$ theory [33, Ch. IX.1] can be used to establish a sharp fixed time $L^2$ estimate. This in turn implies the positive result in Theorem 1.2 after interpolation with easily obtainable estimates involving the $L^1$ and $L^\infty$ spaces. Using standard Sobolev embedding, the estimates at the vertices $Q_1, Q_2, Q_3$ of the quadrilateral ${\mathcal {R}}$ also follow. They are sharp up to endpoints and analogous to the corresponding estimates for the standard sphere in ${{\mathbb {R}}}^{2n+1}$ defined using the Euclidean norm.

This is in sharp contrast to the Korányi sphere with Euclidean translational structure, where the rotational curvature vanishes at the poles and a scaling argument (along the lines of [14]) is needed. Further, the fixed time $L^2$ estimate obtained using scaling is worse than the same estimate for the standard Euclidean sphere, and consequently, so are the corresponding diagonal and off-diagonal estimates.

The Heisenberg estimate at $Q_4$ is indeed affected by the flatness at the poles, as is the corresponding estimate at $P_4$ for the Korányi sphere in the Euclidean case. In both cases, after suitable decomposition and scaling, we need to prove a fixed time $L^2$ estimate for each rescaled piece separately. However, in order to establish these estimates in the Heisenberg case, we need to introduce further localization and make use of an almost $L^2$ orthogonality that exists between the different localized pieces (see Sect. 7.1.1). This is due to the surprising feature that the non-isotropic scaling in the “input variable” y is different from the one which shows up in the “output variable” x, with x, y in (1.2). This phenomenon is absent in the Euclidean version and occurs, again, due to the presence of the bilinear term ${\underline{x}}^\intercal J{\underline{y}}$ in the second term in (1.2) (see Sect. 4, Remark 4.1).

Quite pleasantly, this “imbalanced” scaling argument yields a better estimate at $Q_4$ in the Heisenberg case than the estimate at $P_4$ for the Korányi sphere with Euclidean convolution structure. The former estimate is consistent with a new Knapp-type counterexample Sect. 9.5 which establishes the sharpness of the edge $Q_3Q_4$ in the Heisenberg case. As mentioned already, the edge $P_3P_4$ for the Korányi sphere in the Euclidean setting is also sharp, which can be seen by using a standard Knapp example adapted to a Euclidean degree four homogeneous surface.

The estimates at $Q_4$ (and $P_4$) also rely on an $L^\infty$ bound on the kernel of an associated oscillatory integral operator. To do so, we shall seek to make use of estimates for standard oscillatory integrals of Carleson–Sjölin–Hörmander type, in particular a variant of Stein’s theorem [32] formulated in [20] which relies on the maximal possible number of nonvanishing curvatures for a cone in the fibers of the canonical relation (see (7.12)).

The above discussion was focused on the estimates around the poles. Both the curvature conditions also hold at the equator and in the intermediate region of the Korányi sphere, enabling us to apply Hörmander’s $L^2$ estimate and the aforementioned Stein’s theorem directly. However, their verification is still technically involved in the Heisenberg case due to the additional bilinear term.

We conclude this section by briefly addressing natural questions related to the extension of our results to more general Lie groups and hypersurfaces other than the Korányi sphere.

Remarks 1.4

(i) Extensions to two step nilpotent groups with higher dimensional center: It is likely that the ideas of this paper can be directly extended to prove analogous results for maximal functions associated with Korányi spheres on groups of Heisenberg type [15], or even more generally, on Métivier groups [19]. These are two step nilpotent Lie groups with centers which can have dimension greater than one. Indeed, in [26], the positive results were established in the Metévier setting, but for maximal operators associated to averages over “singular spheres” with codimension one greater than the dimension of the center of the group. One of the main objectives of that paper was to demonstrate the surprising feature that the cinematic curvature of the associated Fourier integral operators remained unaffected by the more singular nature of these surfaces.

In contrast, the problem considered in this paper does not possess the above feature; as any analog of the Korányi sphere for Métivier groups will still be a hypersurface. Thus, a generalization of our results to this setting, even though technically more involved, may not be as striking as the corresponding results in [26]. Further, many of the intriguing features particular to this problem are already exhibited in the case of Heisenberg groups, which is the setting we choose to work in for clarity of exposition.

(ii)Extension to other hypersurfaces on the Heisenberg group: Another pertinent question is whether our methods can be used to establish similar results for other hypersurfaces compatible with the non-isotropic dilation structure on the Heisenberg groups. Indeed, the idea of establishing estimates for the averaging operator associated to a surface by examining the non-vanishing curvatures of the corresponding oscillatory integral operators is very general. That being said, the estimates obtained will depend on the geometric nature of the hypersurface (for example, its degree of flatness). Further, while not every surface might be as amenable to scaling as the Korányi sphere is around the poles where the cinematic curvature vanishes, it is still conceivable to use this argument for surfaces with polynomial defining functions which are radial in the horizontal variables.

1.2 Plan of the paper

The paper is organized as follows:

• Section 2 contains a localization argument for the function f, and a partition of unity argument which streamlines our analysis into three regimes, based on whether the kernel of the Fourier integral operator being considered is supported around the poles, at the equator or in an intermediate region. We also describe the corresponding parametrization of the Korányi sphere (explicitly around the poles and implicitly in the other two regimes).

• In Sect. 3 we list the basic estimates and reduce them to corresponding ones for oscillatory integral operators.

• In Sect. 4 we use a scaling argument to understand the behaviour of our operator around the poles (or the north pole to be more specific, but the behaviour around the south pole is exactly the same due to symmetry).

• Section 5 describes a Stein–Tomas type argument to reduce the estimate at $Q_4$ to an $L^2$ estimate and an $L^\infty$ bound on the kernel of an oscillatory integral operator.

• Section 6 contains a few technical results about matrices and radial functions which will be crucial to calculations verifying the non-vanishing of the rotation curvature and the additional curvature condition (7.12).

• In Sect. 7, we verify these two conditions around the poles (for the rescaled operators). Section 7.1.1 also contains an almost orthogonality argument crucial to the proof of the fixed time $L^2$ estimate for the rescaled operators.

• In Sect. 8, we verify the curvature conditions at the equator and in the intermediate region, thus finishing the proofs of the positive results contained in Theorems 1.2 and 1.1 .

• In Sect. 9 we provide counterexamples to show the sharpness of these theorems.

• In Sect. 10 we briefly discuss their implications on sparse bounds and weighted inequalities for the lacunary and global maximal functions.

• Section 11 contains a technical lemma about the invariance of both the curvature conditions under the Heisenberg group law, and the proof of an interpolation argument used at multiple instances in the paper.

1.3 Notation

Partial derivatives and tangent vectors along the coordinate direction $e_j$ will be denoted by the subscript j for $j\in \{1,2,\ldots , 2n+1\}$. For a function $f: {{\mathbb {R}}}^{2n}\rightarrow {{\mathbb {R}}}^{2n}$, $f''$ shall be used to denote its Hessian which is a $2n\times 2n$ matrix. By $A\lesssim B$ we shall mean that $A\le C\cdot B,$ where C is a positive constant and $A\approx B$ shall signify that $A\lesssim B$ and $B\lesssim A$. $A\lesssim _D B$ shall mean that $A\le C\cdot B$ with the positive constant C depending on the parameter D.

2 Preliminaries

2.1 Localization

Without loss of generality, we can limit our consideration to functions f supported in a small subset of a thin neighborhood of the Korányi sphere of radius one centred at the origin. To see this we use the group translation to tile ${{\mathbb {H}}}^n$. Let $B_0=[-\tfrac{1}{2},\tfrac{1}{2})^{2n+1}$ and, for ${{\mathfrak {n}}}\in {{\mathbb {Z}}}^{2n+1}$, let $B_{{{\mathfrak {n}}}}= {{\mathfrak {n}}}\cdot B_0$, i.e. $B_{{{\mathfrak {n}}}}=\{(\underline{{{\mathfrak {n}}}}+\underline{z},{{\bar{{{\mathfrak {n}}}}}}+{\bar{z}}+\tfrac{1}{2}\underline{{{\mathfrak {n}}}}J\underline{z}): z\in B_0\}$. One then verifies that $\sum _{{{\mathfrak {n}}}\in {{\mathbb {Z}}}^{2n+1} }{\mathbbm {1}}_{B_{{\mathfrak {n}}}}=1$. Moreover, for $t\in [1,2]$, the t-dilates of the measure $\mu$ are supported in $\{w\in {{\mathbb {H}}}^n: |\underline{w}|\le 2, |{\bar{w}}| \le 4\}$, hence in the union of $B_{{\mathfrak {k}}}$ with $|{{\mathfrak {k}}}_j|\le 2$ for $j\le 2n$ and $|{{\mathfrak {k}}}_{2n+1}|\le 6$. Denote this set of indices by ${{\mathfrak {J}}}$. Then

$$\begin{aligned}{\textrm{supp }}\big ({\mathcal {A}}_t[f{\mathbbm {1}}_{ B_{{\mathfrak {n}}}}]\big ) \subset \bigcup _{{{\mathfrak {k}}}\in {{\mathfrak {J}}}} ({{\mathfrak {n}}}\cdot B_0\cdot B_{{\mathfrak {k}}}) \subset \bigcup _{{{\tilde{{{\mathfrak {n}}}}}}\in {{\mathfrak {I}}}({{\mathfrak {n}}})} B_{{{\tilde{{{\mathfrak {n}}}}}}}, \end{aligned}$$

where ${{\mathfrak {I}}}({{\mathfrak {n}}})$ is a set of indices ${{\tilde{{{\mathfrak {n}}}}}}$ with $|{{\mathfrak {n}}}_j-{{\tilde{{{\mathfrak {n}}}}}}_j|\le C(J,n)$ for $j=1,\dots , 2n+1$. This consideration of spatial orthogonality allows us to reduce to the case of functions f supported in the union of a finite number of Heisenberg tiles $B_{{{\mathfrak {n}}}}$ which cover the unit Korányi sphere and its t dilates for $t\in [1,2]$. We can further partition these tiles (resp. the interval [1, 2]) into a finite number of sub-tiles (resp. sub-interval) of small enough size and consider the averaging operator associated to each such piece separately, summing up the estimates at the end. This reduces matters to the case when f is supported in a small subset of a thin neighborhood of the unit Korányi sphere, of length at most $\frac{2^{-400n}}{100n}$ in each coordinate direction. As a consequence, we can also assume that ${\mathcal {A}}_tf$ is supported in a small neighborhood of the origin of length at most $\frac{2^{-400n}}{10n}$ in each direction.

We shall frequently make use of both Taylor approximation and the implicit function theorem to express the phase function of oscillatory integral operators. Both these theorems require the corresponding amplitude to be supported in a set of constant but sufficiently small size. By choosing a suitable smooth partition of unity, we can express ${\mathcal {A}}_t$ as a finite sum of operators, each of which is localized to a small region of the Korányi sphere. It then suffices to prove the required estimates for each such operator independently. We now explicitly describe such a localization.

Given a point $w\in S_K$, let ${\mathcal {Q}}(w):=\{\tilde{w}\in S_K:|w-\tilde{w}|<2^{-400n}\}$. Since the Korányi sphere $S_K$ is a compact submanifold of ${{\mathbb {H}}}^n$, there exists a finite set of points $\{w_{\nu }\}_{\nu }\subseteq S_K$ such that the collection $\{{\mathcal {Q}}_{\nu }:={\mathcal {Q}}(w_{\nu })\}_{\nu }$ forms an open cover of $S_K$. Let $\sum _\nu \eta _\nu$ be a smooth partition of unity subordinate to $\{{\mathcal {Q}}_{\nu }\}_{\nu }$. Define $\mu ^{\nu }:=\mu \eta _{\nu }$ so that $\mu =\sum _\nu \mu ^{\nu }$.

We can rewrite (1.1) as

$$\begin{aligned} {\mathcal {A}}_tf(x):=\sum _\nu \int _{{\mathcal {Q}}_\nu } f({\underline{x}}-t\underline{\omega },{{\bar{x}}}-t^2{{{\bar{\omega }}}}-\tfrac{t}{2} {\underline{x}}^\intercal J\underline{\omega } )\,d\mu ^\nu (\underline{\omega },{{{\bar{\omega }}}}). \end{aligned}$$

Using a change of variables ${\underline{y}}={\underline{x}}-t{\underline{\omega }}$, ${{\bar{y}}}={{\bar{x}}}-t^2{{{\bar{\omega }}}}-\tfrac{t}{2} {\underline{x}}^\intercal J\underline{\omega }$, we can express

$$\begin{aligned} {\mathcal {A}}_tf(x):=\sum _\nu \int _{{\mathcal {Q}}_\nu } f({\underline{y}},{{\bar{y}}})\,d\mu ^\nu \left( \frac{{\underline{x}}-{\underline{y}}}{t},\frac{{{\bar{x}}}-{{\bar{y}}}+\tfrac{1}{2}{\underline{x}}^{\intercal }J{\underline{y}}}{t^2}\right) . \end{aligned}$$

2.2 Parametrization

The defining equation of the Korányi sphere centred at x and of radius t is

$$\begin{aligned} F(x,t,y):=|{\underline{x}}-{\underline{y}}|^4+\left| {{\bar{x}}}-{{\bar{y}}}+\tfrac{1}{2}{\underline{x}}^{\intercal }J{\underline{y}}\right| ^2-t^4=0. \end{aligned}$$

(2.1)

Differentiating, we have

$$\begin{aligned} \nabla _{x,t}F(x,t,y)= & {} 4|{\underline{x}}-{\underline{y}}|^2({\underline{x}}-{\underline{y}})\nonumber \\{} & {} +2\left( {{\bar{x}}}-{{\bar{y}}}+\tfrac{1}{2}{\underline{x}}^{\intercal }J{\underline{y}}\right) \left( \tfrac{1}{2}J{\underline{y}}+e_{2n+1}\right) +4t^3e_{2n+2}. \end{aligned}$$

(2.2)

To parametrize the Korányi sphere as the graph of a smooth function locally, we need to express one of the coordinates of y as a function of the other y-coordinates, x and t. The coordinate we choose, and hence the parametrization, will depend upon the localization neighborhood ${\mathcal {Q}}_\nu$.

Since the Korányi sphere is symmetric about the equator, we can limit our attention to its northern hemisphere. We will choose different parametrizations near the equator and around the north pole. Unlike in the case of the Euclidean sphere, we shall see that the curvature properties depend upon the neighborhood ${\mathcal {Q}}_\nu$ being considered. Heuristically speaking, due to the non-isotropic dilation structure of the Korányi sphere, the region around the equator exhibits similar curvature properties as the Euclidean sphere. However, around the north pole, the Korányi sphere behaves like the surface ${{\bar{y}}}=1+|{\underline{y}}|^4$, with the (cinematic) curvature vanishing at the pole. Other than these two extreme cases, the intermediate region needs to be carefully considered as well.

Remark 2.1

The above discussion only applies to the cinematic curvature condition (see (7.12)). The rotational curvature does not see these different parametrizations, and remains non-vanishing throughout. This has been observed before in the context of the Korányi sphere in [28] and is also to be expected in view of the $L^p$ bounds on the global maximal operator in [10].

Let ${\mathcal {J}}$ denote the indexing set of the cover $\{{\mathcal {Q}}_\nu \}_\nu$. Keeping the above discussion in mind, we partition ${\mathcal {J}}$ into three disjoint sets

$$\begin{aligned} {\mathcal {J}}={\mathcal {J}}_{\text {Eq}}\cup {\mathcal {J}}_{\text {IM}}\cup {\mathcal {J}}_{\text {NP}}, \end{aligned}$$

where

$$\begin{aligned} {\mathcal {J}}_{\text {NP}}&=\{\nu : e_{2n+1}\in {\mathcal {Q}}_\nu \}\\ {\mathcal {J}}_{\text {Eq}}&=\{\nu : \text { there exists } x=({\underline{x}},{{\bar{x}}})\in {\mathcal {Q}}_\nu \text { with } {{\bar{x}}}=0\},\\ {\mathcal {J}}_{\text {IM}}&={\mathcal {J}}\setminus ({\mathcal {J}}_{\text {Eq}}\cup {\mathcal {J}}_{\text {NP}}). \end{aligned}$$

2.2.1 Near the north pole

For $\nu \in {\mathcal {J}}_{\text {NP}}$ and for all $\left( \frac{{\underline{x}}-{\underline{y}}}{t},\frac{{{\bar{x}}}-{{\bar{y}}}+\tfrac{1}{2}{\underline{x}}^{\intercal }J{\underline{y}}}{t^2}\right) \in {\mathcal {Q}_\nu }$, we have $t^{-1}|{\underline{x}}-{\underline{y}}|<2^{-400n}$ and we can write

$$\begin{aligned} {\overline{x}}-{\overline{y}} + \frac{1}{2}{\underline{x}}^T J {\underline{y}}=t^2\sqrt{1-\frac{|{\underline{x}}-{\underline{y}}|^4}{t^4}} \end{aligned}$$

(2.3)

Using Taylor’s remainder formula, we can express

$$\begin{aligned} \sqrt{1-|{{\underline{w}}}|^4}= 1-\tfrac{|{{\underline{w}}}|^4}{2}+|{{\underline{w}}}|^8R({{\underline{w}}}), \end{aligned}$$

(2.4)

where R is a smooth, bounded and radial function such that

$$\begin{aligned} \left| \frac{\partial ^\beta }{\partial {{\underline{w}}}^{\beta }}R({{\underline{w}}})\right| \le 100n\,\, \text { for } |\beta |\in \{0,1,2,3\} \text { and } |{{\underline{w}}}|<\frac{1}{2}. \end{aligned}$$

(2.5)

Plugging (2.4) with ${{\underline{w}}}=\frac{{\underline{x}}-{\underline{y}}}{t}$ into (2.3), we obtain

$$\begin{aligned} {\overline{y}}={\mathcal {G}}(x,t,{\underline{y}}):={\overline{x}} + \frac{1}{2}{\underline{x}}^T J {\underline{y}}-t^2+t^2\frac{|{\underline{x}}-{\underline{y}}|^4}{2t^4}+\frac{|{\underline{x}}-{\underline{y}}|^8}{t^8}R\left( \frac{{\underline{x}}-{\underline{y}}}{t}\right) .\nonumber \\ \end{aligned}$$

(2.6)

2.2.2 Near the equator

Observe that for ${\mathfrak {A}}\in O(2n)$, the action $({\underline{x}},{{\bar{x}}},t,{\underline{y}},{{\bar{y}}})\rightarrow ({\mathfrak {A}}{\underline{x}},{{\bar{x}}},t,{\mathfrak {A}}{\underline{y}},{{\bar{y}}})$ leaves the first and third terms in (2.1) unchanged, and replaces J by $J'={\mathfrak {A}}^{\intercal }J{\mathfrak {A}}$ in the second term. Since $J'$ is still a skew-symmetric matrix with $J'^2=-I_{2n}$, as long as our analysis is uniform in the set of all matrices satisfying these two properties, using horizontal rotations, it suffices to consider only those $\nu \in {\mathcal {J}}_{\text {Eq}}$ such that the corresponding neighborhood ${\mathcal {Q}}_\nu$ contains $e_1$. Let ${\mathcal {J}}_{\text {Eq}}'$ denote the collection of all such $\nu$.

For any $\left( \frac{{\underline{x}}-{\underline{y}}}{t},\frac{{{\bar{x}}}-{{\bar{y}}}+\tfrac{1}{2}{\underline{x}}^{\intercal }J{\underline{y}}}{t^2}\right)$ in such a neighborhood ${\mathcal {Q}}_{\nu }$, $t^{-1}|x_1-y_1|>1-2^{-400n}$ and $t^{-2}|{{\bar{x}}}-{{\bar{y}}}+\tfrac{1}{2}{\underline{x}}^{\intercal }J{\underline{y}}|<2^{-400n}$, and in view of (2.2), this implies that $\left| \frac{\partial F}{\partial y_1}\right| >t(1-2^{-400n})$. Write

$$\begin{aligned}y=({\underline{y}},{{\bar{y}}})=(y_1,y',{{\bar{y}}}),\end{aligned}$$

with $y'\in {{\mathbb {R}}}^{2n-1}$. We can use the implicit function theorem to express

$$\begin{aligned} y_1= & {} H_1(x,t,y',{{\bar{y}}}), \text { with }\nabla _{(x,t)}H_1(x,t,y',{{\bar{y}}})\nonumber \\= & {} -\left[ \left( \frac{\partial F}{\partial y_1}\right) ^{-1}\nabla _{x,t}F\right] \bigg |_{(x,t,H_1(x,t,y',{{\bar{y}}}), y',{{\bar{y}}})}. \end{aligned}$$

(2.7)

2.2.3 Intermediate region

Again, using a horizontal rotation argument as above, it suffices to consider only those $\nu \in {\mathcal {J}}_{\text {IM}}$ such that the corresponding neighborhood ${\mathcal {Q}}_\nu$ contains $s_1e_1+s_{2n+1}e_{2n+1}$, with $\min (s_1, s_{2n+1})>2^{-200n}$ and $s_1^4+s_2^2=1$. Let ${\mathcal {J}}_{\text {IM}}'$ denote the collection of all such $\nu$.

For any $\left( \frac{{\underline{x}}-{\underline{y}}}{t},\frac{{{\bar{x}}}-{{\bar{y}}}+\tfrac{1}{2}{\underline{x}}^{\intercal }J{\underline{y}}}{t^2}\right)$ in such a neighborhood ${\mathcal {Q}}_{\nu }$, $\min \big (t^{-1}|x_1-y_1|, t^{-2}|{{\bar{x}}}-{{\bar{y}}}+\frac{1}{2}{\underline{x}}^{\intercal }J{\underline{y}}|\big )>2^{-400n}$ and $t^{-1}|x'-y'|<2^{-400n}$. In view of (2.2), this implies that $2^{-200n-1}<t^{-2}\left| \frac{\partial F}{\partial {{\bar{y}}}}\right| <1$. We can use the implicit function theorem again, this time to express

$$\begin{aligned} {{\bar{y}}}=\bar{H}(x,t,{\underline{y}}) \text { with }\nabla _{(x,t)}\bar{H}(x,t,{\underline{y}})=-\left[ \left( \frac{\partial F}{\partial {{\bar{y}}}}\right) ^{-1}\nabla _{x,t}F\right] \bigg |_{(x,t, {\underline{y}},\bar{H}(x,t,{\underline{y}}))}.\nonumber \\ \end{aligned}$$

(2.8)

3 Main estimates

Keeping the above parametrizations in mind, we are led to consider three types of generalized Radon transforms, associated to incidence relations given by the Eqs. (2.6), (2.7) and (2.8), with kernels supported in ${\mathcal {Q}}_\nu$ for $\nu$ in ${\mathcal {J}}_{\text {NP}}$, ${\mathcal {J}}_{\text {Eq}'}$ and ${\mathcal {J}}_{\text {IM}}'$. We choose $\nu _\text {NP}, \nu _\text {Eq}, \nu _\text {IM}$ in ${{\mathcal {J}}}_\text {NP}', {{\mathcal {J}}}_\text {Eq}'$ and ${{\mathcal {J}}}_\text {IM}'$ respectively, and fix them for the rest of the paper. For $\square \in \{\text {NP}, \text {Eq}, \text {IM}\}$, we define $\mu ^\square :=\mu ^{\nu _\square }$ and $\eta _\square :=\eta _{\nu _\square }$.

Recall that we have reduced consideration to functions f with small support around a thin neighborhood of the Korányi sphere, so that ${\mathcal {A}}_tf$ is supported around a small neighborhood of the origin. We can choose a smooth function $\chi _1$ supported in a slightly larger neighborhood of the origin so that $\chi _1(x){\mathcal {A}}_tf(x)={\mathcal {A}}_tf(x)$ for all $x\in {{\mathbb {R}}}^{2n+1}$. Let

$$\begin{aligned} \Theta _t(x,y):=\left( \frac{{\underline{x}}-{\underline{y}}}{t},\frac{{{\bar{x}}}-{{\bar{y}}}+\tfrac{1}{2}{\underline{x}}^{\intercal }J{\underline{y}}}{t^2}\right) . \end{aligned}$$

We now define smooth cut-off functions $\tilde{\chi }_\text {NP}(x,t,{\underline{y}}):=\chi _1(x) \eta _\text {NP}\circ \Theta _t(x,{\underline{y}}, {\mathcal {G}}(x,t,{\underline{y}}))$, $\chi _\text {Eq}(x,t,y', {{\bar{y}}}) :=\chi _1(x) \eta _{\text {Eq}}\circ \Theta _t\left( x,H_1(x,t,y'), y', {{\bar{y}}}\right)$ and $\chi _\text {IM}(x,t,{\underline{y}}):=\chi _1(x) \eta _{\text {IM}}\circ \Theta _t\left( x,{\underline{y}}, \bar{H}(x,t,y')\right)$. Observe that x and $y'$ are small in the supports of all three functions, $y_1$ is small in the support of $\tilde{\chi }^\text {NP}$ and ${{\bar{y}}}$ is small in the support of $\chi _\text {Eq}$. Further, due to our assumptions on the supports of $\eta _\text {NP}, \eta _\text {Eq}, \eta _\text {IM}$, it follows that ${\mathcal {G}}(x,t,{\underline{y}})$, $H_1(x,t,y',{{\bar{y}}})$ and $\bar{H}(x,t,{\underline{y}})$ are of size about one in the support of $\tilde{\chi }_\text {NP}, \chi _\text {Eq}$ and $\chi _\text {IM}$ respectively.

