Joint functional calculi and a sharp multiplier theorem for the Kohn Laplacian on spheres

Abstract

Let $\Box _b$ be the Kohn Laplacian acting on (0, j)-forms on the unit sphere in $\mathbb {C}^n$. In a recent paper of Casarino, Cowling, Sikora and the author, a spectral multiplier theorem of Mihlin–Hörmander type for $\Box _b$ is proved in the case $0<j<n-1$. Here we prove an analogous theorem in the exceptional cases $j=0$ and $j=n-1$, including a weak type (1, 1) endpoint estimate. We also show that both theorems are sharp. The proof hinges on an abstract multivariate multiplier theorem for systems of commuting operators.

1 Introduction

Let $(X,\mu )$ be a measure space, $\mathcal {E}$ be a complex vector bundle over X with a hermitian metric, and $\mathfrak {L}$ be a (possibly unbounded) self-adjoint operator on the space $L^2(\mathcal {E})$ of $L^2$-sections of $\mathcal {E}$. By the spectral theorem, we can write

$$\begin{aligned} \mathfrak {L}= \int _\mathbb {R}\lambda \, dE_\mathfrak {L}(\lambda ) \end{aligned}$$

(1)

for a projection-valued measure $E_\mathfrak {L}$, called the spectral resolution of $\mathfrak {L}$. A functional calculus for $\mathfrak {L}$ is then defined via spectral integration and, for all Borel functions $F : \mathbb {R}\rightarrow \mathbb {C}$, the operator

$$\begin{aligned} F(\mathfrak {L}) = \int _\mathbb {R}F(\lambda ) \, dE_\mathfrak {L}(\lambda ) \end{aligned}$$

is bounded on $L^2(\mathcal {E})$ if and only if F is an $E_{\mathfrak {L}}$-essentially bounded function.

Characterizing, or just giving nontrivial sufficient conditions for the $L^p$-boundedness of $F(\mathfrak {L})$ for some $p \ne 2$ in terms of properties of the “spectral multiplier” F is a much harder problem. This question has been particularly studied in the case $\mathfrak {L}$ is the Laplace operator on $\mathbb {R}^d$, or some analogue thereof acting on sections of a vector bundle over a smooth d-manifold. For the Laplacian $\mathfrak {L}= -{\varDelta }$ on $\mathbb {R}^d$, the classical Mihlin–Hörmander multiplier theorem [36, 53] tells us that $F(\mathfrak {L})$ is of weak type (1, 1) and $L^p$-bounded for all $p \in (1,\infty )$ whenever the multiplier F satisfies the scale-invariant local Sobolev condition

$$\begin{aligned} \Vert F\Vert _{L^q_{s,\mathrm {sloc}}} \mathrel {:=}\sup _{t\ge 0} \Vert F(t \, \cdot ) \,\chi \Vert _{L^q_s(\mathbb {R})} < \infty , \end{aligned}$$

(2)

for $q=2$ and some $s > d/2$; here $L^q_s(\mathbb {R})$ is the $L^q$ Sobolev space of (fractional) order s and $\chi \in C^\infty _c((0,\infty ))$ is any nontrivial cutoff function (different choices of $\chi$ give rise to equivalent local Sobolev norms). This result is sharp, i.e., the threshold d / 2 on the order of smoothness s required on the multiplier F cannot be lowered. Note that the above-defined $\Vert F\Vert _{L^q_{s,\mathrm {sloc}}}$ majorizes |F(0)|, because the supremum in (2) includes $t=0$; actually, the value $t=0$ may be dispensed with in the case of the Laplacian on $\mathbb {R}^d$, where $E_\mathfrak {L}(\{0\}) = 0$, but not in other cases discussed below.

Here we are concerned with the case where $\mathfrak {L}= \Box _b$ is the Kohn Laplacian acting on sections of the bundle ${\varLambda }^{0,j} \mathbb {S}$ of (0, j)-forms $(0 \le j \le n-1$) associated to the tangential Cauchy–Riemann complex on the unit sphere $\mathbb {S}$ in $\mathbb {C}^n$, where $n \ge 2$. The sphere $\mathbb {S}$ and the conformally equivalent Heisenberg group have been long studied as models for more general strictly pseudoconvex CR manifolds of hypersurface type [26, 28, 32]. The problem of obtaining a spectral multiplier theorem of Mihlin–Hörmander type for $\Box _b$ has been recently considered in [7], where the following result is proved.