We can thus write

$$\begin{aligned} f*\mu ^{\text {Eq}}_t(x)= & {} \int \chi _{\text {Eq}}(x,t,y', {{\bar{y}}}) f\left( H_1(x,t,y',{{\bar{y}}}), y',{{\bar{y}}}\right) \,d y'd{{\bar{y}}},\,\, f*\mu ^{\text {IM}}_t(x) \nonumber \\= & {} \int \chi _{\text {IM}}(x,t,{\underline{y}}) f\left( \bar{H}(x,t,{\underline{y}}), {\underline{y}}\right) \,d{\underline{y}}\end{aligned}$$

(3.1)

and

$$\begin{aligned} f*\mu ^{\text {NP}}_t(x)=\int \tilde{\chi }_{\text {NP}}(x,t,{\underline{y}}) f\left( {\underline{y}}, {\mathcal {G}}(x,t,{\underline{y}})\right) \,d{\underline{y}}. \end{aligned}$$

When considering the last integral above (at the north pole), it will be convenient to introduce a shear transformation in the x-variables (depending smoothly on t)

$$\begin{aligned} {{\underline{{{\mathfrak {x}}}}}}(x,t)={\underline{x}}, \,\,\, {{\overline{{{\mathfrak {x}}}}}}(x,t)={{\bar{x}}}+t^2. \end{aligned}$$

By this change of variables

$$\begin{aligned} f*\mu ^{\text {NP}}_t({{\underline{{{\mathfrak {x}}}}}}, \overline{{{\mathfrak {x}}}}-t^2)=\int \chi _{\text {NP}}({{\mathfrak {x}}},t,{\underline{y}}) f\left( {\underline{y}}, G({{\mathfrak {x}}},t,{\underline{y}})\right) \,d{\underline{y}}, \end{aligned}$$

(3.2)

with

$$\begin{aligned} G(x,t,{\underline{y}}):={\overline{x}} + \frac{1}{2}{\underline{x}}^T J {\underline{y}}+t^2\frac{|{\underline{x}}-{\underline{y}}|^4}{2t^4}+\frac{|{\underline{x}}-{\underline{y}}|^8}{t^8}R\left( \frac{{\underline{x}}-{\underline{y}}}{t}\right) \end{aligned}$$

(3.3)

and $\chi ^\text {NP}(x,t,y)=\tilde{\chi }_\text {NP}({\underline{x}},{{\bar{x}}}+t^2, y)$. Note that while the shear transform leaves the Lebesgue space estimates unchanged, the smooth cut off $\chi ^\text {NP}$ is now supported in the set where both ${{\bar{x}}}$ and $G(x,t,{{\bar{y}}})$ are of size about one, while ${\underline{x}}$ and ${\underline{y}}$ are small.

The right hand sides of (3.1) and (3.2) represent operators with Schwartz kernel K of three types depending on the neighborhood of localization:

$$\begin{aligned} {\textbf {Near the Equator: }} K(x,t,y)&=\chi _{\text {Eq}}(x,t,y')\delta _0\left( H_1(x,t,y',{{\bar{y}}})-y_1\right) . \\ {\textbf {In the Intermediate Region: }} K(x,t,y)&=\chi _{\text {IM}}(x,t,{\underline{y}})\delta _0\left( \bar{H}(x,t,{\underline{y}})-{{\bar{y}}}\right) .\\ {\textbf {Near the North Pole: }} K(x,t,y)&= \chi _{\text {NP}}(x,t,{\underline{y}})\delta _0\left( G(x,t,{\underline{y}})-{{\bar{y}}}\right) . \end{aligned}$$

By applying the Fourier transform to $\delta _0$, we can express

$$\begin{aligned} K(x,t,y)=\chi (x,t,\tilde{y})\int _{\theta \in {{\mathbb {R}}}}e^{i\psi (x,t,y,\theta )}\,\frac{d\theta }{2\pi }, \end{aligned}$$

where the phase function around the north pole, the equator and in the intermediate region is given by $\psi ^\text {NP}(x,t,y,\theta ):=\theta (G(x,t,{\underline{y}})-{{\bar{y}}})$, $\psi ^\text {Eq}(x,t,y,\theta ):=\theta (H_1(x,t,y',{{\bar{y}}})-y_1)$ and $\psi ^\text {IM}(x,t,y,\theta ):=\theta (\bar{H}(x,t,{\underline{y}})-{{\bar{y}}})$ respectively; the smooth cut-off $\chi$ can be $\chi _\text {NP}, \chi _\text {Eq}$ or $\chi _\text {IM}$, and $\tilde{y}$ can be ${\underline{y}}$ or $(y', {{\bar{y}}})$ depending on the support of the kernel. Then K is well defined as an oscillatory integral distribution.

Next, we perform a dyadic decomposition in the frequency variable $\theta$. Let $\zeta _0$ be a smooth radial function on ${\mathbb {R}}$ with compact support in $\{|\theta |< 1\}$ such that $\zeta _0(\theta )=1$ for $|\theta |\le 1/2$. We set $\zeta _1(\theta )=\zeta _0(\theta /2)-\zeta _0(\theta )$ and $\zeta _k(\theta )= \zeta _1(2^{1-k}\theta )$.

For any $k\ge 0$, $\psi$ of any of the three forms above and corresponding $\chi$, we define

$$\begin{aligned} A^k f(x, t)= A^k_{t} f(x) = \int _{{{\mathbb {R}}}^{2n+1}} \chi (x,t, \tilde{y})\int _{\theta \in {{\mathbb {R}}}}\zeta _k(\theta ) e^{i\psi (x,t,y,\theta )} \tfrac{d\theta }{(2\pi )}\, f(y) dy, \end{aligned}$$

where $y=({\underline{y}}, {{\bar{y}}})=(y_1, y',{{\bar{y}}})$, with ${\underline{y}}\in {{\mathbb {R}}}^{2n}$, $y_1\in {{\mathbb {R}}}$ and $y'\in {{\mathbb {R}}}^{2n-1}$; and $\tilde{y}$ can be ${\underline{y}}$ or $(y',{{\bar{y}}})$ depending on the support of the kernel. Further, let

$$\begin{aligned} M^kf(x)= \sup _{t\in [1,2]} |A^k_{t} f(x)|. \end{aligned}$$

Following are the main estimates for $M^k$, which hold uniformly for all three possible forms of the phase function $\psi$ in the support of the corresponding smooth cut off $\chi$.

Proposition 3.1

• (i) For $1\le p\le \infty$,

  $$\begin{aligned} \Vert M^k f\Vert _p \lesssim 2^{\frac{k}{p}} 2^{-k(2n)\min (\frac{1}{p},\frac{1}{p'})}\Vert f\Vert _p. \end{aligned}$$

  (3.4)

• (ii) For $2\le q\le \infty$,

  $$\begin{aligned} \Vert M^k f\Vert _{q} \lesssim 2^{k (1-\frac{2n+1}{q})} \Vert f\Vert _{q'}. \end{aligned}$$

  (3.5)

• (iii) Set $q_5:=\tfrac{2(n+2)}{n}$. For every $\epsilon >0$, there exists $C_\epsilon >0$ such that

  $$\begin{aligned} \Vert M^k f \Vert _{L^{q_5, \infty }} \le C_\epsilon 2^{-k\left( \frac{n^2-1}{n+2}-\epsilon \right) }\Vert f\Vert _2. \end{aligned}$$

  (3.6)

Proof of Theorem 1.1, given Proposition 3.1

It suffices to show the required bounds for $M f(x):= \sum _{k\ge 0} M^k f$.

We shall show that M is of restricted weak type (p, q) for $(\tfrac{1}{p}, \tfrac{1}{q})\in \{Q_2, Q_3\}$. Observe that for $n\ge 2$, we have $1-\tfrac{2n+1}{2}<0$. To deduce the required restricted weak type estimates for M at $Q_2, Q_3,$ we recall the Bourgain interpolation argument ( [6, 7, §6.2]):

Suppose we are given sublinear operators $T_k$ so that for $k\ge 1$,

$$\begin{aligned} \Vert T_k\Vert _{L^{p_0,1}\rightarrow L^{q_0,\infty }} \lesssim 2^{k a_0}\quad \text {and}\quad \Vert T_k\Vert _{L^{p_1,1}\rightarrow L^{q_1,\infty }} \lesssim 2^{-k a_1}\end{aligned}$$

for some $p_0, q_0, p_1, q_1\in [1,\infty ], a_0, a_1>0$. Then the operator $\sum _{k\ge 1} T_k$ is of restricted weak type (p, q), where

$$\begin{aligned} \left( \tfrac{1}{p}, \tfrac{1}{q} \right) = (1-\vartheta ) \left( \tfrac{1}{p_0},\tfrac{1}{q_0}\right) + \vartheta \left( \tfrac{1}{p_1},\tfrac{1}{q_1}\right) \end{aligned}$$

and $\vartheta =\tfrac{a_0}{a_0+a_1}\in (0,1)$.

Section 11.2 contains a proof of the above for the convenience of the reader. The restricted weak type estimate for M at $Q_2=(\tfrac{2n}{2n+1},\tfrac{2n}{2n+1})$ now follows from the above result and (3.4). Similarly, the restricted weak type bound at $Q_3=(\tfrac{2n}{2n+1},\tfrac{1}{2n+1})$ follows from (3.5). Interpolation between the estimates at $Q_1, Q_2$ and $Q_3$ yields the $L^p\rightarrow L^q$ boundedness of M for $(\tfrac{1}{p}, \tfrac{1}{q})$ in the interior of $\Delta Q_1Q_2Q_3$, on the open boundary segment $(Q_2,Q_3)$, and on the half open boundary segment $[Q_1,Q_2)$

At

$$\begin{aligned}Q_4= \left( \tfrac{1}{p_4},\tfrac{1}{q_4}\right) =\left( \tfrac{n(2n+1)}{2n^2+2n+1},\tfrac{n}{2n^2+2n+1}\right) ,\end{aligned}$$

due to the loss of a $2^{k\epsilon }$ factor on the right hand side of (3.6), we only have

$$\begin{aligned}\left( \tfrac{1}{p_4}-\epsilon ,\tfrac{1}{q_4}+\tfrac{n\epsilon }{n+2},-\epsilon '\right) =(\vartheta +2\epsilon )\left( \tfrac{1}{2},\tfrac{n}{2(n+2)}, b+\epsilon \right) +(1-\vartheta -2\epsilon )(1,0,1),\end{aligned}$$

with with $b=-\tfrac{n^2-1}{n+2}$, $\vartheta =\tfrac{2(n+2)}{2n^2+2n+2}\in (0,1)$ and $\epsilon '=2\epsilon (1-\epsilon )-(2b+\vartheta )>0$ for $n\ge 2$. Letting $Q_{4, \epsilon }=\left( \tfrac{1}{p_\epsilon },\tfrac{1}{q_\epsilon }\right) =\left( \tfrac{1}{p_4}-\epsilon ,\tfrac{1}{q_4}+\tfrac{n\epsilon }{n+2}\right)$, standard interpolation between (3.6) and the case $q=\infty$ of (3.5) implies that

$$\begin{aligned}\Vert M f\Vert _{L^{q_\epsilon }}\lesssim _\epsilon \sum _{k\ge 0}2^{-k\epsilon '}\Vert f\Vert _{L^{p_\epsilon }}\lesssim _\epsilon \Vert f\Vert _{L^{p_\epsilon }}.\end{aligned}$$

Interpolating again between the estimates at $Q_1, Q_2, Q_3$ and $Q_{4, \epsilon }$ yields the $L^p\rightarrow L^q$ boundedness of M for $(\tfrac{1}{p}, \tfrac{1}{q})$ in the interior of the quadrilateral formed by those vertices. Letting $\epsilon \rightarrow 0$, we conclude the same for $(\tfrac{1}{p}, \tfrac{1}{q})$ in the interior of ${\mathcal {R}}$. Since bounds for M imply bounds for ${\mathcal {M}}$, this concludes the proof of part (i) of Theorem 1.1. $\square$

3.1 Reduction to estimates on the averaging operator

We consider a discrete dyadic cover of the interval [1, 2]. Given a non-negative integer k, let ${{\mathcal {Z}}}_k$ denote the set of left endpoints of all dyadic intervals of the form $(\nu 2^{-k}, (\nu +1)2^{-k})$ (with $\nu \in {\mathbb {Z}}$) which intersect [1, 2], endowed with the counting measure. It is clear that $\#({{\mathcal {Z}}}_k)= 2^k$. An application of the fundamental theorem of calculus gives the following pointwise bound

$$\begin{aligned} M^{k}f(x)\le \sup _{t\in {{\mathcal {Z}}}_{k}}|A^{k}_{t}f(x)|+\int _0^{2^{-k}}|\partial _sA^{k}_{t+s}f(x)|\,ds. \end{aligned}$$

(3.7)

Taking the $L^q$ norm on both sides, using the triangle inequality, then dominating the supremum by the $\ell ^q({{\mathcal {Z}}}_k)$ norm and using the Minkowski inequality for the second term, we obtain

$$\begin{aligned} \Vert M^{k} f\Vert _{L^{q}}\le \Vert A^{k} f\Vert _{L^{q}({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})}+\int _0^{2^{-k}}\Vert \partial _sA^{k}_{t+s}f(x)\Vert _{L^{q}({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})}\,ds. \end{aligned}$$

By a change of variables and using homegeneity (in $\theta$) for $k\ge 1$, we have

$$\begin{aligned} A^k_t f(x) = 2^k \int _{{{\mathbb {R}}}^{2n+1}}\int _{{{\mathbb {R}}}} e^{i2^k \psi (x,t,y,\theta )} b(x,t,\tilde{y},\theta ) f(y)\, d\theta \,dy \end{aligned}$$

(3.8)

where the symbol b is compactly supported in ${{\mathbb {R}}}^{2n+1}\times {{\mathbb {R}}}\times {{\mathbb {R}}}^{2n-1}\times {{\mathbb {R}}}$. It can be of the form $b_\text {NP}, b_\text {Eq}$ or $b_\text {IM}$, with $\text {supp}(b_\text {NP})\subseteq \text {supp}(\chi _\text {NP})\times \text {supp}(\zeta _1)$ and $b_\text {Eq}, b_\text {IM}$ defined analogously ($\tilde{y}$ might stand for ${\underline{y}}$ or $(y',{{\bar{y}}})$ depending on the support). In particular, $|\theta |\in [1/2,2]$ in the support of all three types of symbols.

Using homegeneity, it is not hard to see that the operator $2^{-k} \frac{d}{dt} A^k_t$ is a linear combination of expressions involving operators which obey the same Lebesgue space estimates as the operator $A^k_t$. Thus parts (i) and (ii) of Proposition 3.1 are a consequence of the following fixed time estimates.

Proposition 3.2

Let $t\in [1,2]$. (i) For $1\le p\le \infty$

$$\begin{aligned} \Vert {A}^k_tf \Vert _{p} \lesssim 2^{-k (2n)\min (\frac{1}{p},\frac{1}{p'})}\Vert f\Vert _p, \end{aligned}$$

(3.9)

with the implicit constant independent of t.

(ii) For $2\le q\le \infty$,

$$\begin{aligned} \Vert {A}^k_t f\Vert _{ q} \lesssim 2^{k (1-\frac{2n+2}{q})}\Vert f\Vert _{q'} ,\end{aligned}$$

(3.10)

with the implicit constant independent of t.

The above fixed time estimates are sufficient to establish the positive result in Theorem 1.2.

Proof of Theorem 1.2, given Proposition 3.2

Clearly, $\Vert f*\mu \Vert _{L^1}\lesssim \Vert f\Vert _{L^1}$ and $\Vert f*\mu \Vert _{L^\infty }\lesssim \Vert f\Vert _{L^\infty }$. For $1<p<\infty$, (3.9) implies that $\Vert \sum _{k\ge 0}A^k_t\Vert _{L^p\rightarrow L^p}\lesssim \sum _{k\ge 0}2^{-k(2n)\min (\tfrac{1}{p}, \tfrac{1}{p'})}\lesssim 1$.

Further, since $1-\frac{2n+2}{q}=0$ for $q=\tfrac{1}{2n+2}$, (3.10) implies that $\Vert A^k_t\Vert _{L^{q'}\rightarrow L^q}\lesssim 1$ uniformly in k. To combine the $A^k_t$, we use standard applications of Littlewood–Paley theory (see [29, Lemma 2.1] for a precise statement and proof), to express $A^k_t = L_kA^k_t L_k +E_k$, where the $L_k$ satisfy Littlewood–Paley inequalities

for $1<r<\infty$ and the error term $E_k$ has $L^p\rightarrow L^q$ operator norm $O(2^{-k})$ for all $1\le p,q\le \infty$. To deduce the endpoint estimate at $(\tfrac{1}{q'}, \tfrac{1}{q})=(\frac{2n+1}{2n+2}, \frac{1}{2n+2})$, since $q'\le 2\le q$, we can apply the Littlewood–Paley inequalities in conjunction with Minkowski’s inequalities as follows

$$\begin{aligned} \left\| \sum _{k\ge 0} L_k A^k_t L_k f\right\| _q&\lesssim \left\| \left( \sum _{k\ge 0} | A^k_tL_kf|^2\right) ^{1/2}\right\| _q \le \left( \sum _{k\ge 0} \Vert A^k_tL_kf\Vert _q^2\right) ^{1/2}\\&\lesssim \left( \sum _{k\ge 0} \Vert L_kf\Vert _{q'}^2\right) ^{1/2}\le \left\| \left( \sum _{k\ge 0} | L_kf|^2\right) ^{1/2}\right\| _{q'}\lesssim \Vert f\Vert _{q'}. \end{aligned}$$

A further interpolation yields the positive result in the closure of the triangle formed by (0, 0), (1, 1) and $(\frac{2n+1}{2n+2}, \frac{1}{2n+2})$, thus finishing the proof. $\square$

To prove the estimate at $q_5:=\tfrac{2(n+2)}{n}$, we take an $L^{q_5, \infty }$ norm on both sides of (3.7) and conclude that

$$\begin{aligned} \Vert M^{k} f\Vert _{L^{q_5, \infty }}\le \Vert A^{k}_{t} f\Vert _{L^{q_5, \infty }({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})}+\int _0^{2^{-k}}\Vert \partial _sA^{k}_{t+s}f(x)\Vert _{L^{q_5, \infty }({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})}\,ds. \end{aligned}$$

Thus part (iii) of Proposition 3.1 is a consequence of the following estimate.

Proposition 3.3

Let $q_5=\frac{2(n+2)}{n}$. We have

$$\begin{aligned} \Vert A^{k} \Vert _{L^2({{\mathbb {R}}}^{2n+1})\rightarrow L^{q_5, \infty }({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})} \lesssim k2^{-k \left( \frac{n^2-1}{n+2}\right) }. \end{aligned}$$

(3.11)

For all $k\ge 1$, we have the following estimates

$$\begin{aligned}{} & {} \Vert A_{t}^k\Vert _{L^1\rightarrow L^1} + \Vert A_{t}^k \Vert _{L^\infty \rightarrow L^\infty } \lesssim 1, \end{aligned}$$

(3.12)

$$\begin{aligned}{} & {} \Vert A_{t}^k\Vert _{L^1\rightarrow L^\infty } \lesssim 2^{k}. \end{aligned}$$

(3.13)

In view of the above it suffices in what follows to consider the case of large k. The bounds (3.9), (3.10) then follow by an interpolation argument using (3.12), (3.13) and the fixed time $L^2$ estimate

$$\begin{aligned} \Vert A_{t}^k f\Vert _{L^2({\mathbb {R}}^{2n+1})} \lesssim 2^{-k \frac{2n}{2} } \Vert f\Vert _2. \end{aligned}$$

(3.14)

3.2 Reduction to oscillatory integral estimates

To prove (3.14) we use oscillatory integral operators of the form

$$\begin{aligned} T^k_t f(x)=T^k f(x,t) := \int _{{\mathbb {R}}^{2n+1}} e^{i 2^k \Phi (x,t,y)} b(x,t,y) f(y) dy, \end{aligned}$$

(3.15)

where $\Phi$ can be of the form $\Phi ^\text {NP}(x,t,y){=}{{\bar{y}}}G(x,t,{\underline{y}})$, $\Phi ^\text {Eq}(x,t,y){=} y_1H_1(x,t,y',{{\bar{y}}}),$ and $\Phi ^\text {IM}(x,t,y)={{\bar{y}}}\bar{H}(x,t,{\underline{y}})$, with $G, H_1$ and $\bar{H}$ as defined in Eqs. (3.3), (2.7) and (2.8). The corresponding amplitude b is compactly supported in ${{\mathbb {R}}}^{2n+1}\times {{\mathbb {R}}}\times {{\mathbb {R}}}^{2n-1}\times {{\mathbb {R}}}$, and can be of the form $b^\text {NP}, b^\text {Eq}$ or $b^\text {IM}$ as defined in (3.8) (with the role of $\theta$ played by ${{\bar{y}}}$ or $y_1$, depending on the situation).

To reduce matters to oscillatory integral operators of the above form, we define

$$\begin{aligned} F_k(y)= \int _{{{\mathbb {R}}}} f(y, {\bar{w}}) e^{-i2^k\bar{y}{\bar{w}}} d{\bar{w}} \end{aligned}$$

and

$$\begin{aligned} \tilde{F}_k(y)= \int _{{{\mathbb {R}}}} f(w_1, y',{{\bar{y}}}) e^{-i2^ky_{1}w_{1}} dw_{1}. \end{aligned}$$

The integrals on the right can be interpreted as scaled Fourier transforms in the ${{\bar{y}}}$ and $y_1$ variables respectively. After renaming ${\bar{w}}$ to ${{\bar{y}}}$ and ${{\bar{y}}}$ to $\theta$, we have

$$\begin{aligned} A^k f(x,t)= T^k F_k (x,t) \end{aligned}$$

around the north pole and in the intermediate regions. Similarly, renaming $w_1$ to $y_1$ and $y_1$ to $\theta$, we obtain

$$\begin{aligned} A^k f(x,t)= T^k \tilde{F}_k (x,t) \end{aligned}$$

around the equator. By Plancherel’s theorem $\Vert F_k\Vert _2=\Vert \tilde{F}_k\Vert _2=(2^{-k}2\pi )\Vert f\Vert _2$. Hence (3.14) follows from

Proposition 3.4

For all $f\in L^2({{\mathbb {R}}}^{2n+1})$ and $t\in [1,2]$,

$$\begin{aligned} \Vert T^k_t f\Vert _{ L^2({{\mathbb {R}}}^{2n+1})} \lesssim 2^{-k\frac{2n+1}{2}} \Vert f\Vert _2, \end{aligned}$$

(3.16)

with the implicit constant uniform in $t\in [1,2]$.

The proof will be given using Hörmander’s standard $L^2$ estimate [13] (also see [33, Ch. IX.1]) combined with almost-orthogonality arguments. By the same argument, the $L^2\rightarrow L^{q_5, \infty }$ bound (3.11) is reduced to the following estimate.

Proposition 3.5

Let $q_5=\tfrac{2(n+2)}{n}$. For $f\in L^2({{\mathbb {R}}}^{2n+1})$, we have

$$\begin{aligned} \Vert T^{k}f\Vert _{L^{q_5, \infty }({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})} \lesssim k2^{-k\frac{n(2n+1)}{2(n+2)}}\Vert f\Vert _{2}. \end{aligned}$$

(3.17)

4 A scaling argument at the poles

In this section, we focus on the case when the kernel of the operator $T^k$ is supported around the poles (more specifically, around the north pole but the same argument works for the south pole due to symmetry about the equator). In fact, the $L^p$ improving property at $Q_4$ is completely determined by how the operator behaves around the poles. We shall decompose the kernel of the operator $T^k$ based on the distance from the north pole and use a stretching argument to precisely understand this behaviour.

Let $\rho _0$ be a smooth radial function on ${\mathbb {R}}^{2n}$ with compact support in $\{|{\underline{y}}|< 1/4\}$ such that $\rho _0({\underline{y}})=1$ for $|{\underline{y}}|\le 1/8$. Setting $\rho _1({\underline{y}})=\rho _0({\underline{y}}/2)-\rho _0({\underline{y}})$, $\rho _{\ell }({\underline{y}})= \rho _1(2^{\ell }{\underline{y}})$ for $1\le \ell <\tfrac{k}{4}$ and $\rho _{\ell }({\underline{y}})=\rho _0(2^{\ell }{\underline{y}})$ for $\ell =\frac{k}{4}$, we define

$$\begin{aligned}{} & {} T_t^{k, 0} f(x): = \int _{{\mathbb {R}}^{2n+1}} e^{i 2^k \Phi (x,t,y) } b(x,t,y)\left( 1-\rho _0\left( \frac{{\underline{x}}-{\underline{y}}}{2t}\right) \right) f(y) dy,\,\,\, \text { for } \ell =0,\\{} & {} T_t^{k,\ell } f(x) := \int _{{\mathbb {R}}^{2n+1}} e^{i 2^k \Phi (x,t,y) } b(x,t,y)\rho _{\ell }\left( \frac{{\underline{x}}-{\underline{y}}}{t}\right) f(y) dy,\,\,\, \text { for } 1\le \ell \le \frac{k}{4} \end{aligned}$$

and

$$\begin{aligned}T^{k,\ell } f(x, t):=T_t^{k,\ell } f(x).\end{aligned}$$

Here

$$\begin{aligned}\Phi =\Phi ^\text {NP}:={{\bar{y}}}G(x,t,{\underline{y}}) ={{\bar{y}}}\left( {\overline{x}}+ \frac{1}{2}{\underline{x}}^T J {\underline{y}}+t^2\tfrac{|{\underline{x}}-{\underline{y}}|^4}{2t^4}+t^2\frac{|{\underline{x}}-{\underline{y}}|^8}{t^8}R\left( \frac{{\underline{x}}-{\underline{y}}}{t}\right) \right) ,\end{aligned}$$

with R as defined in (2.5), and $b=b^\text {NP}$ with $\text {supp}(b_\text {NP})\subseteq \text {supp}(\chi _\text {NP})\times \text {supp}(\zeta _1)$. Then $T_t^{k} f=\sum _{\ell =0}^{\frac{k}{4}}T_t^{k,\ell } f$. In fact, since b is supported in the set where $\frac{|{\underline{x}}-{\underline{y}}|}{t}<2^{-400n}$, the above decomposition is relevant for large values of $\ell$ (say $\ell \ge 400n$). We make a change of variables $2^{\ell }{\underline{y}}\rightarrow {\underline{y}}$, which gives

$$\begin{aligned}T_t^{k,\ell } f(x) = 2^{-2n\ell }\int _{{\mathbb {R}}^d} e^{i 2^k \Phi (x,t,2^{-\ell }{\underline{y}},{{\bar{y}}}) } b(x,t,2^{-\ell }{\underline{y}},{{\bar{y}}})\rho _{1}(2^{\ell }{\underline{x}}-{\underline{y}}) f(2^{-\ell }{\underline{y}},{{\bar{y}}}) dy.\end{aligned}$$

Observe that

$$\begin{aligned} \Phi (x,t,2^{-\ell }{\underline{y}},{{\bar{y}}})&={\overline{y}}\left( {\overline{x}}+ {\underline{x}}^T J (2^{-\ell }{\underline{y}})+2^{-4\ell }t^2\tfrac{|2^{\ell }{\underline{x}}-{\underline{y}}|^4}{t^4}+2^{-8\ell }t^2\tfrac{|2^{\ell }{\underline{x}}-{\underline{y}}|^8}{t^8}R\left( \tfrac{{\underline{x}}-2^{-\ell }{\underline{y}}}{t}\right) \right) \\&=2^{-4\ell }{\overline{y}}\left( 2^{4\ell }{\overline{x}}+ \frac{1}{2}{\underline{x}}^T 2^{3\ell }J{\underline{y}}+t^2\tfrac{|2^{\ell }{\underline{x}}-{\underline{y}}|^4}{t^4}+2^{-4\ell }t^2\tfrac{|2^{\ell }{\underline{x}}-{\underline{y}}|^8}{t^8}R\left( \tfrac{2^{-\ell }(2^{\ell }{\underline{x}}-{\underline{y}})}{t}\right) \right) . \end{aligned}$$

We define

$$\begin{aligned} g({{\underline{w}}}):=\frac{|{{\underline{w}}}|^4}{2}\quad \text {and}\quad \tilde{g}_{\ell }({{\underline{w}}}):=|{{\underline{w}}}|^8R(2^{-\ell }{{\underline{w}}}), \end{aligned}$$

(4.1)

so that

$$\begin{aligned} g_{\ell }({{\underline{w}}}):=g({{\underline{w}}})+2^{-4\ell }\tilde{g}_{\ell }({{\underline{w}}}). \end{aligned}$$

(4.2)

Due to condition (2.5) on R, we have

$$\begin{aligned} \left| \frac{\partial ^\beta }{\partial {{\underline{w}}}^{\beta }}\tilde{g}_\ell ({{\underline{w}}})\right| \lesssim 100n\cdot 2^{-4\ell }|{{\underline{w}}}|^{8-|\beta |}\,\, \text { for } |\beta |\in \{0,1,2,3\} \text { and } |{{\underline{w}}}|\le \frac{1}{2}. \end{aligned}$$

(4.3)

In fact, $g_{\ell }$ can be expressed as $g_{\ell }({{\underline{w}}})=u_{\ell }(|{{\underline{w}}}|)$, where u is a smooth function in one variable satisfying the condition

$$\begin{aligned} \left| \left( \frac{d}{dr}\right) ^j\left( u_{\ell }(r)-\frac{r^4}{2}\right) \right| \lesssim 100n\cdot 2^{-4\ell }r^{8-j}\,\, \text { for } j\in \{0,1,2,3\} \text { and } |r|\le \frac{1}{2}.\nonumber \\ \end{aligned}$$

(4.4)

We thus have

$$\begin{aligned}T_t^{k,\ell } f(2^{-3\ell }{\underline{x}},2^{-4\ell }{{\bar{x}}}) = 2^{-2n\ell }\int _{{\mathbb {R}}^{2n+1}} e^{i 2^{k-4\ell } \Phi _{\ell }(x,t,y)} a_{\ell }(x,t,y) f(2^{-\ell }{\underline{y}},{{\bar{y}}}) dy,\end{aligned}$$

where

$$\begin{aligned}\Phi _{\ell }(x,t,2{\underline{y}},{{\bar{y}}})={\overline{y}}\left( {\overline{x}}+ \frac{1}{2}{\underline{x}}^T J {\underline{y}}+t^2g_{\ell }\left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) \right) ,\end{aligned}$$

and for a fixed $t\in [1,2]$, $a_{\ell }(x,t,y)=b(2^{-3\ell }{\underline{x}},2^{-4\ell }{{\bar{x}}},t,2^{-\ell }{\underline{y}},{{\bar{y}}})\rho _{1}(2^{-2\ell }{\underline{x}}-{\underline{y}})$ is supported in the set where $|{\underline{x}}|\lesssim 2^{3\ell }, |{{\bar{x}}}|\lesssim 2^{4\ell }$, $\frac{1}{8}\le |2^{-2\ell }{\underline{x}}-{\underline{y}}|\le \frac{1}{2}$ (resp. $|2^{-2\ell }{\underline{x}}-{\underline{y}}|\le \frac{1}{4}$) for $\ell <\frac{k}{4}$ (resp. $\ell =\frac{k}{4}$) and $|{{\bar{y}}}|\sim 1$.

Remark 4.1

Observe that the input variable ${\underline{y}}$ is scaled by $2^{-\ell }$, while the output variable ${\underline{x}}$ is scaled by a factor of $2^{-3\ell }$. This is due to the presence of the bilinear term involving the matrix J in the phase function. This phenomenon was not present in [14], where a scaling type type argument was used to establish the $L^p$ boundedness of global maximal operators associated to families of flat, finite type curves in ${{\mathbb {R}}}^2$ (also see [18]).

We introduce an operator ${{\mathcal {T}}}_t^{k,\ell }$ defined by

$$\begin{aligned} {{\mathcal {T}}}_t^{k,\ell } f({\underline{x}},{{\bar{x}}}) = \int _{{\mathbb {R}}^d} e^{i 2^{k-4\ell } \Phi _{\ell }(x,t,y)} a_{\ell }(x,t,y) f({\underline{y}},{{\bar{y}}}) dy \end{aligned}$$

(4.5)

and let

$$\begin{aligned} {{\mathcal {T}}}^{k,\ell } f({\underline{x}},{{\bar{x}}}, t):={{\mathcal {T}}}_t^{k,\ell } f({\underline{x}},{{\bar{x}}}). \end{aligned}$$

(4.6)

Further, for $a,b\in {{\mathbb {Z}}}$, let $\tau _{a,b}$ be the dilation defined by $\tau _{a,b}({\underline{x}},{{\bar{x}}})=(2^{-a}{\underline{x}},2^{-b}{{\bar{x}}}),$ so that

$$\begin{aligned} T^{k,\ell }_tf=2^{-2n\ell } {{\mathcal {T}}}^{k,\ell }_t(f\circ \tau _{\ell ,0})\circ \tau _{-3\ell ,-4\ell }. \end{aligned}$$

The following lemma will be useful for $TT^*$ type arguments in the upcoming sections.

Lemma 4.2

Let $a,b,c\in {{\mathbb {Z}}}$ and for $i=1,2$, let ${{\mathcal {U}}}_i,U_i$ be operators defined on ${{\mathbb {R}}}^{2n+1}$ such that

$$\begin{aligned} U_i f=2^{-2nc}{{\mathcal {U}}}_i(f\circ \tau _{c,0})\circ \tau _{-a,-b}. \end{aligned}$$

(4.7)

Then for $1\le p,q\le \infty$,

$$\begin{aligned}\Vert U_1U_2^*\Vert _{L^p\rightarrow L^q}\le 2^{-2n(a+c)}2^{-b}2^{(2na+b)\left( \frac{1}{p}-\frac{1}{q}\right) }\Vert {{\mathcal {U}}}_1{{\mathcal {U}}}_2^*\Vert _{L^p\rightarrow L^q}.\end{aligned}$$

Proof

As a consequence of (4.7), we have

$$\begin{aligned}U_2^*f= 2^{-(2na+b)}{{\mathcal {U}}}_2^*(f\circ \tau _{a,b})\circ \tau _{-c,0}. \end{aligned}$$

It follows that

$$\begin{aligned} U_1U_2^*f&=2^{-2nc}{{\mathcal {U}}}_1[(U_2^*f)\circ \tau _{c,0}]\circ \tau _{-a,-b}\\&=2^{-2n(a+c)}2^{-b}{{\mathcal {U}}}_1[{{\mathcal {U}}}_2^*(f\circ \tau _{a,b})\circ \tau _{-c,0}\circ \tau _{c,0}]\circ \tau _{-a,-b}\\&=2^{-2n(a+c)}2^{-b}{{\mathcal {U}}}_1{{\mathcal {U}}}_2^*(f\circ \tau _{a,b})\circ \tau _{-a,-b}. \end{aligned}$$

Taking $L^q$ norms on both sides and changing variables on the right hand side, we get

$$\begin{aligned} \Vert U_1U_2^*f\Vert _{L^q}= & {} 2^{-2n(a+c)}2^{-b}2^{-\frac{2na+b}{q}}\Vert {{\mathcal {U}}}_1{{\mathcal {U}}}_2^*(f\circ \tau _{a,b})\Vert _{L^q}\\\le & {} 2^{-2n(a+c)}2^{-b}2^{-\frac{2na+b}{q}}\Vert {{\mathcal {U}}}_1{{\mathcal {U}}}_2^*\Vert _{L^p\rightarrow L^q}\Vert f\circ \tau _{a,b}\Vert _{L^p}. \end{aligned}$$

Since $\Vert f\circ \tau _{a,b}\Vert _{L^p}=2^{\frac{2na+b}{p}}\Vert f\Vert _{L^p}$, this yields the desired estimate. $\square$

To prove (3.17), we shall also need a fixed time $L^2$ estimate for each scaled piece ${{\mathcal {T}}}^{k,\ell }_t$.

Proposition 4.3

Let ${{\mathcal {T}}}^{k,\ell }_t$ be as defined in (4.5). For all $f\in L^2({{\mathbb {R}}}^{2n+1})$ and $t\in [1,2]$,

$$\begin{aligned} \Vert {{\mathcal {T}}}^{k,\ell }_t f\Vert _{ L^2({{\mathbb {R}}}^{2n+1})} \lesssim 2^{-(k-4\ell )\frac{2n+1}{2}} \Vert f\Vert _2, \end{aligned}$$

with the implicit constant uniform in $t\in [1,2]$.

The above proposition shall be proven in Sect. 8.3. Combining it with Lemma 4.2 for $a=3\ell , b=4\ell$ and $c=\ell$ yields the following.

Corollary 4.4

For $t,t'\in [1,2]$,

$$\begin{aligned}\Vert T^{k,\ell }_t(T^{k,\ell }_{t'})^*\Vert _{L^2({{\mathbb {R}}}^{2n+1})\rightarrow L^2({{\mathbb {R}}}^{2n+1})}\lesssim 2^{-2n(3\ell +\ell )}2^{-4\ell }2^{-(k-4\ell )(2n+1)}=2^{-k(2n+1)},\end{aligned}$$

Remark 4.5

The above corollary implies that $\Vert T^{k,\ell }_t\Vert _{L^2({{\mathbb {R}}}^{2n+1})\rightarrow L^2({{\mathbb {R}}}^{2n+1})}\lesssim 2^{-k(\frac{2n+1}{2})}$. If we sum this up for $0\le \ell \le \frac{k}{4}$, we would obtain the fixed time $L^2$ estimate (3.16) for $T^k$ around the poles but up to a logarithmic loss in $2^k$. However, as we shall see in Sect. 8.3, the rotational curvature remains non-vanishing at the poles and thus (3.16) can be directly obtained without incurring any logarithmic loss for the estimates at $Q_1, Q_2$ and $Q_3$.

5 A Stein–Tomas argument

In this section, we shall reduce the estimate (3.17), for the operators $T^k$ (around the equator and in the intermediate region) and $T^{k,\ell }$ (around the poles), to an $L^{q_5', 1}\rightarrow L^{q_5, \infty }$ estimate for the respective operator precomposed with its adjoint via the $TT^*$ technique. Using a Stein–Tomas type argument, the said estimate will then be obtained by interpolating between an $L^2\rightarrow L^2$ and an $L^1\rightarrow L^\infty$ estimate. We first give the details when the kernel of $T^k$ is supported around the poles (more precisely, the north pole), where the argument needs to be run separately for each piece $T^{k,\ell }$.

5.1 Around the north pole

Since $T^k f=\sum _{\ell =0}^{\frac{k}{4}}T^{k,\ell } f$, to prove (3.17), it suffices to show

$$\begin{aligned} \Vert T^{k, \ell }\Vert _{L^2({{\mathbb {R}}}^{2n+1})\rightarrow L^{q_5, \infty }({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})} \lesssim 2^{-k\frac{n(2n+1)}{2(n+2)}} \end{aligned}$$

for each $0\le \ell \le \frac{k}{4}$. Using a $TT^*$ argument (and the fact that the dual space of $L^{q', 1}$ is $L^{q, \infty }$), the above estimate is a consequence of

$$\begin{aligned} \Vert T^{k, \ell } (T^{k, \ell })^*\Vert _{L^{q_5', 1}({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})\rightarrow L^{q_5, \infty }({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})} \lesssim 2^{-k\frac{n(2n+1)}{(n+2)}}. \end{aligned}$$

(5.1)

We define the operator $S^{k,\ell }$ acting on functions $g:{\mathbb {R}}^{2n+1}\times {\mathcal {Z}}_{k}\rightarrow {{\mathbb {C}}}$ by

$$\begin{aligned} S^{k,\ell }g(x,t) := \sum _{t'\in {\mathcal {Z}}_{k}} T^{k,\ell }_{t} (T^{k,\ell }_{t'})^* [g(\cdot , t')](x). \end{aligned}$$

(5.2)

Then (5.1) follows from

$$\begin{aligned} \Vert S^{k, \ell }g\Vert _{L^{q_5, \infty }({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})} \lesssim 2^{-k\frac{n(2n+1)}{(n+2)}}\Vert g\Vert _{L^{q_5', 1}({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})}. \end{aligned}$$

(5.3)

For $m\ge 0$ and $t\in {\mathcal {Z}}_{k}$, let

$$\begin{aligned} {\mathcal {Z}}_{k}^m(t) = \left\{ t'\in {\mathcal {Z}}_{k}\,:\,2^{-k+m}\le |t-t'|\le 2^{-k+m+1} \right\} . \end{aligned}$$

Note that ${\mathcal {Z}}_{k}^m(t)$ is empty, if $m>k+4$. Further, we define

$$\begin{aligned} S^{k,\ell }_m g(x,t) := \sum _{t'\in {\mathcal {Z}}_{k}^m(t)} T^{k,\ell }_t (T^{k,\ell }_{t'})^* [g(\cdot , t')](x). \end{aligned}$$

(5.4)

We observe

$$\begin{aligned} \left( \tfrac{1}{q_5^\prime },\tfrac{1}{q_5},-\tfrac{n(2n+1)}{n+2},0\right) =\vartheta \left( \tfrac{1}{2},\tfrac{1}{2},-(2n+1),1\right) +(1-\vartheta )\left( 1,0,0,-\tfrac{n}{2}\right) , \end{aligned}$$

with $\tfrac{\vartheta }{2}=\tfrac{1}{q_5}=\tfrac{n}{2(n+2)}$. Thus, using the Bourgain interpolation trick for $S^{k,\ell }=\sum _{m\ge 0}S^{k,\ell }_m$, (5.3) is a consequence of

$$\begin{aligned} \Vert S^{k, \ell }_m g\Vert _{L^{2}({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})} \lesssim 2^{-k(2n+1)}2^m\Vert g\Vert _{L^2({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})} \end{aligned}$$

(5.5)

and

$$\begin{aligned} \Vert S^{k, \ell }_m g\Vert _{L^{\infty }({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})} \lesssim 2^{-\frac{nm}{2}}\Vert g\Vert _{L^1({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})}. \end{aligned}$$

(5.6)

To show (5.5), using the Cauchy–Schwarz inequality, we have

$$\begin{aligned} \Vert S^{k,\ell }_mg\Vert _{L^2({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})}&=\left( \sum _{t\in {{\mathcal {Z}}}_{k}}\int \left| \sum _{t'\in {\mathcal {Z}}_{k}^m(t)} T^{k,\ell }_{t} (T^{k, \ell }_{t'})^* [g(\cdot , t')](x)\right| ^2\,dx\right) ^{1/2}\\&\le \left( \sum _{t\in {{\mathcal {Z}}}_{k}}\#({\mathcal {Z}}_{k}^m(t))\int \sum _{t'\in {\mathcal {Z}}_{k}^m(t)} |T^{k,\ell }_{t} (T^{k,\ell }_{t'})^* [g(\cdot , t')](x)|^2\,dx\right) ^{1/2}\\&\lesssim 2^{-k(2n+1)}2^{m}\Vert g\Vert _{L^2({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})}, \end{aligned}$$

where we have used Corollary 4.4 and the fact that $\#({\mathcal {Z}}_{k}^m(t))\sim 2^{m}$ for all $t\in {{\mathcal {Z}}}_{k}$.

To prove (5.6), we need the following estimate on the kernel of the operator ${{\mathcal {T}}}^{k,\ell }_t({{\mathcal {T}}}^{k,\ell }_{t'})^*$.

Proposition 5.1

Let ${{\mathcal {K}}}^{k,\ell }_{t,t'}$ denote the kernel of ${{\mathcal {T}}}^{k,\ell }_t({{\mathcal {T}}}^{k,\ell }_t)^*$, where ${{\mathcal {T}}}^{k,\ell }_t$ and ${{\mathcal {T}}}^{k,\ell }_{t'}$ are as defined in (4.5) for $t,t'\in [1,2]$. Then for $0\le \ell <\frac{k}{4}$,

$$\begin{aligned} \Vert {{\mathcal {K}}}^{k,\ell }_{t,t'}\Vert _{L^\infty ({{\mathbb {R}}}^{2n+1})}\lesssim (1+2^{k-4\ell }|t-t'|)^{-n}. \end{aligned}$$

(5.7)

Postponing the proof of the above proposition to Sect. 7.2, we use it for now to prove (5.6). The above estimate implies that $\Vert {{\mathcal {T}}}^{k,\ell }_t({{\mathcal {T}}}^{k,\ell }_{t'})^*\Vert _{L^1({{\mathbb {R}}}^{2n+1})\rightarrow L^\infty ({{\mathbb {R}}}^{2n+1})}\lesssim (1+2^{k-4\ell }|t-t'|)^{-n}$. Using Lemma 4.2 with $a=3\ell , b=4\ell , c=\ell$, we obtain

$$\begin{aligned} \Vert T^{k,\ell }_t(T^{k,\ell }_{t'})^*\Vert _{L^1({{\mathbb {R}}}^{2n+1})\rightarrow L^{\infty }({{\mathbb {R}}}^{2n+1})}&\lesssim 2^{-2n(3\ell +\ell )}2^{-4\ell }2^{(6n+4)\ell }(1+2^{k-4\ell }|t-t'|)^{-n}\\&= 2^{-2n\ell }(1+2^{k-4\ell }|t-t'|)^{-n}. \end{aligned}$$

For $m\ge 4\ell$, using the estimate above, we have

$$\begin{aligned}{} & {} \Vert S^{k,\ell }_m g\Vert _{L^1({{\mathbb {R}}}^{2n+1}\times {{\mathcal {Z}}}_k)\rightarrow L^{\infty }({{\mathbb {R}}}^{2n+1}\times {{\mathcal {Z}}}_k)}\\{} & {} \quad \lesssim 2^{-2n\ell }(2^{k-4\ell }\times 2^{-k+m})^{-n}=2^{2n\ell }2^{-nm}\le 2^{-\frac{nm}{2}}. \end{aligned}$$

On the other hand, for $m\le 4\ell$ or $\ell =\frac{k}{4}$, we can use the trivial bound of $2^{-2n\ell }$ (using the fact that $|{\mathcal {K}}^{k, \ell }_{t,t'}|\lesssim 1$ for $\ell =\frac{k}{4}$) to conclude that

$$\begin{aligned} \Vert S^{k,\ell }_m\Vert _{L^1({{\mathbb {R}}}^{2n+1}\times {{\mathcal {Z}}}_k)\rightarrow L^{\infty }({{\mathbb {R}}}^{2n+1}\times {{\mathcal {Z}}}_k)} \lesssim 2^{-2n\ell }\le 2^{-\frac{nm}{2}}. \end{aligned}$$

This implies (5.6), and concludes the proof of (3.11).

5.2 Around the equator and in the intermediate region

The proof of estimate (3.17) when the operator $T^k_t$ has kernel supported away from the north pole proceeds in the exact manner as above. The following replaces Proposition 5.1 in this setting.

Proposition 5.2

Let $K^{k}_{t,t'}$ denote the kernel of $T^{k}_t(T^{k}_t)^*$, where $T^{k}_t$ and $T^{k}_{t'}$ are as defined in (3.15) for $t,t'\in {{\mathcal {Z}}}_k$, with $\Phi =\Phi ^\text {Eq}$ or $\Phi =\Phi ^\text {IM}$. Then

$$\begin{aligned} \Vert K^{k}_{t,t'}\Vert _{L^\infty ({{\mathbb {R}}}^{2n+1})}\lesssim (1+2^{k}|t-t'|)^{-n}. \end{aligned}$$

(5.8)

The above Proposition shall be proved in Sects. 8.1.2 and 8.2.2. Below we show how it implies (3.17). Using the $TT^*$ technique, it suffices to prove that

$$\begin{aligned} \Vert T^{k} (T^{k})^*\Vert _{L^{q_5', 1}({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})\rightarrow L^{q_5, \infty }({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})} \lesssim 2^{-k\frac{n(2n+1)}{(n+2)}}. \end{aligned}$$

We define $S^{k}$ and $S^k_m$ as in (5.2) and (5.4), with $T^{k,\ell }_t, T^{k,\ell }_{t'}$ replaced with $T^k_t, T^k_{t'}$ respectively. The above estimate then follows from

$$\begin{aligned} \Vert S^{k}g\Vert _{L^{q_5, \infty }({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})} \lesssim 2^{-k\frac{n(2n+1)}{(n+2)}}\Vert g\Vert _{L^{q_5', 1}({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})}, \end{aligned}$$

which by Bourgain’s interpolation trick is a consequence of the estimates

$$\begin{aligned} \Vert S^{k}_m g\Vert _{L^{2}({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})} \lesssim 2^{-k(2n+1)}2^m\Vert g\Vert _{L^2({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})} \end{aligned}$$

(5.9)

and

$$\begin{aligned} \Vert S^{k}_m g\Vert _{L^{\infty }({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})} \lesssim 2^{-\frac{nm}{2}}\Vert g\Vert _{L^1({\mathbb {R}}^{2n+1}\times {{\mathcal {Z}}}_{k})}. \end{aligned}$$

(5.10)

Estimate (5.9) follows like in the previous case, using the Cauchy–Schwarz inequality and the fixed time estimate (3.16) for $T^k$ with kernel supported around the equator or in an intermediate region.