Theorem 1.1

[7] Let $\Box _b$ be the Kohn Laplacian on (0, j)-forms on the unit sphere $\mathbb {S}$ in $\mathbb {C}^n$, where $0< j < n-1$. For all bounded Borel functions $F : \mathbb {R}\rightarrow \mathbb {C}$, if $\Vert F\Vert _{L^2_{s,\mathrm {sloc}}} < \infty$ for some $s > (2n-1)/2$, then the operator $F(\Box _b)$ is of weak type (1, 1) and $L^p$-bounded for all $p \in (1,\infty )$, and moreover

$$\begin{aligned} \Vert F(\Box _b)\Vert _{L^1 \rightarrow L^{1,\infty }} \le C \Vert F\Vert _{L^2_{s,\mathrm {sloc}}}. \end{aligned}$$

A key feature of this result is the threshold $(2n-1)/2$ in the smoothness condition, i.e., half the topological dimension $d = 2n-1$ of the sphere $\mathbb {S}$. In fact, by means of quite general theorems [10, 19], it is fairly straightforward to prove the above result under the stronger assumption that “$\Vert F\Vert _{L^\infty _{s,\mathrm {sloc}}} <\infty$ for some $s > Q/2$”, where $Q=2n$ is the so-called homogeneous dimension associated with the control distance for $\Box _b$. The fact that $Q > d$ is connected with the lack of ellipticity of $\Box _b$ (cf. [24, 27]) and the problem of obtaining sharp multiplier theorems of Mihlin–Hörmander type for nonelliptic, subelliptic operators is still widely open (see, e.g., [50]). Most of the analysis in [7] is devoted to proving a “weighted Plancherel-type estimate” that allows weakening the assumption on the multiplier, by replacing Q / 2 with d / 2 (and $L^\infty _s$ with $L^2_s$).

The cases $j=0$ and $j=n-1$ are not treated in [7]. These cases are exceptional because the orthogonal projection onto the kernel of $\Box _b$ (which is the Szegő projection in the case $j=0$) is not $L^1$-bounded [43]. This constitutes a serious obstruction to the application of the “standard machinery” of [10, 19], on which [7] is based, and moreover puts some limits on the results that can be expected. Indeed, when $0< j < n-1$, the Bochner–Riesz means $(1-t\Box _b)_+^\alpha$ of order $\alpha > (d-1)/2$ are $L^1$-bounded for all $t > 0$ [7, Theorem 1.2]. The analogous statement in the case $j\in \{0,n-1\}$ is simply false, independently of the order $\alpha$.

An alternative approach to this problem is developed in [60], where a multiplier theorem of Mihlin–Hörmander type for $\Box _b$ in the case $j=0$ is proved for a fairly general class of compact CR manifolds. However in [60] the more restrictive smoothness condition that “$\Vert F\Vert _{L^2_{s,\mathrm {sloc}}} < \infty$ for some $s > (Q+1)/2$” is required and the technique used seems not to yield a weak type (1, 1) bound.

In contrast, the main result of this paper, which extends Theorem 1.1 to the missing cases $j = 0$ and $j=n-1$, requires a smoothness condition $s>d/2$ on the multiplier and includes the weak type (1, 1) endpoint.

Theorem 1.2

Let $\Box _b$ be the Kohn Laplacian on (0, j)-forms on the unit sphere $\mathbb {S}$ in $\mathbb {C}^n$, where $j \in \{0,n-1\}$. For all bounded Borel functions $F : \mathbb {R}\rightarrow \mathbb {C}$, if $\Vert F\Vert _{L^2_{s,\mathrm {sloc}}} < \infty$ for some $s > (2n-1)/2$, then the operator $F(\Box _b)$ is of weak type (1, 1) and $L^p$-bounded for all $p \in (1,\infty )$, and moreover

$$\begin{aligned} \Vert F(\Box _b)\Vert _{L^1 \rightarrow L^{1,\infty }} \le C \Vert F\Vert _{L^2_{s,\mathrm {sloc}}}. \end{aligned}$$

Note that this implies the weak type (1, 1) and $L^p$-boundedness for $p \in (1,\infty )$ of the Bochner–Riesz means $(1-t\Box _b)_+^\alpha$ for all $\alpha > (d-1)/2$ and $t>0$.