(5.10) is a straightforward consequence of (5.8), for we have

$$\begin{aligned}\Vert S^{k,\ell }_m g\Vert _{L^1({{\mathbb {R}}}^{2n+1}\times {{\mathcal {Z}}}_k)\rightarrow L^{\infty }({{\mathbb {R}}}^{2n+1}\times {{\mathcal {Z}}}_k)} \lesssim (2^{k}\times 2^{-k+m})^{-n}=2^{-nm}\le 2^{-\frac{nm}{2}}. \end{aligned}$$

This concludes the proof of (3.17).

Thus, in order to prove the positive results in Theorems 1.1 and 1.2, we need to establish Propositions 4.3 and 5.1 around the poles (proven in Sects. 7.1 and 7.2 respectively), Proposition 5.2 around the equator and in the intermediate region (contained in Sects. 8.1.2 and 8.2.2 respectively) and Proposition 3.4 in all three cases (proven in Sects. 8.1.1, 8.2.1 and 8.3).

6 Results about radial functions and matrix inverses

This section contains a few results about the Hessian of a radial function g and its interaction with a skew symmetric matrix J satisfying $J^2=-I_d$, which will aid us in the calculations to follow. Depending on their taste, the reader may choose to directly skip to the latter sections, and consult these results when required.

Lemma 6.1

Let $d\in {{\mathbb {N}}}$, $u:{{\mathbb {R}}}\rightarrow {{\mathbb {R}}}$ be a smooth function and let $g:{{\mathbb {R}}}^d\rightarrow {{\mathbb {R}}}$ be given by $g(w)=u(|w|)$. Then

• (i) $\nabla g(w)=\frac{u'(|w|)}{|w|}w$.

• (ii) $\det g''(w)=\left( \frac{u'(|w|)}{|w|}\right) ^{d-1}u''(|w|)$.

Proof

The first part is a simple application of the chain rule, so we focus on the proof of the second part. Setting $r=|w|$ and taking partial derivatives twice, we have

$$\begin{aligned}\partial ^2_{ij}g(w)=u''(r)\frac{w_iw_j}{r^2}+u'(r)\left( \delta _{ij}\frac{1}{r}-\frac{w_iw_j}{r^3}\right) \end{aligned}$$

for $1\le i,j\le d$. Thus

$$\begin{aligned} g''(w)=\frac{u'(r)}{r}I_d+\left( \frac{u''(r)}{r^2}-\frac{u'(r)}{r^3}\right) ww^\intercal . \end{aligned}$$

(6.1)

Here $I_d$ denotes the identity matrix of order d. Taking determinants on both sides, we have

$$\begin{aligned}\det g''(w)=\left( \frac{u'(r)}{r}\right) ^d \det \left[ I_d+\left( \frac{u''(r)}{u'(r)r}-\frac{1}{r^2}\right) ww^\intercal \right] .\end{aligned}$$

We claim that for any $\sigma \in {{\mathbb {R}}}$,

$$\begin{aligned} \det (I_d+\sigma ww^{\intercal })=1+\sigma |w|^2=1+\sigma r^2. \end{aligned}$$

(6.2)

The claim then implies the desired equality, for then it follows that

$$\begin{aligned}\det g''(w)=\left( \frac{u'(r)}{r}\right) ^d\left[ 1+\left( \frac{u''(r)}{u'(r)r}-\frac{1}{r^2}\right) r^2\right] =\left( \frac{u'(|w|)}{|w|}\right) ^{d-1}u''(|w|).\end{aligned}$$

We now prove (6.2) by induction on the dimension d. It can be easily verified for $d=1,2$. Suppose now that the claim is true for $d-1$ with $d\ge 3$. Expressing $w\in {{\mathbb {R}}}^d$ as $w=(w',w_d)$ with $w'\in {{\mathbb {R}}}^{d-1}$, we set $r'=|w'|$. Using Schur’s complement, we have

$$\begin{aligned} \det (I_d+\sigma ww^{\intercal })&=\det \begin{pmatrix} 1+\sigma w_1^2 &{}\quad \sigma w_1w_2 &{}\quad \ldots &{}\quad \sigma w_1w_d\\ w_1w_2 &{}\quad 1+\sigma w_2^2 &{}\quad \ldots &{}\quad \sigma w_2w_d\\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ \sigma w_1w_d &{}\quad \ldots &{}\quad \ldots &{}\quad 1+\sigma w_d^2 \end{pmatrix}\\&=(1+\sigma w_d^2)\det \left[ I_{d-1}+\sigma w'w'^{\intercal }-\frac{\sigma ^2w_d^2}{(1+\sigma w_d^2)} w'w'^{\intercal }\right] \\&=(1+\sigma w_d^2)\det \left[ I_{d-1}+\frac{\sigma }{(1+\sigma w_d^2)}w'w'^{\intercal }\right] . \end{aligned}$$

Using the induction hypothesis, we conclude that

$$\begin{aligned}\det (I_d+\sigma ww^{\intercal })= & {} (1+\sigma w_d^2)\left( 1+\frac{\sigma }{(1+\sigma w_d^2)}|w'|^2\right) \\= & {} 1+\sigma (|w'|^2+|w_d|^2)=1+\sigma |w|^2.\end{aligned}$$

$\square$

Next we have a formula for the inverses of matrices of the form $J+g''(w)$ where J is a skew-symmetric matrix of dimension d with $J^2=-I_{d}$ and $w\in {{\mathbb {R}}}^{d}$.

Lemma 6.2

Let J be a skew-symmetric matrix with $J^2=-I_{d}$ and let $w\in {{\mathbb {R}}}^{d}$. For any $\sigma ,\lambda , \kappa , \gamma \in {{\mathbb {R}}}$, we have

$$\begin{aligned}{} & {} (\sigma I_{d}+\lambda ww^{\intercal }+\kappa J+\gamma w(Jw)^{\intercal })^{-1}\nonumber \\{} & {} \quad =\frac{1}{\sigma ^2+\kappa ^2}\bigg [\sigma I_{d}-\kappa J+\frac{\gamma \kappa -\sigma \lambda }{\sigma ^2+\kappa ^2+(\sigma \lambda -\gamma \kappa )|w|^2}\left( \sigma ww^{\intercal }-\kappa (Jw)w^{\intercal }\right) \nonumber \\{} & {} \qquad +\frac{\gamma \sigma +\lambda \kappa }{\sigma ^2+\kappa ^2+(\sigma \lambda -\gamma \kappa )|w|^2}\left( -\sigma w(Jw)^{\intercal }+\kappa (Jw)(Jw)^{\intercal }\right) \bigg ]. \end{aligned}$$

(6.3)

Proof

This can be verified directly, using the facts that $J^2=-I_{d}$, $ww^\intercal w w^\intercal =|w|^2ww^\intercal$ and $w^\intercal J w=0$ (due to J being skew-symmetric). $\square$

The following corollaries shall be useful in later calculations.

Corollary 6.3

Suppose there exist positive constants c, C such that

$$\begin{aligned}c<\min \{\sigma ^2+\kappa ^2, \sigma ^2+\kappa ^2+(\sigma \lambda -\gamma \kappa )|w|^2\} \end{aligned}$$

and

$$\begin{aligned}\max \{|\alpha |, |\lambda |, |\kappa |, |\gamma |, |w|\}<C.\end{aligned}$$

Then it is clear from (6.3) that the elements of the inverse of $\sigma I_{d}+\lambda ww^{\intercal }+\kappa J+\gamma w(Jw)^{\intercal }$ are bounded above by an absolute constant depending on C and c. We can thus conclude that

$$\begin{aligned}\Vert (\sigma I_{d}+\lambda ww^{\intercal }+\kappa J+\gamma w(Jw)^{\intercal })^{-1}\Vert \lesssim _{c,C,d} 1,\end{aligned}$$

where the norm above denotes the spectral norm of the matrix. We also have that

$$\begin{aligned} |\det \left[ \sigma I_{d}+\lambda ww^{\intercal }+\kappa J+\gamma w(Jw)^{\intercal }\right] ^{-1}|\lesssim _{c, C, d} 1. \end{aligned}$$

and thus

$$\begin{aligned} |\det \left[ \sigma I_{d}+\lambda ww^{\intercal }+\kappa J+\gamma w(Jw)^{\intercal }\right] |\gtrsim _{c, C, d} 1. \end{aligned}$$

Corollary 6.4

Observe that $(Jw)^{\intercal }w=0$ as J is skew-symmetric. Thus, using (6.3), we conclude that

$$\begin{aligned}&(\sigma I_{d}+\lambda ww^{\intercal }+\kappa J+\gamma w(Jw)^{\intercal })^{-1}w\\&\quad =\frac{1}{\sigma ^2+\kappa ^2}\bigg [(\sigma I_{d}-\kappa J)w+\frac{\gamma \kappa -\sigma \lambda }{\sigma ^2+(\sigma \lambda -\gamma \kappa )|w|^2+\kappa ^2} (\sigma ww^{\intercal }- \kappa (Jw)w^{\intercal })w \bigg ]\\&\quad =\frac{1}{\sigma ^2+(\sigma \lambda -\gamma \kappa )|w|^2+\kappa ^2}(\sigma I_{d}-\kappa J)w. \end{aligned}$$

7 Proof of Propositions 4.3 and 5.1: estimates at the poles

We begin by proving Proposition 4.3 about the fixed time $L^2$ bound on operators ${{\mathcal {T}}}^{k,\ell }_t$ for $400n\le \ell \le \frac{k}{4}$.

7.1 Proof of Proposition 4.3

Recall that for $0\le \ell \le \frac{k}{4}$, the operator ${{\mathcal {T}}}^{k,\ell }_t$ is defined as follows

$$\begin{aligned} {{\mathcal {T}}}_t^{k,\ell } f({\underline{x}},{{\bar{x}}}) = \int _{{\mathbb {R}}^{2n+1}} e^{i 2^{k-4\ell } \Phi _{\ell }(x,t,y)} a_{\ell }(x,t,y) f({\underline{y}},{{\bar{y}}}) dy. \end{aligned}$$

Here

$$\begin{aligned} \Phi _{\ell }(x,t,{\underline{y}},{{\bar{y}}})={\overline{y}}\left( {\overline{x}}+ \frac{1}{2}{\underline{x}}^T J {\underline{y}}+t^2g_{\ell }\left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) \right) \end{aligned}$$

(7.1)

and for a fixed $t\in [1,2]$, $a_{\ell }(x,t,y)=b(2^{-3\ell }{\underline{x}},2^{-4\ell }{{\bar{x}}},t,2^{-\ell }{\underline{y}},{{\bar{y}}})\rho _{1}(2^{-2\ell }{\underline{x}}-{\underline{y}})$ is supported in the set where $|{\underline{x}}|\lesssim 2^{3\ell }, |{{\bar{x}}}|\lesssim 2^{4\ell }$, $\frac{1}{8}\le |2^{-2\ell }{\underline{x}}-{\underline{y}}|\le \frac{1}{2}$ (resp. $|2^{-2\ell }{\underline{x}}-{\underline{y}}|\le \frac{1}{4}$) for $\ell <\frac{k}{4}$ (resp. $\ell =\frac{k}{4}$), and $|{{\bar{y}}}|\sim 1$.

We seek to apply Hörmander’s classical $L^2$ bound [33, ch. IX.1]. For a fixed phase function $\Phi$ supported in ${{\mathbb {R}}}^{2n+1}\times {{\mathbb {R}}}^{2n+1}$, this bound guarantees the fixed time estimate in Proposition 4.3, provided that the rank of the $(2n+1)\times (2n+1)$ mixed Hessian matrix $\Phi _{(x,y)}''$ is equal to $2n+1$. The implicit constant in the inequality depends on the lower bound for the determinant of $\Phi _{(x,y)}''$, the upper bound on finitely many derivatives of $\Phi$ and the amplitude a, and the size of the support of a.

In our case, we require uniform $L^2$ bounds for a family of operators with phase functions $\Phi _{\ell }$ and amplitudes $a_{\ell }$ for $400n\le \ell \le \frac{k}{4}$. From the definition of $a_\ell$, it is clear that for large $\ell$, the derivatives of $a_\ell$ up to a finite order are bounded from above by an absolute constant times the derivatives of a. Coming to the phase functions, recall that $g_\ell$ (in the definition of $\Phi _\ell$ above) is given by

$$\begin{aligned} g_{\ell }({{\underline{w}}})=g({{\underline{w}}})+2^{-4\ell }{\tilde{g}}_\ell ({{\underline{w}}}), \end{aligned}$$

with $\tilde{g}_\ell$ satisfying property (4.3), namely

$$\begin{aligned} \left| \frac{\partial ^\beta }{\partial {{\underline{w}}}^{\beta }}\tilde{g}_\ell ({{\underline{w}}})\right| \le 100n\cdot 2^{-4\ell }|{{\underline{w}}}|^{8-|\beta |}\,\, \text { for } |\beta |\in \{0,1,2,3\} \text { and } |{{\underline{w}}}|\le \frac{1}{2}. \end{aligned}$$

The above estimate is also true for higher order derivatives, perhaps with an absolute constant greater than 100n. Thus for large enough $\ell$ (say $\ell \ge 400n$), $g_{\ell }$ and its higher order derivatives essentially behave like g and consequently $\Phi _{\ell }$ is a smooth perturbation of $\Phi (x,t,{\underline{y}},{{\bar{y}}})={\overline{y}}\left( {\overline{x}}+ {\underline{x}}^T J {\underline{y}}+t^2g(\tfrac{-{\underline{y}}}{t}) \right)$.

However, the size of the x support of the amplitude $a_\ell$ is no longer uniform in $\ell$, since $a_\ell$ is supported in the set where $|{\underline{x}}|\lesssim 2^{3\ell }$ and $|{{\bar{x}}}|\lesssim 2^{4\ell }$. Hence we need to introduce spatial localization in the x variable. To this effect, let $\tilde{\rho }$ be a smooth, non-negative, compactly supported function in ${{\mathbb {R}}}^{2n+1}$ such that $\sum _{\mu \in {{\mathbb {Z}}}^{2n+1}}\tilde{\rho }(x-\mu )=1$ for all $x\in {{\mathbb {R}}}^{2n+1}$. Define

$$\begin{aligned}{{\mathcal {T}}}^{k,\ell }_{t,\mu }f(x):=\tilde{\rho }(x-\mu )\int _{{\mathbb {R}}^{2n+1}} e^{i 2^{k-4\ell } \Phi _{\ell }(x,t,y)} a_{\ell }(x,t,y) f({\underline{y}},{{\bar{y}}}) dy.\end{aligned}$$

Then we can write

$$\begin{aligned}{{\mathcal {T}}}^{k,\ell }_{t}f(x)=\sum _{\mu \in {{\mathbb {Z}}}^{2n+1}}{{\mathcal {T}}}^{k,\ell }_{t,\mu }f(x).\end{aligned}$$

The kernel of each ${{\mathcal {T}}}^{k,\ell }_{t,\mu }$ is now supported in a set of constant size. Since Hörmander’s estimate is stable under smooth perturbations, it can be applied to show that $\Vert {{\mathcal {T}}}^{k,\ell }_{t,\mu }\Vert _{L^2({{\mathbb {R}}}^{2n+1})}\lesssim 2^{-(k-4\ell )\frac{2n+1}{2}}$ uniformly in $\mu$ and t, provided we establish that

$$\begin{aligned} {\textrm{rank }}(\Phi _\ell )_{(x,y)}''= 2n+1 \end{aligned}$$

(7.2)

with $|\det (\Phi _\ell )_{(x,y)}''|$ bounded from below uniformly in $\ell , t$ and $\mu$. Let

$$\begin{aligned} \Xi ^\ell (x,t,y):=\nabla _{x,t} \Phi _\ell (x,t,y). \end{aligned}$$

(7.3)

We calculate

$$\begin{aligned} \Xi ^\ell (x,t,y)= {\overline{y}}\begin{pmatrix} \frac{1}{2}J{\underline{y}}+2^{-2\ell }t\nabla g_{\ell } \left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) \\ 1\\ 2tg\left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) -\nabla g\left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) \cdot (2^{-2\ell }{\underline{x}}-{\underline{y}})+2^{-4\ell }\Upsilon _{\ell }\left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) \end{pmatrix}, \end{aligned}$$

where $\Upsilon _{\ell }\left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) :=\frac{\partial }{\partial t}\left( t^2\tilde{g}_\ell \left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) \right)$ and satisfies

$$\begin{aligned} \left| \frac{\partial ^\beta }{\partial {{\underline{w}}}^{\beta }}\Upsilon _\ell ({{\underline{w}}})\right| \le 4\cdot 100n\cdot 2^{-4\ell }|{{\underline{w}}}|^{8-|\beta |}\,\, \text { for } |\beta |\in \{0,1,2,3\} \text { and } |{{\underline{w}}}|\le \frac{1}{2}.\nonumber \\ \end{aligned}$$

(7.4)

The partial derivative with respect to t is not needed at the moment but will be helpful for calculations in the next subsection. Note that since g is a homogeneous polynomial of degree 4, we have

$$\begin{aligned} \nabla g\left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) \cdot (2^{-2\ell }x-y)=4tg\left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) . \end{aligned}$$

Using the above in the last co-ordinate, and subscripts $1,2, \ldots , 2n, 2n+1$ to denote partial derivatives in $y_1, y_2, \ldots , y_{2n}$ and ${{\bar{y}}}$ respectively, we have

$$\begin{aligned} \Xi _{j}^{\ell }&={{\bar{y}}}\begin{pmatrix} \frac{1}{2}Je_j-2^{-2\ell }\partial _j\nabla g_{\ell }\left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) \\ 0\\ -2\partial _jg\left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) +2^{-4\ell }t^{-1}\partial _j\Upsilon _\ell \left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) \end{pmatrix},\\ \Xi _{2n+1}^{\ell }&=\begin{pmatrix} \frac{1}{2}J{\underline{y}}+2^{-2\ell }t\nabla g_{\ell } \left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) \\ 1\\ -2tg\left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) +2^{-4\ell }\Upsilon _{\ell }\left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) \end{pmatrix}. \end{aligned}$$

Thus, for a fixed $t\in [1,2]$, the mixed Hessian $(\Phi _\ell )''_{(x,y)}$ is of the form

$$\begin{aligned} \begin{pmatrix} \frac{J}{2}-2^{-2\ell }g''_\ell \left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) &{}\quad *\\ 0&{}\quad 1 \end{pmatrix}. \end{aligned}$$

Using (6.1), we get

$$\begin{aligned}\frac{J}{2}-2^{-2\ell }g''\left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) =\frac{J}{2}-2^{-2\ell }\frac{u_\ell '(r)}{r}I_{2n}-2^{-2\ell }\left( \frac{u_\ell ''(r)}{r^2}-\frac{u_\ell '(r)}{r^3}\right) {{\underline{w}}}{{\underline{w}}}^\intercal ,\end{aligned}$$

where ${{\underline{w}}}= \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}$ and $r=|{{\underline{w}}}|$.

Let $\lambda _{\ell }(r):=\frac{u_{\ell }''(r)}{r^2}-\frac{u_{\ell }'(r)}{r^3}$, $\sigma _{\ell }(r):=\frac{u_{\ell }'(r)}{r}$ and $\kappa =\tfrac{1}{2}$. Since $u_\ell$ satisfies (4.4), for $\ell \ge 400n$ and $|r|\le \frac{1}{2}$, we have

$$\begin{aligned}\kappa ^2+2^{-4\ell }\sigma _{\ell }\lambda _{\ell }r^2+2^{-4\ell }\sigma _{\ell }^2=\frac{1}{4}+2^{-4\ell } u_{\ell }''(r)\sigma _\ell \gtrsim \frac{1}{4}. \end{aligned}$$

Moreover, $2^{-4\ell }\sigma _{\ell }^2+\kappa ^2\ge \frac{1}{4}$ and $\max \{2^{-2\ell }|\sigma _\ell |, 2^{-2\ell }|\lambda _\ell |, \kappa , |{{\underline{w}}}|\}\le 1$.

We now use Corollary 6.3, with $\lambda =-2^{-2\ell }\lambda _\ell (r)$, $\sigma =-2^{-2\ell }\sigma _\ell (r)$, $\kappa =\frac{1}{2}$ and $\gamma =0$ to conclude that

$$\begin{aligned}\left\| \left[ \frac{J}{2}-2^{-2\ell }g''\left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) \right] ^{-1}\right\| \lesssim _n 1.\end{aligned}$$

It follows that $(\Phi _\ell )''_{(x,y)}$ is invertible with $\Vert [(\Phi _\ell )''_{(x,y)}]^{-1}\Vert \lesssim _n 1$ and

$$\begin{aligned}\left| \det (\Phi _\ell )''_{(x,y)}\right| \gtrsim _n 1.\end{aligned}$$

7.1.1 Almost orthogonality

We have thus verified (7.2) and can conclude using Hörmander’s $L^2$ estimate that

$$\begin{aligned} \Vert {{\mathcal {T}}}^{k,\ell }_{t,\mu }\Vert _{L^2({{\mathbb {R}}}^{2n+1})}\lesssim 2^{-(k-4\ell )\frac{2n+1}{2}} \end{aligned}$$

(7.5)

for each $\mu \in {{\mathbb {Z}}}^{2n+1}$. To avoid cumbersome notation, we suppress the dependence on $k, \ell$ and t and denote ${{\mathcal {T}}}^{k,\ell }_{t,\mu }$ by ${{\mathcal {T}}}_\mu$. In order to obtain the required $L^2$ estimate for ${{\mathcal {T}}}^{k,\ell }_{t}=\sum _{\mu \in {{\mathbb {Z}}}^{2n+1}}{{\mathcal {T}}}_{\mu }$, by the Cotlar–Stein lemma, it suffices to show that

$$\begin{aligned} \sup _{\mu ,\nu \in {{\mathbb {Z}}}^{2n+1}}\{\Vert {{\mathcal {T}}}_\mu {{\mathcal {T}}}_\nu ^*\Vert _{L^2}, \Vert {{\mathcal {T}}}_\mu ^*{{\mathcal {T}}}_\nu \Vert _{L^2}\}\le c_N 2^{-(k-4\ell )\frac{2n+1}{2}}(1+|\mu -\nu |)^{-N} \end{aligned}$$

(7.6)

for large enough N. Because of (7.5), it is enough to consider the cases when $\mu \ne \nu$. The kernel of ${{\mathcal {T}}}_\mu ^*{{\mathcal {T}}}_\nu$ is of the form

$$\begin{aligned} \int _{{\mathbb {R}}^{2n+1}} e^{i 2^{k-4\ell }[\Phi _{\ell }(x,t,y)-\Phi _{\ell }(x,t,\breve{y})]} a_{\ell }(x,t,y)\overline{a_{\ell }(x,t,\breve{y})}\tilde{\rho }(x-\mu )\tilde{\rho }(x-\nu ) dx \end{aligned}$$

and is non-zero only for $|\mu -\nu |\le M$ for some constant M depending on $\rho$. In view of (7.5), this implies the desired estimate for $\Vert {{\mathcal {T}}}_\mu {{\mathcal {T}}}_\nu ^*\Vert _{L^2}$ for a large enough constant $c_N$.

We now consider ${{\mathcal {T}}}_\mu {{\mathcal {T}}}_\nu ^*$ whose kernel is given by

$$\begin{aligned} {{\mathcal {K}}}_{\mu \nu ^*}(x,\breve{x}):=\tilde{\rho }(x-\mu )\tilde{\rho }(\breve{x}-\nu ) \int _{{\mathbb {R}}^{2n+1}} e^{i 2^{k-4\ell } [\Phi _{\ell }(x,t,y)-\Phi _{\ell }(\breve{x},t,y)]} a_{\ell }(x,t,y)\overline{a_{\ell }(\breve{x},t, y)}dy. \end{aligned}$$

This kernel is supported in the set where $\max \left( |{{\bar{x}}}|,|\bar{\breve{x}}|\right) \lesssim 2^{4\ell }$, $\max \big (|2^{-2\ell }{\underline{x}}-{\underline{y}}|,|2^{-2\ell }\underline{\breve{x}}-{\underline{y}}| \big )\le \frac{1}{2}$, hence $|{{\bar{x}}}-\bar{\breve{x}}|\lesssim 2^{4\ell }$ and $|{\underline{x}}-\underline{\breve{x}}|\le 2^{2\ell }$. Recall that the phase function is given by

$$\begin{aligned} \Phi _{\ell }(x,t,{\underline{y}},{{\bar{y}}})={\overline{y}}\left( {\overline{x}}+ \frac{1}{2}{\underline{x}}^T J {\underline{y}}+t^2g_{\ell }(\tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}) \right) . \end{aligned}$$

Differentiating twice with respect to ${\underline{x}}$, we have

$$\begin{aligned} (\Phi _\ell )_{{\underline{x}}{\underline{x}}}''(x,t,y)=2^{-4\ell }{{\bar{y}}}g_\ell ''(\tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}), \end{aligned}$$

(7.7)

while

$$\begin{aligned} \partial _{{{\bar{x}}}}(\nabla _{{\underline{x}}}\Phi _\ell )(x,t,y)=0,\ \partial _{{{\bar{x}}}}^2(\Phi _\ell )(x,t,y)=0. \end{aligned}$$

(7.8)

Thus, using Taylor expansion of $\Phi _\ell$ around x and differentiating with respect to y, we have

$$\begin{aligned} \nabla _y[\Phi _{\ell }(x,t,y)-\Phi _{\ell }(\breve{x},t, y)]=(\Phi _{\ell })''_{x,y}(x,t,y)\cdot (x-\breve{x})+ 2^{-4\ell }\tilde{R}_\ell ({\underline{x}},\underline{\breve{x}},t,y),\nonumber \\ \end{aligned}$$

(7.9)

with $\tilde{R}_\ell$ satisfying the condition

$$\begin{aligned}|\tilde{R}_\ell ({\underline{x}},\underline{\breve{x}},t,y)|\lesssim |{\underline{x}}-\underline{\breve{x}}|^2 \end{aligned}$$

on account of (4.3), (7.7), (7.8) and the fact that $\Vert g_\ell ''\left( \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) \Vert \lesssim 1$ for $\left| \tfrac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right| \le \frac{1}{2}$. Since $|{\underline{x}}-\underline{\breve{x}}|\lesssim 2^{2\ell }$ and $(\Phi _\ell )_{x,y}''(x,y,t)$ is invertible with $\Vert [(\Phi _\ell )''_{(x,y)}(x,t,y)]^{-1}\Vert \lesssim _n 1$ (see the calculation above), for $\ell \ge 400n$ we can conclude that

$$\begin{aligned}|\nabla _y[\Phi _{\ell }(x,t,y)-\Phi _{\ell }(\breve{x},t, y)]|\gtrsim |x-\breve{x}|.\end{aligned}$$

Similarly, for any 2n-dimensional multindex $\alpha$ with $|\alpha |\ge 2$, we have

$$\begin{aligned} \partial ^\alpha _y[\Phi _{\ell }(x,t,y)-\Phi _{\ell }(\breve{x},t, y)]=\partial ^\alpha _y(\nabla _x\Phi _{\ell })(x,t,y)\cdot (x-\breve{x})+ 2^{-4\ell }\tilde{R}_{\ell ,\alpha }({\underline{x}},\underline{\breve{x}},t,y)\nonumber \\ \end{aligned}$$

(7.10)

with $\tilde{R}_{\ell ,\alpha }$ also satisfying the condition

$$\begin{aligned} |\tilde{R}_{\ell ,\alpha }({\underline{x}},\underline{\breve{x}},t,y)|\lesssim |{\underline{x}}-\underline{\breve{x}}|^2. \end{aligned}$$

Now

$$\begin{aligned}|\partial ^\alpha _y(\nabla _{\underline{x}}\Phi _{\ell })(x,t,y)|\lesssim 1,\end{aligned}$$

while any derivative of $\partial _{{\bar{x}}}\Phi _\ell (x,t,y)$ of order two or higher (in the y variables) is zero.

Since $|{\underline{x}}-\breve{{\underline{x}}}|\lesssim 2^{2\ell }$, we conclude using (7.10) that

$$\begin{aligned}\left| \partial ^\alpha _y[\Phi _{\ell }(x,t,y)-\Phi _{\ell }(\breve{x},t, y)]\right| \lesssim |{\underline{x}}-\breve{{\underline{x}}}|.\end{aligned}$$

Integrating by parts N times yields

$$\begin{aligned}|{{\mathcal {K}}}_{\mu \nu ^*}(x,\breve{x})|\lesssim _N\tilde{\rho }(x-\mu )\tilde{\rho }(\breve{x}-\nu )(1+2^{k-4\ell }|x-\breve{x}|)^{-N}.\end{aligned}$$

Thus

$$\begin{aligned}{} & {} \int _{{{\mathbb {R}}}^{2n+1}}|{{\mathcal {K}}}_{\mu \nu ^*}(x,\breve{x})|dx\lesssim _{N}2^{-(k-4\ell )N}\int _{|x-\breve{x}|\sim |\mu -\nu |}\rho (x-\mu )|x-\breve{x}|^{-N}dx\\{} & {} \quad \lesssim _N 2^{-(k-4\ell )N}(1+|\mu -\nu |)^{-N+2n}. \end{aligned}$$

By symmetry we have the same estimate for $\int _{{{\mathbb {R}}}^{2n+1}}|{{\mathcal {K}}}_{\mu \nu ^*}(x,\breve{x})|d\breve{x}$ and thus it follows by the Schur test that

$$\begin{aligned}\Vert {{\mathcal {T}}}_\mu ^*{{\mathcal {T}}}_\nu \Vert _{L^2}\lesssim _N 2^{-(k-4\ell )\frac{2n+1}{2}}(1+|\mu -\nu |)^{-N+2n}.\end{aligned}$$

The above estimate implies (7.6) with N replaced by $N-2n$, which is still good enough for large enough N (say $N>4n+2$). An application of the Cotlar–Stein lemma concludes the proof of Proposition 4.3.

7.2 Proof of Proposition 5.1

Let $\Xi ^\ell =\nabla _{x,t}\Phi _\ell$ as in (7.3), $N\in {\mathbb {R}}^{2n+2}$ be a unit vector, and let $\mathscr {C}^\ell (x,t,y)$ be the $(2n+1)\times (2n+1)$ cinematic curvature matrix with respect to N given by

$$\begin{aligned} {\mathscr {C}}_{ij}^\ell = \frac{\partial ^2}{\partial y_i\partial y_j} \langle N,\Xi ^\ell \rangle \end{aligned}$$

(7.11)

We would like to argue as in the proof of [20, Proposition 3.4] using a variant of Stein’s theorem [32] according to which the kernel estimate (5.7) holds provided the rank of the mixed Hessian $(\Phi _\ell )_{(x,t),y}''=2n+1$ and the curvature condition

$$\begin{aligned} \langle N, \Xi ^\ell _{j} \rangle =0 ,\,\, j=1,\dots , 2n+1 \quad \implies \quad {\textrm{rank }}\, {\mathscr {C}}^\ell = 2n \end{aligned}$$

(7.12)

is satisfied; i.e. the conic surface $\Sigma ^\ell _{x,t}$ parametrized by $y\mapsto \Xi ^\ell (x,t,y)$ has the maximal number 2n of nonvanishing principal curvatures. The argument would be a consequence of the method of stationary phase applied to the oscillatory integral operator given by ${{\mathcal {T}}}^{k,\ell }_t({{\mathcal {T}}}^{k,\ell }_{t'})^*$.

However, the kernel estimate in [20] was proved for a single oscillatory integral operator with a fixed phase function $\Phi$, whereas we need to apply it for the family of operators ${{\mathcal {T}}}^{k,\ell }_t$ with $\ell <\frac{k}{4}$. The implicit constant in Stein’s theorem depends on the upper bound on finitely many spatial derivatives of the phase function $\Phi _\ell$ and the amplitude $a_\ell$, the size of the y support of $a_\ell$ and the lower bound for the determinant of the invertible $2n\times 2n$ minor of ${\mathscr {C}}^\ell$. Since the argument involves integration by parts, we also need the bounds

$$\begin{aligned} |\nabla _y[\Phi _{\ell }(x,t,y)-\Phi _{\ell }(\breve{x},\breve{t}, y)]|\gtrsim |x-\breve{x}|+|t-\breve{t}| \end{aligned}$$

and

$$\begin{aligned} \left| \partial ^\alpha _y[\Phi _{\ell }(x,t,y)-\Phi _{\ell }(\breve{x},t, y)]\right| \lesssim |{\underline{x}}-\breve{{\underline{x}}}|+|t-\breve{t}| \end{aligned}$$

on the phase function for any 2n-dimensional multindex $\alpha$ with $|\alpha |\ge 2$.

We have already observed in the previous subsection that the spatial derivatives of the amplitude $a_\ell$ are bounded from above by a constant times the corresponding derivatives of a for all large $\ell$. Further, arguing similarly as in the last subsection using property (4.3) of $g_\ell$, it can be seen that $\Phi _\ell$ and $\nabla _{(x,t)}\Phi _\ell$ are smooth perturbations of $\Phi$ and $\nabla _{(x,t)}\Phi$, where $\Phi ={\overline{y}}\left( {\overline{x}}+ {\underline{x}}^T J {\underline{y}}+t^2g(\tfrac{-{\underline{y}}}{t}) \right)$. Moreover, the size of the y support of $a_\ell$ is bounded in $\ell$, and while the same is not true for the size of the x support, arguing as in the previous subsection using using the Taylor expansion of $\Phi _\ell$ around (x, t), we can still conclude the aforementioned estimates on the phase function, required for integration by parts.

Since the estimate in [20] is stable under smooth perturbations, it suffices to show that ${\textrm{rank }}{{\mathscr {C}}^\ell }=2n$ and the determinant of its invertible $2n\times 2n$ minor is bounded from below by an absolute constant independent of $\ell$.

To this effect, let $N=(\underline{\alpha },{\bar{\alpha }},\alpha _{2n+2})$ be a unit normal vector satisfying $\langle N, \Xi _j^\ell \rangle =0$ for $1\le j\le 2n+1$. The first 2n of these equations can be written down in the matrix form

$$\begin{aligned}{} & {} -\left( \frac{J}{2}-2^{-2\ell }g_{\ell }''\left( \frac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) \right) ^{\intercal } \underline{\alpha }\\{} & {} \quad =\alpha _{2n+2}\left[ -2\nabla g_{\ell }\left( \frac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) +2^{-4\ell }\nabla \Upsilon _\ell \left( \frac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) \right] . \end{aligned}$$

Recall that ${{\underline{w}}}:=-\frac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}$ and $r=|{{\underline{w}}}|$. Let $\lambda _{\ell }(r):=\frac{u_{\ell }''(r)}{r^2}-\frac{u_{\ell }'(r)}{r^3}$ and $\sigma _{\ell }(r):=\frac{u_{\ell }'(r)}{r}$. Using (6.1) for $-\left( \frac{J}{2}-2^{-2\ell }g_{\ell }''\left( \frac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) \right) ^{\intercal } =\left( \frac{J}{2}+2^{-2\ell }g_{\ell }''\left( \frac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}\right) \right)$, we get

$$\begin{aligned} \underline{\alpha }&=-\alpha _{2n+2}\left( \frac{J}{2}+2^{-2\ell }\sigma _{\ell }(r)I_{2n}+2^{-2\ell }\lambda _{\ell }(r){{\underline{w}}}{{\underline{w}}}^\intercal \right) ^{-1}(2\nabla g_{\ell }({{\underline{w}}})-2^{-4\ell }\nabla \Upsilon _{\ell }({{\underline{w}}})). \end{aligned}$$

Applying Lemma 6.1, part (i) and Remark 6.4 to calculate the gradient and matrix inverse on the right hand side yields

$$\begin{aligned} \underline{\alpha }=\frac{-2\alpha _{2n+2}\sigma _{\ell }}{\left( \tfrac{1}{4}+2^{-4\ell }r^2\sigma _{\ell }\lambda _{\ell }+2^{-4\ell }\sigma _{\ell }^2\right) }\left( 2^{-2\ell }\sigma _{\ell } I_{2n}-\frac{J}{2}\right) {{\underline{w}}}+2^{-4\ell }\alpha _{2n+2}\tilde{{{\underline{w}}}},\qquad \end{aligned}$$

(7.13)

where $\tilde{{{\underline{w}}}}:=-\left( \frac{J}{2}+2^{-2\ell }\sigma _{\ell }(r)I_{2n}+2^{-2\ell }\lambda _{\ell }(r){{\underline{w}}}{{\underline{w}}}^\intercal \right) ^{-1}2^{-4\ell }\nabla \Upsilon _{\ell }({{\underline{w}}})$. Using (7.4) and Corollary 6.3, it is easy to see that $|\tilde{{{\underline{w}}}}|\lesssim 2^{-4\ell }$.

We compute

$$\begin{aligned} \Xi _{ij}^{\ell }&={{\bar{y}}}\begin{pmatrix} 2^{-2\ell }t^{-1}\partial ^2_{ij}\nabla g_{\ell }({{\underline{w}}})\\ 0\\ 2t^{-1}\partial ^2_{ij}g_{\ell }({{\underline{w}}})-2^{-4\ell }t^{-1} \partial ^2_{ij}\Upsilon _{\ell }({{\underline{w}}}) \end{pmatrix}\,\, \text { for } 1\le i,j\le 2n,\\ \Xi _{ij}^{\ell }&={{\bar{y}}}^{-1}\Xi _{j}^{\ell }\,\, \text { for } i=2n+1. \end{aligned}$$

Using (4.1) and (4.2), we can write

$$\begin{aligned} \Xi _{ij}^{\ell }={{\bar{y}}}\begin{pmatrix} 2^{-2\ell }t^{-1}\partial ^2_{ij}\nabla g({{\underline{w}}})\\ 0\\ 2t^{-1}\partial ^2_{ij}g_{\ell }({{\underline{w}}}) \end{pmatrix}+{{\bar{y}}}2^{-4\ell }\tilde{\Xi }_{ij}^{\ell }({{\underline{w}}})\,\, \text { for } 1\le i,j\le 2n, \end{aligned}$$

with $|\tilde{\Xi }_{ij}^{\ell }({{\underline{w}}})|\le 4\cdot 100n\cdot 2^{-4\ell }|{{\underline{w}}}|^{6}$, using (7.4).

We now use Eq. (7.13) to calculate the elements of the cinematic curvature matrix:

$$\begin{aligned} \frac{t}{{{\bar{y}}}}\langle \Xi _{ij}^{\ell }, N\rangle&=\frac{-2^{-2\ell +1}\alpha _{2n+2}\sigma _{\ell }}{\tfrac{1}{4}+2^{-4\ell }\sigma _{\ell }\lambda _{\ell }|{{\underline{w}}}|^2+2^{-4\ell }\sigma _{\ell }^2}\left\langle \left( 2^{-2\ell }\sigma _{\ell } I_{2n}-\frac{J}{2}\right) {{\underline{w}}},\partial _{ij}^2\nabla g({{\underline{w}}}) \right\rangle \\&\quad +2\alpha _{2n+2} \partial ^2_{ij}g({{\underline{w}}})+\alpha _{2n+2}2^{-4\ell }a_{ij}({{\underline{w}}}), \end{aligned}$$

where $|a_{ij}({{\underline{w}}})|\lesssim 2^{-4\ell }$. Since $\langle {{\underline{w}}},\partial _{ij}^2\nabla g({{\underline{w}}})\rangle =2\partial ^2_{jk}g({{\underline{w}}})$ and $\frac{1}{4}+2^{-4\ell }\sigma _{\ell }\lambda _{\ell }r^2+2^{-4\ell }\sigma _{\ell }^2=\frac{1}{4}+2^{-4\ell } u_{\ell }''\sigma _\ell$, we have

$$\begin{aligned} \frac{t}{{{\bar{y}}}}\langle \Xi _{ij}^{\ell }, N\rangle&=\alpha _{2n+2}\left( 2-\frac{2^{-4\ell +2}\sigma _{\ell }^2}{\tfrac{1}{4}+2^{-4\ell } u_{\ell }''\sigma _\ell }\right) \partial ^2_{ij}g({{\underline{w}}})\\&\quad +\frac{2^{-2\ell }\alpha _{2n+2}\sigma _{\ell }}{\frac{1}{4}+2^{-4\ell } u_{\ell }''\sigma _\ell }\left\langle J{{\underline{w}}},\partial _{jk}^2\nabla g({{\underline{w}}}) \right\rangle +\alpha _{2n+2}2^{-4\ell }a_{ij}({{\underline{w}}}). \end{aligned}$$

Because u satisfies (4.4), it follows that for sufficiently large $\ell$ (for example $\ell \ge 2$) and $|r|< \frac{1}{2}$,

$$\begin{aligned} \max \left( \frac{\sigma _{\ell }(r)}{\tfrac{1}{4}+2^{-4\ell }u_{\ell }''\sigma _{\ell }},\sigma _{\ell }(r)\right) \lesssim 1. \end{aligned}$$

Thus, for $1\le i,j\le 2n$ and $|{{\underline{w}}}|<\frac{1}{2}$,

$$\begin{aligned} \left| \frac{t}{{{\bar{y}}}}\langle \Xi _{ij}^{\ell }, N\rangle -2\alpha _{2n+2}\partial ^2_{ij}g({{\underline{w}}})\right| \lesssim 2^{-2\ell }. \end{aligned}$$

(7.14)

From (7.13), we infer that $|{\bar{\alpha }}|\lesssim |\alpha _{2n+2}|$. This, combined with the equation $\langle N, \Xi _{2n+1}\rangle =0$, further implies that $|{\bar{\alpha }}|\lesssim |\alpha _{2n+2}|$. Since N is a unit vector, it follows that $|\alpha _{2n+2}|\gtrsim 1$.

The second part of Lemma 6.1 implies that

$$\begin{aligned} \det g''({{\underline{w}}})=\left( \frac{2r^3}{r}\right) ^{2n-1}6r^2, \end{aligned}$$

which is uniformly bounded away from 0, since $\frac{1}{8}\le r=|{{\underline{w}}}|\le \frac{1}{2}$ when ${{\underline{w}}}=\frac{2^{-2\ell }{\underline{x}}-{\underline{y}}}{t}$ is in the support of $a_\ell$ for $\ell <\frac{k}{4}$. Hence, we conclude that $g''({{\underline{w}}})$ is invertible.

Let $\bar{{\mathscr {C}}}_{\ell }(x,t,y)$ denote the $2n\times 2n$ matrix with the (i, j)-th entry given by $\langle \Xi _{ij}^{\ell }, N\rangle$ for $1\le i,j\le 2n$. Using (7.14), we conclude that $\bar{{\mathscr {C}}}_{\ell }(x,t,y)$ is invertible as well, with its determinant bounded from below by an absolute constant independent of $\ell$ for $\ell \ge 400n$.

Further, observe that for $i=2n+1$, $\langle N,\Xi _{ij}\rangle =\langle N, {{\bar{y}}}^{-1}\Xi _{j}\rangle =0$ for $1\le j\le 2n+1$. Since the cinematic curvature matrix at (x, t, y) takes the form

$$\begin{aligned}{\mathscr {C}}^\ell (x,t,y)=2\frac{\alpha _{2n+2}}{{{\bar{y}}}}\begin{pmatrix} \bar{{\mathscr {C}}}_\ell &{} &{} 0\\ 0&{} &{} 0 \end{pmatrix}, \end{aligned}$$

we conclude that it is of rank 2n. This finishes the proof of Proposition 5.1.

8 Proof of Propositions 3.4 and 5.2

In this section, we prove the fixed time $L^2$ estimate (3.4) for the operator $T^k_t$ and the estimate (5.8) for the kernel $K^k_{tt'}$ around the equator and in the intermediate region. We shall observe that the former estimate is stable in the intermediate region and thus implies the $L^2$ estimate for $T^k$ around the north pole. We plan to use Hörmander’s $L^2$ estimate [33, ch. IX.1] and the curvature condition in [20, Proposition 3.4] again to prove Propositions 3.16 and 5.2 respectively. Since these results are applied for a fixed oscillatory operator (and not a family of them as in the previous section), they are directly applicable.

8.1 Estimates around the equator

Recall that the phase function of $T_k$ around the equator is given by $\Phi =\Phi ^\text {Eq}=y_1H_1(x,t,y',{{\bar{y}}})$ with

$$\begin{aligned} \nabla _{(x,t)}H_1(x,t,y',{{\bar{y}}})=-\left[ \left( \frac{\partial F}{\partial y_1}\right) ^{-1}\nabla _{x,t}F\right] \bigg |_{(x,t,H_1(x,t,y',{{\bar{y}}}), y', {{\bar{y}}})}. \end{aligned}$$

Here F(x, t, y) is the defining function of the Korányi sphere centred at x and of radius t

$$\begin{aligned} F(x,t,y)=|{\underline{x}}-{\underline{y}}|^4+|{{\bar{x}}}-{{\bar{y}}}+\tfrac{1}{2}{\underline{x}}^{\intercal }J{\underline{y}}|^2-t^4. \end{aligned}$$

The amplitude $b=b_\text {Eq}(x,t,y)$ is compactly supported in a set where x is small, $|y_1|\in [1/2,2]$, $t^{-1}|x'-y'|<2^{-400n}$, $t^{-2}|{{\bar{x}}}-{{\bar{y}}}+\frac{1}{2}{\underline{x}}^{\intercal }J(H_1(x,t,y',{{\bar{y}}}),y')|<2^{-400n}$ and $t^{-1}|x_1-H_1(x,t,y',{{\bar{y}}})|>1-2^{-400n}$.

8.1.1 Proof of Proposition 3.4 at the Equator

By Hörmander’s $L^2$ estimate, it suffices to prove that the mixed Hessian $\Phi _{x,y}''$ is of rank $2n+1$ in the support of b, with the determinant bounded from below by an absolute constant independent of t.

Let $\Theta (x,y):=({\underline{x}}-{\underline{y}}, {{\bar{x}}}-{{\bar{y}}}+\tfrac{1}{2}{\underline{x}}^\intercal J {\underline{y}})$. Since $\det \Phi _{x,y}''(x,t,y)=\det \Phi _{x,y}''(0,t, \Theta (x,y))$ (see Sect. 11.1) , it suffices to verify the curvature condition under the assumptions $x=0$, $|{{\bar{y}}}|\in [1/2,2]$, $t^{-1}|y'|<2^{-400n}$, $t^{-2}|{{\bar{y}}}|<2^{-400n}$ and $t^{-1}|H_1(0,t,y',{{\bar{y}}})|>t(1-2^{-400n})$.

Let

$$\begin{aligned} \Xi (x,t,y)=\nabla _{(x,t)}\Phi ^\text {Eq}(x,t,y). \end{aligned}$$

We calculate

$$\begin{aligned} \Xi (0,t,y)= & {} y_1\nabla _{(x,t)}H_1(0,t,y',{{\bar{y}}})\\= & {} y_1\left[ \frac{1}{4|{\underline{y}}|^2y_1}(4|{\underline{y}}|^2{\underline{y}}+{{\bar{y}}}(J{\underline{y}}+2e_{2n+1})-4t^3e_{2n+2})\right] \Big |_{|y_1=H_1(0 ,t,y',{{\bar{y}}})}. \end{aligned}$$

Let $\Xi _1:=\frac{\partial }{\partial {y_1}}\Xi =\nabla _{(x,t)}H_1(0,t,y',{{\bar{y}}})$.

Remark 8.1

We have

$$\begin{aligned} \nabla _{(x,t)}H_1(x ,t,y',{{\bar{y}}})=-\left[ \left( \frac{\partial F}{\partial y_1}\right) ^{-1}\nabla _{x,t}F\right] \bigg |_{y_1=H_1(x,t,y',{{\bar{y}}})}. \end{aligned}$$

Taking partial derivatives with respect to $y_j$ and using the chain rule, we obtain

$$\begin{aligned} \frac{\partial }{\partial {y_j}}\nabla _{(x,t)}H_1(0,t,y')&=-\left[ \left( \frac{\partial F}{\partial y_1}\right) ^{-1}\left( \frac{\partial }{\partial y_j}+\frac{\partial H_1}{\partial y_j}\frac{\partial }{\partial y_1}\right) \nabla _{x,t}F\right] \bigg |_{x=0, y_1=H_1(0 ,t,y',{{\bar{y}}})}\\&\quad + \left[ \nabla _{x,t}F\left( \frac{\partial F}{\partial y_1}\right) ^{-2}\left( \frac{\partial }{\partial y_j}+\frac{\partial H_1}{\partial y_j}\frac{\partial }{\partial y_1}\right) \frac{\partial F}{\partial y_1}\right] \bigg |_{x=0, y_1= H_1(0 ,t,y',{{\bar{y}}})} \end{aligned}$$

for $2\le j\le 2n+1$. Observe that the second term is a scalar multiple of $\Xi _1=\nabla _{(x,t)}H_1(0,t,y',{{\bar{y}}})$. Define

$$\begin{aligned}\Xi _j:=-\left[ \left( \frac{\partial F}{\partial y_1}\right) ^{-1}\left( \frac{\partial }{\partial y_j}+\frac{\partial H_1}{\partial y_j}\frac{\partial }{\partial y_1}\right) \nabla _{x,t}F\right] \bigg |_{x=0, y_1=H_1(0 ,t,y',{{\bar{y}}})} \end{aligned}$$

for $2\le j\le 2n+1$. Then it follows that the tangent space of $\Xi$ at (x, t, y) is spanned by $\Xi _1, \ldots , \Xi _{2n+1}$. We shall repeatedly use this kind of observation to simplify the expressions for tangent vectors and their derivatives.

Using the fact that

$$\begin{aligned} \frac{\partial H_1(0,t,y')}{\partial {y_j}}=-\frac{y_j}{H_1(0,t,y',{{\bar{y}}})},\,\,\frac{\partial H_1(0,t,y')}{\partial {{{\bar{y}}}}}=-\frac{{{\bar{y}}}}{2|{\underline{y}}|^2y_1}\bigg |_{y_1=H_1(0,t,y')}, \end{aligned}$$

we calculate

$$\begin{aligned} \Xi _1&=\nabla _{(x,t)}H_1(0,t,y',{{\bar{y}}})\\&=\left[ \frac{1}{4|{\underline{y}}|^2y_1}(4|{\underline{y}}|^2{\underline{y}}+{{\bar{y}}}(J{\underline{y}}+2e_{2n+1})-4t^3e_{2n+2})\right] \bigg |_{y_1=H_1(0 ,t,y',{{\bar{y}}})},\\ \Xi _j&=y_1\left[ \frac{1}{4|{\underline{y}}|^2y_1}\left( 8y_j{\underline{y}}+(4|{\underline{y}}|^2+{{\bar{y}}}J)e_j-\frac{y_j}{y_1}\left( 8y_1{\underline{y}}+(4|{\underline{y}}|^2+{{\bar{y}}}J)e_1\right) \right) \right] \bigg |_{y_1=H_1(0 ,t,y',{{\bar{y}}})},\\&\qquad 2\le j\le 2n,\\ \Xi _{2n+1}&=y_1\left[ \frac{1}{4|{\underline{y}}|^2y_1}\left( J{\underline{y}}+2e_{2n+1}-\frac{{{\bar{y}}}}{2|{\underline{y}}|^2y_1}\left( 8y_1{\underline{y}}+(4|{\underline{y}}|^2+{{\bar{y}}}J)e_1\right) \right) \right] \bigg |_{y_1=H_1(0 ,t,y',{{\bar{y}}})}. \end{aligned}$$

Henceforth, I shall denote the $2n\times 2n$ identity matrix $I_{2n}$. The above discussion implies that the mixed Hessian $\Phi _{(x,y)}''$ at $x=0$ and $y'=0$ has the same determinant as the matrix

$$\begin{aligned} y_1\left[ \frac{1}{4|{\underline{y}}|^2y_1}\begin{pmatrix} I_{y_1}\left( 4|{\underline{y}}|^2I-{{\bar{y}}}J\right) &{} &{}2{{\bar{y}}}e_1\\ &{} &{} \\ -\frac{{{\bar{y}}}}{2|{\underline{y}}|^2y_1}\left( 12|{\underline{y}}|^2e_1+{{\bar{y}}}Je_1\right) ^{\intercal }+(J{\underline{y}})^{\intercal }&{} &{} 2 \end{pmatrix}\right] \Bigg |_{y_1=H_1(0,t,0,{{\bar{y}}}), y'=0}.\nonumber \\ \end{aligned}$$

(8.1)

Here $I_{y_1}$ is the $2n\times 2n$ diagonal matrix with $y_1$ as the first diagonal entry and the rest being 1. Henceforth, we shall abbreviate $H_1(0,t,0,{{\bar{y}}})$ to $H_1$. The determinant of the above matrix is equal to

$$\begin{aligned} y_1^{2n+1}\left( \left[ \frac{1}{4|H_1|^2H_1}\right] ^{2n+1}\det \left( 4|H_1|^2I-{{\bar{y}}}J\right) (2-\gamma )\right) , \end{aligned}$$

with $\gamma :=\left[ \left( -\frac{{{\bar{y}}}}{2|{\underline{y}}|^2y_1}(12|{\underline{y}}|^2e_1+{{\bar{y}}}Je_1)+J{\underline{y}}\right) ^{\intercal }(4|{\underline{y}}|^2I-{{\bar{y}}}J)^{-1}\left( \frac{2{{\bar{y}}}}{y_1}e_1\right) \right] \Big |_{y_1=H_1,y'=0}$.

We use Lemma 6.2 to obtain $\left( 4|{\underline{y}}|^2I-{{\bar{y}}}J\right) ^{-1}=\frac{1}{16|{\underline{y}}|^4+|{{\bar{y}}}|^2}\left( 4|{\underline{y}}|^2I+{{\bar{y}}}J\right)$. Recalling that $|{\underline{y}}|=|H_1e_1|=|H_1|$, we calculate

$$\begin{aligned} \gamma&= \frac{2{{\bar{y}}}}{H_1(16|H_1|^4+|{{\bar{y}}}|^2)}\left( -\frac{1}{2|H_1|^2H_1}\left( 12|H_1|^2{{\bar{y}}}e_1+(|{{\bar{y}}}|^2-2|H_1|^4)Je_1\right) \right) ^{\intercal }\\&\quad \left( 4|H_1|^2e_1+{{\bar{y}}}Je_1\right) \\&=-\frac{{{\bar{y}}}^2}{H_1^4(16|H_1|^4+|{{\bar{y}}}|^2)}\left( 48|H_1|^4-2|H_1|^4+|{{\bar{y}}}|^2\right) \le 0. \end{aligned}$$

Thus the absolute value of the determinant of the matrix in (8.1) greater than or equal to

$$\begin{aligned} |y_1|^{2n+1}\det \left[ \frac{1}{2|H_1|^3}\left( 4|H_1|^2I-{{\bar{y}}}J\right) \right] . \end{aligned}$$

Recall that $1>t^{-1}|H_1|>1-2^{-400n}$ and $t^{-2}|{{\bar{y}}}|<2^{-400n}$. Hence we have $16|H_1|^4+|{{\bar{y}}}|^2\ge 1$. Using Corollary 6.3, with $\sigma =4H_1^2$, $\kappa ={{\bar{y}}}$, $\lambda =\gamma =0$, and the fact that $|y_1|\sim 1$, we conclude that

$$\begin{aligned}|y_1|^{2n+1}\det \left[ \frac{1}{2|H_1|^3}\left( 4|H_1|^2I-{{\bar{y}}}J\right) \right] \gtrsim 1,\end{aligned}$$

with the implicit constant being independent of $t\in [1,2]$. It follows that the same is true for $\Phi _{x,y}''(0,t, H_1,0)$ or in other words, the mixed Hessian of $\Phi$ is of rank $2n+1$. This finishes the proof of Proposition 3.4 around the equator.

8.1.2 Proof of Proposition 5.2 around the Equator

Let $N=(\underline{\alpha },{\bar{\alpha }},\alpha _{2n+2})$ be a unit normal vector such that $\langle N, \Xi _j\rangle =0$ for $1\le j\le 2n+1$. Let ${\mathscr {C}}(x,t,y)$ be the $(2n+1)\times (2n+1)$ cinematic curvature matrix with respect to N given by

$$\begin{aligned} {\mathscr {C}}_{ij} = \frac{\partial ^2}{\partial y_i\partial y_j} \langle N,\Xi \rangle \end{aligned}$$

(8.2)

We argue as in the proof of [20, Proposition 3.4] according to which Proposition 5.2 holds provided that ${\textrm{rank }}{\Phi _{(x,t),y}''}=2n+1$ and the additional curvature condition

$$\begin{aligned} \langle N, \Xi _{j} \rangle =0 ,\,\, j=1,\dots , 2n+1 \quad \implies \quad {\textrm{rank }}\, {\mathscr {C}} = 2n \end{aligned}$$

(8.3)

is satisfied; i.e. the conic surface $\Sigma _{x,t}$ parametrized by $y\mapsto \Xi (x,t,y)$ has the maximal number 2n of nonvanishing principal curvatures. It remains to verify (8.3).

Since ${\textrm{rank }}{{\mathscr {C}}(x,t,y)}={\textrm{rank }}{{\mathscr {C}}(0,t,\Theta (x,y))}$, we may again assume that $x=0$ (see Sect. 11.1). Further, when $y_i=0$ for $2\le i\le 2n$, the equations $\langle N, \Xi _j\rangle =0$ for $1\le j\le 2n$ can be expressed as

$$\begin{aligned} \left( 4|{\underline{y}}|^2I-{{\bar{y}}}J\right) \underline{\alpha }\big |_{y_1=H_1}= -\frac{1}{y_1}(2{{\bar{y}}}{\bar{\alpha }}-4t^3\alpha _{2n+2})e_1:=Te_1, \end{aligned}$$

Solving, we get

$$\begin{aligned} \underline{\alpha }=T\left( 4|{\underline{y}}|^2I-{{\bar{y}}}J\right) ^{-1}e_1\big |_{y_1=H_1}=\frac{T}{16|{\underline{y}}|^4+|{{\bar{y}}}|^2}\left( 4|{\underline{y}}|^2I+{{\bar{y}}}J\right) e_{1}\Big |_{y_1=H_1}.\nonumber \\ \end{aligned}$$

(8.4)

Remark 8.2

Equation (8.4) implies that $|\underline{\alpha }|\lesssim T$. Further, since $\left| \frac{{{\bar{y}}}}{2|{\underline{y}}|^2H_1}\right| \lesssim 2^{-400n}$, the equation $\langle N, \Xi _{2n+1}\rangle =0$ implies that $|{\bar{\alpha }}|\lesssim |\underline{\alpha }|$ and thus $|{\bar{\alpha }}|\lesssim T$ as well. Using the definition of T, we have that $T\gtrsim |\alpha _{2n+2}|-2^{-400n}|{\bar{\alpha }}|$ and therefore $|\alpha _{2n+2}|\lesssim T$. We conclude that $|N|\lesssim T$ and since N is a unit vector, it follows that $|T|\gtrsim 1$.

We now calculate the derivatives of the tangent vectors, using subscripts to denote the corresponding partial derivatives. Observe that $\Xi _{1j}$ is a scalar multiple of $\Xi _j$ for $2\le j\le 2n+1$, while $\Xi _{11}=0$. Thus $\langle \Xi _{1j},N\rangle =0$ for $1\le j\le 2n+1$.

Let $\delta _{ij}:=\delta _0(i-j)$ where $\delta _0$ denotes the Dirac measure at the origin in ${{\mathbb {R}}}$. Recall that $H_1=H_1(0,t,0,{{\bar{y}}})$ and thus $|{\underline{y}}|=|H_1|$ when $y'=0$. For $2\le i,j\le 2n+1$, we have

$$\begin{aligned} \langle \Xi _{ij}, N\rangle&=\delta _{ij}\frac{y_1}{4|{\underline{y}}|^2y_1}\left[ 8{\underline{y}}-\frac{1}{y_1}\left( 8y_1{\underline{y}}+(4|{\underline{y}}|^2I+{{\bar{y}}}J)e_1\right) \right] ^{\intercal }\bigg |_{y_1=H_1}\underline{\alpha }\\&=-\delta _{ij}\frac{y_1}{4|H_1|^2H_1^2}\left( 4|H_1|^2I+{{\bar{y}}}J)e_1\right) ^{\intercal }\underline{\alpha }. \end{aligned}$$

Here we have used that $y_j=0$ for $2\le j\le 2n$ and also the fact that the dot product of N with terms involving the first derivatives of $\nabla _{(x,t)}F$ is zero since they lie in the tangent space spanned by vectors $\Xi _1, \ldots , \Xi _{2n+1}$. Using (8.4) and the properties $J^2=-I_{2n}$ and $J^\intercal =-J$, we calculate

$$\begin{aligned} \langle \Xi _{ij}, N\rangle = -y_1\delta _{ij}\frac{1}{4|H_1|^2H_1^2}\frac{T}{16|H_1|^4+|{{\bar{y}}}|^2}({16|H_1|^4+|{{\bar{y}}}|^2}) =-y_1\delta _{ij}\frac{T}{4|H_1|^4}. \end{aligned}$$

For $i=2n+1$ and $2\le j\le 2n$, we have

$$\begin{aligned} \langle \Xi _{ij},N\rangle&=\frac{y_1}{4|H_1|^2H_1}\left[ Je_j-\frac{{{\bar{y}}}}{2|H_1|^2H_1}(8H_1e_j)\right] ^{\intercal }\underline{\alpha }\\&=\frac{y_1}{4|H_1|^3}\left[ \frac{4T}{|H_1|^2(16|H_1|^4+|{{\bar{y}}}|^2)}\left( |H_1|^4+|{{\bar{y}}}|^2\right) \right] e_1Je_j \end{aligned}$$

and

$$\begin{aligned} \langle \Xi _{ii},N\rangle&=\frac{y_1}{4|H_1|^2H_1}\left( -\frac{1}{2|H_1|^2H_1}\right) \left[ \frac{2|H_1|^4+{{\bar{y}}}^2}{|H_1|^2}\left( 6e_1+\frac{3{{\bar{y}}}}{2|H_1|^2}Je_1\right) \right] ^{\intercal }N\\&=\frac{y_1}{4|H_1|^2H_1}\left( -\frac{3T}{4|H_1|^4H_1}\right) (2|H_1|^4+{{\bar{y}}}^2). \end{aligned}$$

Let $P:{{\mathbb {R}}}^{2n}\rightarrow {{\mathbb {R}}}^{2n-1}$ be the projection onto the orthogonal complement of $e_1$. At the equator, when $x=0$, $y'=0$ and ${{\bar{y}}}=0$, we have $H_1=t$ and the cinematic curvature matrix with respect to N is given by

$$\begin{aligned} {\mathscr {C}}\Big |_{x=0,y_1=te_1,{{\bar{y}}}=0}=\frac{T}{4t^3}\begin{pmatrix} 0&{} &{}\quad 0&{} &{}\quad 0\\ 0&{} &{}\quad -I_{2n-1}&{} &{} \quad \frac{1}{4t}Pe_1Je_j\\ 0&{} &{}\quad \frac{1}{4t}(e_1Je_j)^{\intercal }P^\intercal &{} &{}\quad -\frac{3}{2} \end{pmatrix}, \end{aligned}$$

Since $|T|\gtrsim 1$, the rank of the above matrix is 2n, provided $\left| -\frac{3}{2}+\frac{1}{16t^2}|e_1J|^2\right| \ne 0$. This is true, for

$$\begin{aligned} \frac{1}{16t^2}|e_1J|^2=\frac{1}{16t^2}\le \frac{1}{16} \end{aligned}$$

when $t\in [1,2]$. This verifies (8.3), and finishes the proof of Proposition 5.2 at the equator.

8.2 Estimates in the intermediate region

Recall that the phase function of $T_k$ in the intermediate region is given by $\Phi =\Phi ^\text {IM}={{\bar{y}}}\bar{H}(x,t,{\underline{y}})$ with

$$\begin{aligned} \nabla _{(x,t)}\bar{H}(x,t,y',{{\bar{y}}})=-\left[ \left( \frac{\partial F}{\partial {{\bar{y}}}}\right) ^{-1}\nabla _{x,t}F\right] \bigg |_{(x,t,\bar{H}(x,t,{\underline{y}}), {\underline{y}})}, \end{aligned}$$

where F(x, t, y) is the defining function of the Korányi sphere of radius t. The amplitude $b=b_\text {IM}$ is supported in a small neighborhood of the set where x is small, $|{{\bar{y}}}|\in [1/2,2]$, $t^{-1}|x'-y'|<2^{-400n}$ and $\min \left( t^{-1}|x_1-y_1|, t^{-2}|{{\bar{x}}}-\bar{H}(x,t,{\underline{y}})+\tfrac{1}{2}{\underline{x}}^{\intercal }J{\underline{y}}|\right) >2^{-400n}$.

8.2.1 Proof of proposition 3.16 in the intermediate region

By Hörmander’s $L^2$ estimate, it again suffices to prove that the mixed Hessian $\Phi _{x,y}''$ is of rank $2n+1$ in the support of b, with the determinant bounded from below by an absolute constant independent of t.

Because of the fact that $\det \Phi _{x,y}''(x,t,y)=\det \Phi _{x,y}''(0,t, \Theta (x,y))$ (with $\Theta (x,y):=({\underline{x}}-{\underline{y}}, {{\bar{x}}}-{{\bar{y}}}+\tfrac{1}{2}{\underline{x}}^\intercal J {\underline{y}})$, see Sect. 11.1), it again suffices to verify the curvature condition under the assumptions that $x=0$, $|{{\bar{y}}}|\in [1/2,1]$, $t^{-1}|y'|<2^{-400n}$ and $\min \left( t^{-1}|y_1|, t^{-2}|\bar{H}(x,t,{\underline{y}})|\right) >2^{-400n}$.

Let

$$\begin{aligned} \Xi (x,t,y)=\nabla _{(x,t)}\Phi ^\text {Eq}(x,t,y). \end{aligned}$$

We calculate

$$\begin{aligned} \Xi (0,t,y)= & {} {{\bar{y}}}\nabla _{(x,t)}\bar{H}(0,t,y',{{\bar{y}}})\\= & {} {{\bar{y}}}\left[ \frac{1}{2{{\bar{y}}}}(4|{\underline{y}}|^2{\underline{y}}+{{\bar{y}}}J{\underline{y}}+2{{\bar{y}}}e_{2n+1})-4t^3e_{2n+2})\right] \bigg |_{{{\bar{y}}}=\bar{H}(0 ,t, {\underline{y}})}.\end{aligned}$$

The tangent vectors at $x=0$ are spanned by

$$\begin{aligned} \Xi _j&=\frac{{{\bar{y}}}}{2\bar{H}(0,t,{\underline{y}})}\left[ 8y_j{\underline{y}}+(4|{\underline{y}}|^2+\bar{H}(0,t,{\underline{y}})J)e_j-\frac{4|{\underline{y}}|^2y_j}{\bar{H}(0,t,{\underline{y}})}\left( \frac{1}{2}J{\underline{y}}+e_{2n+1}\right) \right] ,\\&\qquad 1\le j\le 2n,\\ \Xi _{2n+1}&=\nabla _{(x,t)}H(0,t,{\underline{y}}). \end{aligned}$$

Here we have used the same idea as in Remark 8.1 to get a simplified form for the tangent vectors. We abbreviate $\bar{H}(0,t,y_1e_1)=\bar{H}$. The mixed Hessian of $\Phi$ when $x=0$ and $y_j=0$ for $2\le j\le 2n$ has determinant equal to that of the matrix

$$\begin{aligned} \frac{{{\bar{y}}}}{\bar{H}}\begin{pmatrix} 4{\underline{y}}{\underline{y}}^{\intercal }+2|{\underline{y}}|^2I-\frac{\bar{H}}{2} J-\frac{|{\underline{y}}|^2}{\bar{H}}{\underline{y}}(J{\underline{y}})^{\intercal }&{} &{}-\frac{2|{\underline{y}}|^2}{\bar{H}} {\underline{y}}\\ &{} &{} \\ \frac{1}{\bar{H}}\left[ (2|{\underline{y}}|^2I+\frac{\bar{H}}{2} J){\underline{y}}\right] ^{\intercal }&{} &{} 1 \end{pmatrix}. \end{aligned}$$

Using Schur’s complement, the absolute value of the determinant of the above matrix is

$$\begin{aligned} \left| \frac{{{\bar{y}}}}{\bar{H}}\right| ^{2n+1}\det X \left[ 1-\frac{1}{\bar{H}}\left( 2|{\underline{y}}|^2I+\frac{\bar{H}}{2} J){\underline{y}}\right) ^{\intercal }X^{-1}\left( -\frac{2|{\underline{y}}|^2}{\bar{H}}{\underline{y}}\right) \right] , \end{aligned}$$

where

$$\begin{aligned} X=4{\underline{y}}{\underline{y}}^{\intercal }+2|{\underline{y}}|^2I-\frac{\bar{H}}{2} J-\frac{|{\underline{y}}|^2}{\bar{H}}{\underline{y}}(J{\underline{y}})^{\intercal }. \end{aligned}$$

Using Corollary 6.4 with $\sigma =2|{\underline{y}}|^2, \lambda =4, \kappa =-\frac{\bar{H}}{2}$ and $\gamma =-\frac{|{\underline{y}}|^2}{\bar{H}}$, we get $X^{-1}{\underline{y}}=\frac{4}{46|{\underline{y}}|^4+|\bar{H}|^2}(2|{\underline{y}}|^2I+\frac{\bar{H}}{2} J){\underline{y}}$. Thus

$$\begin{aligned} -\frac{1}{\bar{H}}\left( 2|{\underline{y}}|^2I+\frac{\bar{H}}{2} J){\underline{y}}\right) ^{\intercal }X^{-1}\left( -\frac{2|{\underline{y}}|^2}{\bar{H}}{\underline{y}}\right) =\frac{2|{\underline{y}}|^2}{\bar{H}^2}\frac{16|{\underline{y}}|^4+|\bar{H}|^2}{46|{\underline{y}}|^4+|\bar{H}|^2}|{\underline{y}}|^2\ge 0. \end{aligned}$$

Hence

$$\begin{aligned}\left| \det X \left[ 1-\frac{1}{\bar{H}}\left( 2|{\underline{y}}|^2I+\frac{\bar{H}}{2} J){\underline{y}}\right) ^{\intercal }X^{-1}\left( -\frac{2|{\underline{y}}|^2}{\bar{H}}{\underline{y}}\right) \right] \right| \ge |\det X|. \end{aligned}$$

Using Corollary 6.3 for X with $\sigma ^2+\kappa ^2=4|{\underline{y}}|^4+\frac{|\bar{H}|^2}{4}\ge \frac{t^4}{4}\ge \frac{1}{4}$ and

$$\begin{aligned} \sigma ^2+\kappa ^2+(\sigma \lambda -\gamma \kappa )|{\underline{y}}|^2= & {} 4|{\underline{y}}|^4+\frac{|\bar{H}|^2}{4} +\left( 8|{\underline{y}}|^2-\frac{|{\underline{y}}|^2}{2}\right) |{\underline{y}}|^2\\= & {} \frac{46|{\underline{y}}|^4+|\bar{H}|^2}{4}\ge \frac{1}{4},\end{aligned}$$

we conclude that $|\det X|\gtrsim _n 1$ uniformly in $t\in [1,2]$.

Since $\left| \frac{{{\bar{y}}}}{\bar{H}}\right| ^{2n+1}\gtrsim 1$ in the support of $b=b_\text {IM}$, it follows that $\det \Phi _{x,y}''\gtrsim _n 1$ uniformly in $t\in [1,2]$ as well. In other words, we have shown that the mixed Hessian has rank $2n+1$, which finishes the proof.

8.2.2 Proof of Proposition 5.2 in the intermediate region

Let $N=(\underline{\alpha },{\bar{\alpha }},\alpha _{2n+2})$ be a unit normal vector satisfying $\langle N, \Xi _j\rangle =0$ for $1\le j\le 2n+1$. Let ${\mathscr {C}}(x,t,y)$ be the $(2n+1)\times (2n+1)$ cinematic curvature matrix with respect to N. To prove Proposition 5.2 in the intermediate region, we need to verify the additional curvature condition (8.3) for ${\mathscr {C}}(x,t,y)$. We may again assume without loss of generality that $x=0$.

When $y_i=0$ for $2\le i\le 2n$, the equations $\langle N, \Xi _j\rangle =0$ for $1\le j\le 2n$ can be written down in the matrix form as

$$\begin{aligned} X\underline{\alpha }=\left( 4{\underline{y}}{\underline{y}}^{\intercal }+2|{\underline{y}}|^2I-\frac{\bar{H}}{2} J-\frac{|{\underline{y}}|^2}{\bar{H}}{\underline{y}}(J{\underline{y}})^{\intercal }\right) \underline{\alpha }= {\bar{\alpha }}\frac{2|{\underline{y}}|^2}{\bar{H}} {\underline{y}}. \end{aligned}$$

Thus

$$\begin{aligned} \underline{\alpha }={\bar{\alpha }}\frac{2|{\underline{y}}|^2}{\bar{H}} X^{-1}{\underline{y}}=\frac{2{\bar{\alpha }}|{\underline{y}}|^2}{\bar{H}(46|{\underline{y}}|^4+|\bar{H}|^2)}(8|{\underline{y}}|^2I+2\bar{H}J){\underline{y}}. \end{aligned}$$

(8.5)

Remark 8.3

Arguing as in Remark 8.2, using the equations above and the fact that $|N|=1$, it is not difficult to see that $|{\bar{\alpha }}|\gtrsim 1$.

We now calculate the derivatives of the tangent vectors, denoting them by subscripts. Observe that for $i=2n+1$, $\Xi _{ij}$ is a scalar multiple of $\Xi _j$ for $1\le j\le 2n$, while $\Xi _{ii}=0$. Thus $\langle \Xi _{ij},N\rangle =0$ for $1\le j\le 2n+1$. Further, for $2\le i,j\le 2n$, we have

$$\begin{aligned} \langle \Xi _{ij}, N\rangle =\frac{{{\bar{y}}}}{\bar{H}}\delta _{ij}\left[ 4{\underline{y}}-\frac{2|{\underline{y}}|^2}{\bar{H}}\left( \frac{1}{2}J{\underline{y}}+e_{2n+1}\right) \right] ^{\intercal }N. \end{aligned}$$

Here we use the fact that $y_j=0$ for $2\le j\le 2n$ and the dot product of N with terms involving the first derivatives of $\nabla _{(x,t)}F$ disappears since these terms lie in the space spanned by the tangent vectors. For $2\le i\le 2n$, using (8.5), we calculate

$$\begin{aligned} \langle \Xi _{ij}, N\rangle = \frac{{{\bar{y}}}}{\bar{H}}\delta _{ij}\frac{2{\bar{\alpha }}|{\underline{y}}|^2}{\bar{H}}\left( \frac{30|{\underline{y}}|^4}{46|{\underline{y}}|^4+|\bar{H}|^2}-1\right) =-\frac{{{\bar{y}}}}{\bar{H}}\delta _{ij}\frac{2{\bar{\alpha }}|{\underline{y}}|^2}{\bar{H}}\left( \frac{16|{\underline{y}}|^4+|\bar{H}|^2}{46|{\underline{y}}|^4+|\bar{H}|^2}\right) . \end{aligned}$$

For $i=1$ and $2\le j\le 2n$, using the fact that $\frac{\partial }{\partial y_1}\bar{H}(0,t,y_1e_1)=-\frac{2|{\underline{y}}|^2y_1}{\bar{H}}$, we obtain

$$\begin{aligned} \langle \Xi _{ij},N\rangle&=\frac{{{\bar{y}}}}{\bar{H}}\left[ 4y_1e_j-\frac{|{\underline{y}}|^2y_1}{\bar{H}}Je_j\right] ^{\intercal }N =-\frac{{{\bar{y}}}}{\bar{H}}\frac{16{\bar{\alpha }}|{\underline{y}}|^2}{|\bar{H}|^2}y_1{\underline{y}}Je_j\left( \frac{|\bar{H}|^2+|{\underline{y}}|^4}{46|{\underline{y}}|^4+|\bar{H}|^2}\right) \end{aligned}$$

and

$$\begin{aligned} \langle \Xi _{ii},N\rangle&=\frac{{{\bar{y}}}}{\bar{H}}\left[ 12{\underline{y}}-\frac{1}{2}\left( \frac{10|{\underline{y}}|^2}{\bar{H}}+\frac{4|{\underline{y}}|^6}{|\bar{H}|^3}\right) J{\underline{y}}-\frac{1}{\bar{H}}\left( 6|{\underline{y}}|^2+\frac{4|{\underline{y}}|^6}{|\bar{H}|^2}\right) e_{2n+1}\right] ^{\intercal }N\\&=\frac{2{\bar{\alpha }}{{\bar{y}}}|{\underline{y}}|^2}{\bar{H}^2}\left[ \frac{96|{\underline{y}}|^4}{{46|{\underline{y}}|^4+|\bar{H}|^2}}-\frac{|{\underline{y}}|^4}{|\bar{H}|^2}\left( \frac{10|\bar{H}|^2+4|{\underline{y}}|^4}{46|{\underline{y}}|^4+|\bar{H}|^2}\right) -\frac{3|\bar{H}|^2+2|{\underline{y}}|^4}{|\bar{H}|^2}\right] . \end{aligned}$$

Simplifying, we get

$$\begin{aligned}{} & {} \frac{96|{\underline{y}}|^4}{{46|{\underline{y}}|^4+|\bar{H}|^2}}-\frac{|{\underline{y}}|^4}{|\bar{H}|^2}\left( \frac{10|\bar{H}|^2+4|{\underline{y}}|^4}{46|{\underline{y}}|^4+|\bar{H}|^2}\right) -\frac{3|\bar{H}|^2+2|{\underline{y}}|^4}{|\bar{H}|^2}\\{} & {} \quad =\frac{12|{\underline{y}}|^4}{{46|{\underline{y}}|^4+|\bar{H}|^2}}\left( 7-8\frac{|{\underline{y}}|^4}{|\bar{H}|^2}\right) -3:=D. \end{aligned}$$

Thus the cinematic curvature matrix at $y=y_1e_1+\bar{H}e_{2n+1}$ is given by

$$\begin{aligned} \frac{2{\bar{\alpha }}{{\bar{y}}}|{\underline{y}}|^2}{\bar{H}^2}\begin{pmatrix} D&{} &{}\quad B^{\intercal }P^\intercal &{} &{}\quad 0\\ PB&{} &{}\quad -\lambda I_{2n-1}&{} &{}\quad 0\\ 0&{} &{}\quad 0 &{} &{}\quad 0 \end{pmatrix}, \end{aligned}$$

(8.6)

where $\lambda =\frac{16|{\underline{y}}|^4+|\bar{H}|^2}{46|{\underline{y}}|^4+|\bar{H}|^2}$, $B=-\frac{8y_1}{\bar{H}}\left( \frac{|\bar{H}|^2+|{\underline{y}}|^4}{46|{\underline{y}}|^4+|\bar{H}|^2}\right) {\underline{y}}^{\intercal }J$, $P:{{\mathbb {R}}}^{2n}\rightarrow {{\mathbb {R}}}^{2n-1}$ is the projection onto the orthogonal complement of $e_1$ and D is as defined above. Since $|{\underline{y}}|^4+|\bar{H}|^2=t^4$ with $t\in [1,2]$, we have that $\frac{1}{46}\le \lambda \le 1$. Further, $\min (|{\underline{y}}|, |{{\bar{y}}}|, |\bar{H}|)\ge 2^{-400n}$ and $|\bar{H}|\le 4$. Thus, the rank of the above matrix is 2n, provided $|D+\lambda ^{-1} B^{\intercal }B|$ is uniformly bounded from below by a positive constant. We have

$$\begin{aligned} |D+\lambda ^{-1}B^{\intercal }B|\ge & {} 3-\frac{12|{\underline{y}}|^4}{{46|{\underline{y}}|^4+|\bar{H}|^2}}\left( 7-8\frac{|{\underline{y}}|^4}{|\bar{H}|^2}\right) \\{} & {} -\frac{64|y_1|^2}{|\bar{H}|^2}\left( \frac{(|\bar{H}|^2+|{\underline{y}}|^4)^2}{(46|{\underline{y}}|^4+|\bar{H}|^2)(16|{\underline{y}}|^4+|\bar{H}|^2)}\right) |{\underline{y}}|^2. \end{aligned}$$

Now

$$\begin{aligned}&\frac{12|{\underline{y}}|^4}{{46|{\underline{y}}|^4+|\bar{H}|^2}}\left( 7-8\frac{|{\underline{y}}|^4}{|\bar{H}|^2}\right) +\frac{64|y_1|^2}{|\bar{H}|^2}\left( \frac{(|\bar{H}|^2+|{\underline{y}}|^4)^2}{(46|{\underline{y}}|^4+|\bar{H}|^2)(16|{\underline{y}}|^4+|\bar{H}|^2)}\right) |{\underline{y}}|^2\\&\quad \le \frac{12|{\underline{y}}|^4}{{46|{\underline{y}}|^4+|\bar{H}|^2}}\left( 7-8\frac{|{\underline{y}}|^4}{|\bar{H}|^2}\right) +64|{\underline{y}}|^4\left( \frac{1+\frac{|{\underline{y}}|^4}{|\bar{H}|^2}}{46|{\underline{y}}|^4+|\bar{H}|^2}\right) \\&\quad \le \frac{|{\underline{y}}|^4}{{46|{\underline{y}}|^4+|\bar{H}|^2}}\left( 148-\frac{32|{\underline{y}}|^4}{|\bar{H}|^2}\right) . \end{aligned}$$

We claim that

$$\begin{aligned} \frac{|{\underline{y}}|^4}{{46|{\underline{y}}|^4+|\bar{H}|^2}}\left( 148-\frac{32|{\underline{y}}|^4}{|\bar{H}|^2}\right) =\frac{\tfrac{|{\underline{y}}|^4}{|\bar{H}|^2}}{46\tfrac{|{\underline{y}}|^4}{|\bar{H}|^2}+1}\left( 148-\frac{32|{\underline{y}}|^4}{|\bar{H}|^2}\right) \le 2.81.\end{aligned}$$

This can be seen by optimizing the function

$$\begin{aligned}f(u)=\frac{u(148-32u)}{46u+1}\end{aligned}$$

on the interval $[0,\infty ].$ The equation $f'(u)=0$ has a unique solution in this interval, namely $u_0=\frac{3\sqrt{95}}{92}-\frac{1}{46}$. Further, $f''(u)=-\frac{13680}{(46u+1)^3}<0$ for $u\in [0,\infty )$ and thus $u_0$ is a point of maximum on the interval. Evaluating, we get

$$\begin{aligned}f(u_0)=\frac{2}{529}(859-12\sqrt{95})\sim 2.8054\end{aligned}$$

Thus $|D+\lambda ^{-1}B^{\intercal }B|\ge 3-2.81=0.19.$ We conclude that the rank of the cinematic curvature matrix is 2n, thus verifying condition (8.3) in the intermediate setting and finishing the proof of Proposition 5.2.

8.3 Proof of Proposition 3.4 at the Poles

We end this section by briefly describing how the calculation in Sect. 8.2.1 also yields the fixed time $L^2$ estimate for $T^k$ at the poles. In Sect. 8.2.1, we showed that the determinant of Hessian of the the phase function is bounded from below by a constant times the determinant of the matrix X given by

$$\begin{aligned}X=4{\underline{y}}{\underline{y}}^{\intercal }+2|{\underline{y}}|^2I-\frac{\bar{H}}{2} J-\frac{|{\underline{y}}|^2}{\bar{H}}{\underline{y}}(J{\underline{y}})^{\intercal }.\end{aligned}$$

Here $\bar{H}$ is defined uniquely such that $|{\underline{y}}|^4+|\bar{H}|^2=t^4$. At the poles, since $|{\underline{y}}|=0$ and $\bar{H}=t^2$, it follows that $X=-\frac{t^2}{2}J$ and

$$\begin{aligned}|\det X|=\frac{t^2}{2} |\det J| \ge \frac{1}{2}\end{aligned}$$

for $t\in [1,2]$ and J satisfying the condition $J^2=-I$. Thus the mixed Hessian of the phase function has rank $2n+1$ at the poles as well, which implies the fixed time estimate (3.16) for the corresponding $T^k$ via Hörmander’s $L^2$ estimate. Observe that this argument works directly without using the Taylor’s expansion (2.4) of the phase function.

Remark 8.4

We also note that arguing directly as above would not work when calculating the rank of the cinematic curvature matrix ${\mathscr {C}}$ around the poles. As is clear from (8.6), ${\mathscr {C}}$ has rank 0 when $|{\underline{y}}|=0$. Thus the scaling argument of Sect. 4 is required to understand the behaviour of the Kóranyi sphere around the poles.

9 Necessary conditions

We provide five counter-examples, corresponding to each edge of the quadrilateral ${\mathcal {R}}$ in Theorem 1.1 and one for the point $Q_2$, which show the necessity of all the conditions in Theorem 1.1. They are suitable modifications of the examples in [1, 25, 27] for the Euclidean case, which were in turn adapted from standard examples for spherical means and maximal functions. The first four counter-examples are similar to the corresponding ones in [26] for the maximal function associated to codimension two spheres on the Heisenberg groups. The fifth one is new and can be viewed as the replacement of a standard Knapp type example. It has the interesting feature that the dimensions of the set associated to the input test function are not equal to the dimensions of the set on which the output function (under the action of the operator) is large. This is quite unlike what is observed for the Euclidean Knapp examples.

We also briefly discuss the counter-examples which imply the necessary condition in Theorem 1.2 for the single average and which can be easily inferred from the counter-examples for the maximal operator.

Recall that for $x=({\underline{x}},{{\bar{x}}})\in {{\mathbb {H}}}^n$, the Korányi norm of x is defined to be

$$\begin{aligned} |x|_K:=\left( |{\underline{x}}|^4+|{{\bar{x}}}|^2\right) ^{\frac{1}{4}}. \end{aligned}$$

9.1 The line connecting $Q_1$ and $Q_2$

This is the necessary condition $p\le q$ imposed by translation invariance and noncompactness of the group ${{\mathbb {H}}}^n$ (see [12] for the analogous argument in the Euclidean case).

9.2 The line connecting $Q_2$ and $Q_3$

Let $B_{\delta }$ be the ball of radius $\delta$ centered at the origin. Let $f_{\delta }$ be the characteristic function of $B_{100\delta }$. Then

$$\begin{aligned}\Vert f_{\delta }\Vert _p\approx \delta ^{(2n+1)/p}.\end{aligned}$$

For $1\le t\le 2$ we consider the set

$$\begin{aligned} R{:=} \{x=(\underline{x}, {\bar{x}}){:}1\le |x|_K\le 2\}.\end{aligned}$$

Then $|R|{\gtrsim } 1$. Let $\Sigma _{x}{=} \{\omega =({\underline{\omega }},{{{\bar{\omega }}}}){:} |\omega |_K{=}1, \left| \left( \tfrac{{\underline{x}}}{|x|_K}, \tfrac{{{\bar{x}}}}{|x|_K^2}\right) -({\underline{\omega }},{{{\bar{\omega }}}})\right| \le \delta /4\}$, which has surface measure $\approx \delta ^{2n}$.

For $x\in R$ and $\omega \in \Sigma _{x}$, we set $t=|x|_K$. Then

$$\begin{aligned}|\underline{x}-t{\underline{\omega }}|\le t|t^{-1}{\underline{x}}-{\underline{\omega }}|\le \delta \end{aligned}$$

and

$$\begin{aligned}|{\bar{x}}-t^2{{{\bar{\omega }}}}- \tfrac{t}{2} \underline{x}^\intercal J{\underline{\omega }}| \le t^2|t^{-2}{{\bar{x}}}-{{{\bar{\omega }}}}|+\frac{1}{2}|{\underline{x}}^{\intercal }J({\underline{x}}-t{\underline{\omega }})| \le 3\delta \end{aligned}$$

(here we have used the skew symmetry of the J). We get

$$\begin{aligned}f_{\delta }*\mu _t(\underline{x},\bar{x})=\int _{|\omega |_K=1} f_{\delta }(\underline{x}-t{{\underline{\omega }}},\bar{x}-t^2{{{\bar{\omega }}}}-\tfrac{t}{2}\underline{x}^\intercal \! J{{\underline{\omega }}})\,d\mu ({\omega })\gtrsim \delta ^{2n}\end{aligned}$$

for $x\in R$ and $t=|x|_K$. Passing to the maximal operator yields the inequality

$$\begin{aligned}\delta ^{2n}\lesssim \delta ^{(2n+1)/p},\end{aligned}$$

and consequently, the necessary condition

$$\begin{aligned} 2n\ge \frac{2n+1}{p}, \end{aligned}$$

that is, (1/p, 1/q) lies on or above the line connecting $Q_2$ and $Q_3$.

9.3 The point $Q_2$

For $p=p_2:=\frac{2n+1}{2n}$ the $L^p\rightarrow L^p$ bound fails. Here one uses a modification of Stein’s example [31] for the Euclidean spherical maximal function. One considers the function $f_\alpha$ defined by $f_\alpha (\underline{v}, {\overline{v}})= |v|^{-\frac{2n+1}{p_2}} |\log |v||^{-\alpha }$ for $|\underline{v}|\le 1/2$, $|\overline{(}v) |\le 1$ which belongs to $L^{p_2}$ for $\alpha >1/p_2$. One finds that if $t(x)=|x|_K$ then for $\alpha <1$ the integrals $f*\sigma _{t(x)}(x)$ are $\infty$ on a set of positive measure.

9.4 The line connecting $Q_1$ and $Q_4$

For this line we just use the counterexample for the individual averaging operator, using it to bound the maximal function from below. Given $t\in [1,2]$, let $g_{\delta ,t}$ be the characteristic function of the set $\{y=(\underline{y},\bar{y}): | |y|_K-t|\le 100\delta \}$. Thus $\Vert g_{\delta ,t}\Vert _p\sim \delta ^{1/p}.$

Let $x=(\underline{x},\bar{x})$ be such that $|\underline{x}|\le \delta$ and $|{\bar{x}}| \le \delta$. For any $\omega =({\underline{\omega }},{{{\bar{\omega }}}})$ with $|\omega |_K=1$, we have $t|\underline{x}^\intercal J{\omega }|\lesssim 2\delta$. Furthermore,

$$\begin{aligned}&|({\underline{x}}-t{\underline{\omega }}, {{\bar{x}}}-t^2{{{\bar{\omega }}}}-\tfrac{t}{2}{\underline{x}}^{\intercal }J{\underline{\omega }}|_K^4-t^4\\&\quad =|{\underline{x}}-t{\underline{\omega }}|^4+|{{\bar{x}}}-t^2{{{\bar{\omega }}}}-\tfrac{t}{2}{\underline{x}}^{\intercal }J{\underline{\omega }}|^2-t^4\\&\quad \le |{\underline{x}}|^4+4t^2|{\underline{x}}|^2|{\underline{\omega }}|^2+2t^2|{\underline{x}}|^2|{\underline{\omega }}|^2+2t|{\underline{x}}|^3|{\underline{\omega }}|+2t^3|{\underline{x}}||{\underline{\omega }}|^3+t^4|{\underline{\omega }}|^4\\&\qquad +|{{\bar{x}}}|^2+t^4|{{{\bar{\omega }}}}|^2+\tfrac{t^2}{4}|\underline{x}^\intercal J{\omega }|^2+2t^2|{{\bar{x}}}||{{{\bar{\omega }}}}|+t^3|{{{\bar{\omega }}}}||\underline{x}^\intercal J{\omega }|+t|{{\bar{x}}}||{\underline{x}}^\intercal J{\underline{\omega }}|-t^4\\&\quad \lesssim \delta +t^4|\omega |_K^4-t^4=\delta . \end{aligned}$$

Thus

$$\begin{aligned}\left| |({\underline{x}}-t{\underline{\omega }}, {{\bar{x}}}-t^2{{{\bar{\omega }}}}-\tfrac{t}{2}{\underline{x}}^{\intercal }J{\underline{\omega }}|_K-t\right| \lesssim \frac{\delta }{t^3}\lesssim \delta , \end{aligned}$$

implying that $|g_{\delta ,t}*\sigma _t(x)|\gtrsim 1$. This yields the inequality $\delta ^{(2n+1)/q}\le \delta ^{1/p}$ which leads to the necessary condition

$$\begin{aligned} \frac{1}{q} \ge \frac{1}{2n+1} \frac{1}{p}, \end{aligned}$$

that is, (1/p, 1/q) lies on or above the line connecting $Q_1$ and $Q_4$.

9.5 The line connecting $Q_3$ and $Q_4$

Let $f_{\delta }$ be the characteristic function of the set

$$\begin{aligned}\{y=({\underline{y}},{{\bar{y}}}):|{\underline{y}}|\lesssim \delta ^{1/4}, |{{\bar{y}}}|\lesssim \delta \}.\end{aligned}$$

Then $\Vert f_{\delta }\Vert _p\sim \delta ^{\frac{2n}{4}+1}=\delta ^{\frac{n}{2}+1}$.

For $t\in [1,2]$, let $R_{\delta ,t}$ denote the set

$$\begin{aligned}R_{\delta ,t}=\{x=({\underline{x}},{{\bar{x}}}):|{\underline{x}}|\lesssim \delta ^{\frac{3}{4}}, |{{\bar{x}}}-t^2|\lesssim \delta \}.\end{aligned}$$

Then $|R_{\delta ,t}|\gtrsim \delta ^{\frac{3n}{2}+1}$. Finally, let $\Sigma _{\delta ,t}$ denote the set

$$\begin{aligned}\Sigma _{\delta ,t}:=\{\omega =({\underline{\omega }},{{{\bar{\omega }}}}):|\omega |_K=1, |{\underline{\omega }}|\lesssim \delta ^{\frac{1}{4}}, {{{\bar{\omega }}}}\ge 0\}.\end{aligned}$$

The above condition on $|{\underline{\omega }}|$ implies that $1-{{{\bar{\omega }}}}\lesssim \delta$, for

$$\begin{aligned}1-{{{\bar{\omega }}}}=\frac{1-|{{{\bar{\omega }}}}|^2}{1+{{{\bar{\omega }}}}}\le 1-|{{{\bar{\omega }}}}|^2=|{\underline{\omega }}|^4\lesssim \delta .\end{aligned}$$

Further $|\Sigma _{\delta ,t}|\sim \delta ^{\frac{2n}{4}}=\delta ^{\frac{n}{2}}$ and we have

$$\begin{aligned} |{\underline{x}}-t{\underline{\omega }}|&\lesssim |{\underline{x}}|+t|{\underline{\omega }}|\lesssim \delta ^{\frac{3}{4}}+\delta ^{\frac{1}{4}}\lesssim \delta ^{\frac{1}{4}},\\ |{{\bar{x}}}-t^2{{{\bar{\omega }}}}-\tfrac{t}{2}{\underline{x}}^{\intercal }J{\underline{\omega }}|&\lesssim |{{\bar{x}}}-t^2|+t^2|(1-{{{\bar{\omega }}}})|+\frac{t}{2}|{\underline{x}}^{\intercal }J{\underline{\omega }}|\lesssim \delta +t^2\delta +\frac{t}{2}\delta ^{\frac{3}{4}}\delta ^{\frac{1}{4}}\lesssim \delta . \end{aligned}$$

As a consequence

$$\begin{aligned}f_{\delta }*\mu _t(\underline{x},\bar{x})=\int _{|\omega |_K=1} f_{\delta }(\underline{x}-t{\omega },\bar{x}-t^2{{{\bar{\omega }}}}-\tfrac{t}{2}\underline{x}^\intercal \! J{\omega })\,d\mu ({\omega })\gtrsim \delta ^{\frac{n}{2}}\end{aligned}$$

for $x\in R_{\delta ,t}$. Passing to the maximal operator yields the inequality $\delta ^{\frac{n}{2}}\delta ^{(\frac{3n}{2}+1-1)\frac{1}{q}}\lesssim \delta ^{(\frac{n}{2}+1)\frac{1}{p}}$ which leads to the necessary condition

$$\begin{aligned} n+\frac{3n}{q}\ge \frac{n+2}{p}, \end{aligned}$$

that is, (1/p, 1/q) lies on or above the line connecting $Q_3$ and $Q_4$.

9.6 Necessary conditions for the averaging operator

The necessary condition $p\ge q$ from Sect. 9.1 corresponds to the line joining (0, 0) and (1, 1), while Sect. 9.4 implies that (1/p, 1/q) lies on or above the line connecting (0, 0) and $(\frac{2n+1}{2n+2}, \frac{1}{2n+2})$.

For the line connecting $(\frac{2n+1}{2n+2}, \frac{1}{2n+2})$ and (1, 1), we use the same example as in Sect. 9.2 but for a single average. More precisely, we set $t=1$ and test the output not on R but on the set

$$\begin{aligned} R_\delta := \{x=(\underline{x}, {\bar{x}}) :||x|_K-1|\le \delta \} \end{aligned}$$

with $|R_\delta |\sim \delta$. As in Sect. 9.2, let $\Sigma _{x}= \Bigg \{\omega =({\underline{\omega }},{{{\bar{\omega }}}}): |\omega |_K=1, \left| \left( \frac{{\underline{x}}}{|x|_K}, \frac{{{\bar{x}}}}{|x|_K^2}\right) \right. \left. -({\underline{\omega }},{{{\bar{\omega }}}})\right| \le \delta /4\Bigg \}$ with surface measure $\approx \delta ^{2n}$.

For $x\in R_\delta$ and $\omega \in \Sigma _{x}$, we have

$$\begin{aligned} |\underline{x}-{\underline{\omega }}|\le |x|_K\left| \frac{{\underline{x}}}{|x|_K}-{\underline{\omega }}\right| +|{\underline{\omega }}|\left| |x|_K-1\right| \le 2\delta \end{aligned}$$

and

$$\begin{aligned} |{\bar{x}}-{{{\bar{\omega }}}}- \tfrac{1}{2}\underline{x}^\intercal J{\underline{\omega }}| \le |x|_K^2\left| \frac{{{\bar{x}}}}{|x|_K^2}-{{{\bar{\omega }}}}\right| +|{{{\bar{\omega }}}}|\left| |x|_K^2-1\right| +\frac{1}{2}|{\underline{x}}^{\intercal }J({\underline{x}}-t{\underline{\omega }})| \le 4\delta . \end{aligned}$$

Here we have used the skew symmetry of the J. We get

$$\begin{aligned}f_{\delta }*\mu (\underline{x},\bar{x})=\int _{|\omega |_K=1} f_{\delta }(\underline{x}-{\omega },\bar{x}-{{{\bar{\omega }}}}-\tfrac{1}{2}\underline{x}^\intercal J{\omega })\,d\mu ({\omega })\gtrsim \delta ^{2n} \end{aligned}$$

for $x\in R_\delta$. This yields the inequality

$$\begin{aligned}\delta ^{2n}\delta ^{1/q}\lesssim \delta ^{(2n+1)/p},\end{aligned}$$

and consequently, the necessary condition

$$\begin{aligned} 2n+\frac{1}{q}\ge \frac{2n+1}{p}, \end{aligned}$$

(9.1)

that is, (1/p, 1/q) lies on or above the line connecting $(\frac{2n+1}{2n+2}, \frac{1}{2n+2})$ and (1, 1).

10 Sparse bounds and weighted inequalities

As mentioned in the introduction, the principal goal of [11] was to derive for the lacunary maximal operator ${{\mathfrak {M}}}^{\text {lac}}:=\sup _{k\ge 0}|{\mathcal {A}}_{2^k}(x)|$ an inequality of the form

$$\begin{aligned} \int _{{{\mathbb {H}}}^n} {{\mathfrak {M}}}^{\text {lac}} f(x) g(x) dx \le C\sup \big \{ \Lambda _{{{\mathcal {S}}}, p_1,p_2} (f,g) :\,{{{\mathcal {S}}}\, \textrm{sparse}} \big \}, \end{aligned}$$

(10.1)

where the supremum is taken over sparse families of nonisotropic Heisenberg cubes (see [11] for precise definitions and constructions) and the sparse form $\Lambda _{{{\mathcal {S}}},p_1,p_2}$ is given by

$$\begin{aligned} \Lambda _{{{\mathcal {S}}},p_1,p_2} (f,g) = \sum _{S\in {{\mathcal {S}}}} |S| \left( \frac{1}{|S|} \int |f|^{p_1}\right) ^{1/p_1} \left( \frac{1}{|S|} \int _S |g|^{p_2} \right) ^{1/p_2}. \end{aligned}$$

By using the sharp $L^p\rightarrow L^q$ bounds in Theorem 1.2 and by appealing to the arguments in [2, 11], we can prove the following theorem.

Theorem 10.1

The sparse bound (10.1) holds if $(1/p_1, 1-1/p_2)$ lies in the interior of the triangle with vertices (0, 0), (1, 1) and $(\tfrac{2n+1}{2n+2}, \frac{1}{2n+2})$.

The above result is sharp up to the boundary and improves upon the main result in [11].

Similarly, by applying the reasoning in [2, 11] and using the $L^p\rightarrow L^q$ bounds in Theorem 1.1, we can prove the following result which is new and also sharp up to the boundary.

Theorem 10.2

Let ${\mathfrak {M}}f(x):=\sup _{t>0}|{\mathcal {A}}_t f(x)|$ be the Kóranyi global maximal function. The sparse bound

$$\begin{aligned} \int _{{{\mathbb {H}}}^n} {{\mathfrak {M}}}f(x) g(x) dx \le C\sup \big \{ \Lambda _{{{\mathcal {S}}}, p_1,p_2} (f,g) :\,{{{\mathcal {S}}}\, \textrm{sparse}} \big \} \end{aligned}$$

holds whenever $(1/p_1, 1-1/p_2)$ lies in the interior of the quadrilateral ${\mathcal {R}}$ in (1.2) (or on the open line segment $Q_1Q_2$).

The proof of sparse bounds for the global maximal operator by using $L^p\rightarrow L^q$ estimates for localized maximal functions was pioneered by Lacey [16] in his work on the Euclidean spherical maximal function. The recent papers [4] and [8] give very general results about this correspondence for Euclidean spaces and spaces of homogeneous type, respectively.

10.1 Weighted inequalities

The connection of sparse forms to weighted inequalities is also well known. We briefly describe some terminology, followed by new results which can be deduced from our estimates on the aforementioned sparse forms on Heisenberg groups (see [16, §6–§7] for an expanded discussion in the Euclidean setting).

For us, a weight w is a non-negative, locally integrable function on ${{\mathbb {H}}}^n$.

Definition 10.3

For $p\in (1,\infty )$, the Muckhenhoupt class of weights $A_p$ on ${{\mathbb {H}}}^n$ is the set of all weights w satisfying the condition

$$\begin{aligned}_{A_p}:=\sup _Q \left( \frac{1}{|Q|} \int w\right) \left( \frac{1}{|Q|} \int w^{1-p'}\right) ^{p-1}<\infty ,\end{aligned}$$

where the supremum is taken over all (non-isotropic) Heisenberg cubes.

Definition 10.4

For $p\in (1,\infty )$, the Reverse-Hölder class of weights $RH_p$ on ${{\mathbb {H}}}^n$ is the set of all weights w satisfying the condition

$$\begin{aligned}_{RH_p}:=\sup _Q \left( \frac{1}{|Q|} \int w\right) ^{-1}\left( \frac{1}{|Q|} \int w^{p}\right) ^{\frac{1}{p}}<\infty ,\end{aligned}$$

where the supremum is taken over all (non-isotropic) Heisenberg cubes.

The following theorem, proven in [5, §6], gives a weighted estimate for sparse forms of the form considered above.

Theorem 10.5

Let $1\le p_1<p_2'\le \infty$, $p\in (p_1, p_2')$ and $\alpha =\max \Big \{\frac{1}{p-1}, \frac{p_2'-1}{p_2'-p}\Big \}$. Then

$$\begin{aligned}\Lambda _{{{\mathcal {S}}}, p_1, p_2}(f,g)\le \{[w]_{A_{p/p_1}}\cdot [w]_{RH_{(p_2'/p)'}}\}^{\alpha }\Vert f\Vert _{L^p(w)}\Vert g\Vert _{L^{p'}(w^{1-p'})}.\end{aligned}$$

We can use the aforementioned estimate to deduce the following corollaries about the weighted boundedness properties of the operators ${{\mathfrak {M}}}^{\text {lac}}$ and ${{\mathfrak {M}}}$, the first of which is an improvement on Theorem 5.8 in [11].

Corollary 10.6

• (i) Let $n\ge 1$ and let $\phi _{\text {lac}}$ be a real valued function such that $\tfrac{1}{\phi _{\text {lac}}}$ is a piecewise linear function on [0, 1] whose graph connects the points (0, 1), (1, 0) and $(\tfrac{2n+1}{2n+2}, \frac{2n+1}{2n+2})$. Then ${{\mathfrak {M}}}^{\text {lac}}$ is bounded on $L^p(w)$ for all $w\in A_{p/p_1}\cap RH_{(\phi _{\text {lac}}(1/p_1)'/p)'}$, where $1<p_1<p<\phi _{\text {lac}}(1/p_1)'$.

• (ii) Let $n\ge 2$ and let $\phi _{\text {full}}$ be a real valued function such that $\tfrac{1}{\phi _{\text {full}}}$ is a piecewise linear function on $[0, \tfrac{2n}{2n+1}]$ whose graph connects the points (0, 1), $(\tfrac{2n}{2n+1}, \frac{2n}{2n+1})$ and $(\tfrac{n(2n+1)}{2n^2+2n+2}, \tfrac{2n^2+n+2}{2n^2+2n+2})$. Then ${{\mathfrak {M}}}$ is bounded on $L^p(w)$ for all $w\in A_{p/p_1}\cap RH_{(\phi _{\text {lac}}(1/p_1)'/p)'}$, where $\frac{2n+1}{2n}<p_1<p<\phi _{\text {full}}(1/p_1)'$.

Appendix

1.1 Translation invariance

Let $\Theta :{{\mathbb {R}}}^{2n+1}\times {{\mathbb {R}}}^{2n+1}\rightarrow {{\mathbb {R}}}^{2n+1}$ denote the Heisenberg group translation map

$$\begin{aligned} \Theta (x,y):= \left( {\underline{x}}-{\underline{y}},{{\bar{x}}}-{{\bar{y}}}+\tfrac{1}{2}{\underline{x}}^{\intercal }J{\underline{y}}\right) . \end{aligned}$$

Let $\Phi$ be a smooth real valued function on ${{\mathbb {H}}}^n\times [1,2]$. For our purpose, we shall think of $\Phi$ as a function from ${{\mathbb {R}}}^{2n+1}\times [1,2]\rightarrow {{\mathbb {R}}}$. Let $\tilde{\Phi }:{{\mathbb {R}}}^{2n+1}\times [1,2]\times {{\mathbb {R}}}^{2n+1}\rightarrow {{\mathbb {R}}}$ be defined as

$$\begin{aligned} \tilde{\Phi }(x,t,y):=\Phi (\Theta (x, y), t)=\Phi ({\underline{x}}-{\underline{y}},{{\bar{x}}}-{{\bar{y}}}+\tfrac{1}{2}{\underline{x}}^{\intercal }J{\underline{y}},t). \end{aligned}$$

(11.1)

We prove the following lemma which says that the rank of the curvature matrices associated to the oscillatory integral operator corresponding to the phase function $\tilde{\Phi }$ is invariant under group translation.

Lemma 11.1

Let $\tilde{\Phi }$ be as defined above and let $t\in [1,2]$.

• (i)
  $$\begin{aligned}{\textrm{rank }}\tilde{\Phi }''_{x,y}(x, t, y)={\textrm{rank }}\tilde{\Phi }''_{x,y}(0,t,\Theta (x,y)). \end{aligned}$$

• (ii) Let ${\mathscr {C}}$ be the cinematic curvature matrix for $\tilde{\Phi }$ as defined in (8.2) with respect to a unit normal vector N. Then

  $$\begin{aligned} {\textrm{rank }}{\mathscr {C}}(x,t,y)={\textrm{rank }}{\mathscr {C}}(0,t,\Theta (x,y)). \end{aligned}$$

Proof

Differentiating (11.1) and using the chain rule, we have

$$\begin{aligned} \nabla _{(x,t)}\tilde{\Phi }(x,t,y)= & {} \nabla \Phi \big |_{(\Theta (x,y),t)}\cdot \begin{pmatrix} \Theta _{(x,t)}'&{} &{}\quad 0\\ 0&{} &{}\quad 1 \end{pmatrix}\\= & {} \nabla \Phi \big |_{(\Theta (x,y),t)}\cdot \begin{pmatrix} I_{2n}&{} &{}\quad 0&{} &{}\quad 0\\ \tfrac{1}{2}(J{\underline{y}})^{\intercal }&{} &{}\quad 1&{} &{}\quad 0\\ 0 &{} &{} \quad 0 &{} &{}\quad 1 \end{pmatrix}.\end{aligned}$$

Let $d=2n+1$. Expanding the matrix product on the right gives

$$\begin{aligned} \nabla _{(x,t)}\tilde{\Phi }(x,t,y)=\left( \nabla _{2n}\Phi +\tfrac{1}{2}\Phi '_{d}J{\underline{y}},\Phi '_{d},\Phi '_{t}\right) \big |_{(\Theta (x,y),t)}. \end{aligned}$$

Here $\nabla _{2n}\Phi$ denotes the partial derivative of $\Phi$ with respect to the first 2n coordinates, while $\Phi '_{d}, \Phi '_{t}$ denote its derivatives with respect to the $2n+1$-th and the last (time) coordinates respectively.

Another application of chain rule yields

$$\begin{aligned}{} & {} \tilde{\Phi }''_{(x,t),y}(x,t,y)\nonumber \\{} & {} \quad =-\begin{pmatrix} D_{2n}^2\Phi +\tfrac{1}{2}\nabla _{2n}\Phi '_{d}(J{\underline{y}})^{\intercal }+\tfrac{1}{2}\Phi _{d}'J&{} &{}\quad \nabla _{2n}\Phi '_{d}+\tfrac{1}{2}\Phi ''_{dd}J{\underline{y}}\\ \nabla _{2n}\Phi '_{d}&{} &{}\quad \Phi ''_{dd}\\ \nabla _{2n}\Phi '_{t}&{} &{}\quad \Phi ''_{tt} \end{pmatrix}\Bigg |_{(\Theta (x,y),t)}\begin{pmatrix} I_{2n}&{} &{}\quad 0\\ \tfrac{1}{2}(J{\underline{x}})^{\intercal }&{} &{}\quad 1 \end{pmatrix}.\nonumber \\ \end{aligned}$$

(11.2)

Here $D_{2n}^2$ denotes the Hessian of $\Phi$ with respect to the first 2n coordinates. In particular, for a fixed t, the mixed Hessian of $\tilde{\Phi }$ at (x, y) is given by

$$\begin{aligned}{} & {} \tilde{\Phi }''_{x,y}(x, t, y)\\{} & {} \quad =-\begin{pmatrix} D_{2n}^2\Phi +\tfrac{1}{2}\nabla _{2n}\Phi '_{d}(J{\underline{y}})^{\intercal }+\tfrac{1}{2}\Phi _{d}'J&{} &{}\quad \nabla _{2n}\Phi '_{d}+\tfrac{1}{2}\Phi ''_{dd}J{\underline{y}}\\ \nabla _{2n}\Phi '_{d}&{} &{}\quad \Phi ''_{dd} \end{pmatrix}\bigg |_{(\Theta (x,y),t)}\begin{pmatrix} I_{2n}&{} &{}\quad 0\\ \tfrac{1}{2}(J{\underline{x}})^{\intercal }&{} &{}\quad 1 \end{pmatrix}. \end{aligned}$$

Since the second matrix in the product on the right hand side has determinant equal to one, the rank of $\tilde{\Phi }''_{x,y}(x,t,y)$ is the same as the rank of

$$\begin{aligned}&-\begin{pmatrix} D_{2n}^2\Phi +\tfrac{1}{2}\nabla _{2n}\Phi '_{d}(J{\underline{y}})^{\intercal }+\tfrac{1}{2}\Phi _{d}'J&{} &{}\quad \nabla _{2n}\Phi '_{d}+\tfrac{1}{2}\Phi ''_{dd}J{\underline{y}}\\ \nabla _{2n}\Phi '_{d}&{} &{}\quad \Phi ''_{dd} \end{pmatrix}\bigg |_{(\Theta (x,y),t)}\\&\quad =-\begin{pmatrix} D_{2n}^2\Phi +\tfrac{1}{2}\nabla _{2n}\Phi '_{d}(J{\underline{y}})^{\intercal }+\tfrac{1}{2}\Phi _{d}'J&{} &{}\quad \nabla _{2n}\Phi '_{d}+\tfrac{1}{2}\Phi ''_{dd}J{\underline{y}}\\ \nabla _{2n}\Phi '_{d}&{} &{}\quad \Phi ''_{dd} \end{pmatrix}\bigg |_{(\Theta (x,y),t)}\begin{pmatrix} I_{2n}&{} &{}\quad 0\\ 0&{} &{}\quad 1 \end{pmatrix}\\&\quad =\tilde{\Phi }''_{x,y}(0,t,\Theta (x,y)). \end{aligned}$$

This proves the first part. For the second part, we consider the surface

$$\begin{aligned} \Sigma _{(x,t)}:=\nabla _{(x,t)}\tilde{\Phi }(x,t,y). \end{aligned}$$

The tangent space of $\Sigma _{(x,t)}$ at a point y is spanned by the column vectors of the matrix $\tilde{\Phi }''_{(x,t),y}(x,t,y)$. Observe that (11.2) implies that the column vectors of the matrix

$$\begin{aligned} \begin{pmatrix} D_{2n}^2\Phi +\tfrac{1}{2}\nabla _{2n}\Phi '_{d}(J{\underline{y}})^{\intercal }+\tfrac{1}{2}\Phi _{2n+1}'J&{} &{}\quad \nabla _{2n}\Phi '_{d}+\tfrac{1}{2}\Phi ''_{dd}J{\underline{y}}\\ \nabla _{2n}\Phi '_{d}&{} &{}\quad \Phi ''_{dd}\\ \nabla _{2n}\Phi '_{t}&{} &{}\quad \Phi ''_{tt} \end{pmatrix}\Bigg |_{(\Theta (x,y),t)}. \end{aligned}$$

also form a basis for the tangent space to $\Sigma _{(x,t)}$ at y. For $1\le j\le 2n+1$, let $\Xi _j$ denote the j-th column vector of the matrix above, and let $N\in {{\mathbb {R}}}^{2n+2}$ be a unit vector normal to $\Sigma _{(x,t)}$, so that $\langle N, \Xi _j\rangle =0$ for $1\le j\le 2n+1$.

Recall that the cinematic curvature matrix ${\mathscr {C}}(x,t,y)$ with respect to N is a $(2n+1)\times (2n+1)$ dimensional matrix with the (i, j)-th entry given by $\left\langle N, \frac{\partial }{\partial y_i}\Xi _j(\Theta (x,y),t)\right\rangle$. We use subscript i with $i\in \{1, \ldots , 2n+1\}$ to denote partial derivative in the direction of $e_i$. By the chain rule, the i-th row of ${\mathscr {C}}(x,t,y)$ is given by

$$\begin{aligned} N^{\intercal }\cdot \begin{pmatrix} {\mathscr {A}}&{} &{}\quad {\mathscr {B}}\\ \nabla _{2n}\Phi ''_{di}&{} &{}\quad \Phi '''_{ddi}\\ \nabla _{2n}\Phi ''_{ti}&{} &{}\quad \Phi '''_{tti} \end{pmatrix}\Bigg |_{(\Theta (x,y),t)} \begin{pmatrix} I_{2n}&{} &{}\quad 0\\ \tfrac{1}{2}(J{\underline{x}})^{\intercal }&{} &{}\quad 1 \end{pmatrix}, \end{aligned}$$

with ${\mathscr {A}}:=D_{2n}^2\Phi '_i+\tfrac{1}{2}\nabla _{2n}\Phi ''_{di}(J{\underline{y}})^{\intercal }+\tfrac{1}{2}\nabla _{2n}\Phi '_{d}(Je_i)^{\intercal }+\tfrac{1}{2}\Phi _{di}''J$ and ${\mathscr {B}}:=\nabla _{2n}\Phi ''_{di}+\tfrac{1}{2}\Phi '''_{ddi}J{\underline{y}}+\tfrac{1}{2}\Phi ''_{dd}Je_i$.

Observe that

$$\begin{aligned} N^{\intercal }\cdot \begin{pmatrix} {\mathscr {A}}&{} &{}\quad {\mathscr {B}}\\ \nabla _{2n}\Phi ''_{di}&{} &{}\quad \Phi '''_{ddi}\\ \nabla _{2n}\Phi ''_{ti}&{} &{}\quad \Phi '''_{tti} \end{pmatrix}\Bigg |_{(\Theta (x,y),t)} = N^{\intercal }\cdot \begin{pmatrix} {\mathscr {A}}&{} &{}\quad {\mathscr {B}}\\ \nabla _{2n}\Phi ''_{di}&{} &{}\quad \Phi '''_{ddi}\\ \nabla _{2n}\Phi ''_{ti}&{} &{}\quad \Phi '''_{tti} \end{pmatrix}\Bigg |_{(\Theta (x,y),t)}\begin{pmatrix} I_{2n}&{} &{}\quad 0\\ 0&{} &{}\quad 1 \end{pmatrix}, \end{aligned}$$

which is the i-th row of the matrix ${\mathscr {C}}(0,t,\Theta (x,y))$. Thus we have

$$\begin{aligned}{\mathscr {C}}(x,t,y)={\mathscr {C}}(0,t,\Theta (x,y))\begin{pmatrix} I_{2n}&{} &{}\quad 0\\ \tfrac{1}{2}(J{\underline{x}})^{\intercal }&{} &{}\quad 1 \end{pmatrix}.\end{aligned}$$

This clearly implies that the rank of the cinematic curvature matrix for $\tilde{\Phi }$ at (x, t, y) is equal to the cinematic curvature matrix rank at $(0,t,\Theta (x,y)).$ $\square$

1.2 Bourgain’s interpolation argument

We derive an interpolation lemma using a trick introduced by Bourgain [6] in his proof of an end-point bound for the Euclidean spherical maximal function. This type of interpolation argument occurs quite frequently in several works on maximal functions and we give a proof in our setting for the convenience of the reader, referring to [7, §6.2] for an abstract analogue.

Lemma 11.2

Suppose we are given sublinear operators $T_k$ so that for $k\ge 1$,

$$\begin{aligned} \Vert T_k\Vert _{L^{p_0,1}\rightarrow L^{q_0,\infty }} \lesssim 2^{k a_0}\quad \text {and}\quad \Vert T_k\Vert _{L^{p_1,1}\rightarrow L^{q_1,\infty }} \lesssim 2^{-k a_1} \end{aligned}$$

for some $p_0, q_0, p_1, q_1\in [1,\infty ], a_0, a_1>0$. Then the operator $\sum _{k\ge 1} T_k$ is of restricted weak type (p, q), where

$$\begin{aligned} \left( \tfrac{1}{p}, \tfrac{1}{q}\right) = (1-\vartheta ) \left( \tfrac{1}{p_0},\tfrac{1}{q_0}\right) + \vartheta \left( \tfrac{1}{p_1},\tfrac{1}{q_1}\right) \end{aligned}$$

(11.3)

with $\vartheta =\tfrac{a_0}{a_0+a_1}\in (0,1)$.

Proof

Given a set A, we use |A| to denote its measure. Let E be a measurable set and let $\lambda >0$ be a real number. We need to show that

$$\begin{aligned} | \{x: \bigg | \sum _{k\ge 1}T_k{\mathbbm {1}}_E(x)\bigg |>\lambda \}|\lesssim \lambda ^{-q}|E|^{\frac{q}{p}}, \end{aligned}$$

(11.4)

where the implicit constant is independent of E and $\lambda$. For any non-negative real number N, we have

$$\begin{aligned}|\{x:\bigg |\sum _{k\ge 1}T_k{\mathbbm {1}}_E(x)\bigg |>\lambda \}|\le & {} |\{x:\bigg |\sum _{1\le k\le N}T_k{\mathbbm {1}}_E(x)\bigg |>\lambda /2\}|\\{} & {} +|\{x:|\sum _{k\ge N}T_k{\mathbbm {1}}_E(x)|>\lambda /2\}|.\end{aligned}$$

To bound the first term on the right, we use the $L^{p_0,1}\rightarrow L^{q_0, \infty }$ boundedness of the operators $T_k$ to get

$$\begin{aligned} |\{x:\bigg |\sum _{1\le k\le N}T_k{\mathbbm {1}}_E(x)\bigg |>\lambda /2\}|{} & {} \lesssim \bigg \Vert \sum _{1\le k\le N}T_k\bigg \Vert _{L^{p_0,1}\rightarrow L^{q_0, \infty }}^{q_0}\lambda ^{-q_0}|E|^{\frac{q_0}{p_0}}\\{} & {} \lesssim \left( \sum _{1\le k\le N}2^{ka_0}\right) ^{q_0}\lambda ^{-q_0}|E|^{\frac{q_0}{p_0}}. \end{aligned}$$

We thus conclude that

$$\begin{aligned}|\{x:\bigg |\sum _{1\le k\le N}T_k{\mathbbm {1}}_E(x)\bigg |>\lambda /2\}|\lesssim 2^{Na_0q_0}\lambda ^{-q_0}|E|^{\frac{q_0}{p_0}}.\end{aligned}$$

Similarly, we can use our assumption on the $L^{p_1,1}\rightarrow L^{q_1, \infty }$ boundedness of the operators $T_k$ to obtain

$$\begin{aligned}|\{x:\bigg |\sum _{k\ge N}T_k{\mathbbm {1}}_E(x)\bigg |>\lambda /2\}|\lesssim 2^{-Na_1q_1}\lambda ^{-q_1}|E|^{\frac{q_1}{p_1}}.\end{aligned}$$

Consequently

$$\begin{aligned}|\{x:\bigg |\sum _{k\ge 1}T_k{\mathbbm {1}}_E(x)\bigg |>\lambda \}|\lesssim 2^{Na_0q_0}\lambda ^{-q_0}|E|^{\frac{q_0}{p_0}}+ 2^{-Na_1q_1}\lambda ^{-q_1}|E|^{\frac{q_1}{p_1}}.\end{aligned}$$

The two terms on the right hand side are equal precisely when

$$\begin{aligned}2^N=\left( \lambda ^{q_0-q_1}|E|^{\frac{q_1}{p_1}-\frac{q_0}{p_0}}\right) ^{\frac{1}{a_0q_0+a_1q_1}}.\end{aligned}$$

We substitute this into the left hand side of the above inequality to get

$$\begin{aligned}|\{x:\bigg |\sum _{k\ge 1}T_k{\mathbbm {1}}_E(x)\bigg |>\lambda \}|\lesssim \left( \lambda ^{\frac{a_0(q_0-q_1)}{a_0q_0+a_1q_1}-1}|E|^{\frac{\frac{q_1}{p_1}-\frac{q_0}{p_0}}{a_0q_0+a_1q_1}+\frac{1}{p_0}}\right) ^{q_0}.\end{aligned}$$

Using (11.3) and rearranging terms on the right hand side yields the desired inequality (11.4). This finishes the proof. $\square$