Our proof of Theorem 1.2 could be easily adapted to the case of Heisenberg groups, equipped with the standard strictly pseudoconvex structure. In fact, the proof there would be even simpler because of the translation-invariance and homogeneity of $\Box _b$ on Heisenberg groups. However, there is no need to do this, in the sense that the result on the Heisenberg group can be directly obtained from the corresponding result on the sphere by transplantation.

Corollary 1.3

Theorems 1.1 and 1.2 hold also when the sphere $\mathbb {S}$ is replaced by the $(2n-1)$-dimensional Heisenberg group $H_{n-1}$.

The idea of transplanting estimates from complex spheres to Heisenberg groups has been used several times in the literature (see, e.g., [6, 14, 15, 57]). Here however we propose a different approach, along the lines of [42], which does not require any group or symmetric space structure on the manifold, or group-invariance of the operator. This general transplantation technique (Theorem 5.2) applies to arbitrary self-adjoint differential operators on a vector bundle over a smooth manifold and allows transplanting weak type as well as strong type bounds.

The same technique, combined with an argument of [50], yields the sharpness of the above multiplier theorems. In fact, thanks to the analysis of [3], we can prove a more general result for the Kohn Laplacian on any non-Levi-flat CR manifold of hypersurface type (see [3, 5, 16] for definitions).

For a general nonnegative self-adjoint operator $\mathfrak {L}$ on $L^2(\mathcal {E})$ as in (1), define the sharp Mihlin–Hörmander threshold $\varsigma (\mathfrak {L})$ as the infimum of the $s \in (0,\infty )$ such that

$$\begin{aligned} \exists C \in (0,\infty ) :\forall F \in \mathfrak {B} :\Vert F(\mathfrak {L})\Vert _{L^{2} \rightarrow L^2} + \Vert F(\mathfrak {L})\Vert _{L^{1} \rightarrow L^{1,\infty }} \le C \, \Vert F \Vert _{L^2_{s,\mathrm {sloc}}}, \end{aligned}$$

where $\mathfrak {B}$ is the set of bounded Borel functions $F : \mathbb {R}\rightarrow \mathbb {C}$. Clearly

$$\begin{aligned} \varsigma (\mathfrak {L}) \ge \varsigma _-(\mathfrak {L}), \end{aligned}$$

where $\varsigma _-(\mathfrak {L})$ is the infimum of the $s \in (0,\infty )$ such that

$$\begin{aligned} \forall p \in (1,\infty ) :\exists C \in (0,\infty ) :\forall F \in \mathfrak {B} :\Vert F(\mathfrak {L})\Vert _{L^p \rightarrow L^p} \le C \, \Vert F \Vert _{L^\infty _{s,\mathrm {sloc}}}. \end{aligned}$$

Note that Theorems 1.1 and 1.2 and Corollary 1.3 can be restated as follows:

$$\begin{aligned} \varsigma (\Box _b) \le (2n-1)/2 \end{aligned}$$

for the Kohn Laplacian $\Box _b$ on a sphere or Heisenberg group of dimension $2n-1$.

Theorem 1.4

Let M be a non-Levi-flat CR manifold of hypersurface type and dimension $2n-1$, with a compatible hermitian metric. Let $\Box _b$ be any self-adjoint extension of the Kohn Laplacian on (0, j)-forms on M, where $0 \le j \le n-1$. Then

$$\begin{aligned} \varsigma _-(\Box _b) \ge (2n-1)/2. \end{aligned}$$

In particular, Theorems 1.1 and 1.2 and Corollary 1.3 are sharp.

Apart from sharpness and weak type endpoint, another reason of interest for Theorem 1.2 is the technique used in its proof. To prove that $F(\Box _b)$ is of weak type (1, 1), here we show that $F(\Box _b)$ is a singular integral operator, satisfying the “averaged Hörmander condition” of [18, Theorem 1] (see also [10, Theorem 3.3]):

$$\begin{aligned} \sup _{r > 0} \mathop {\hbox {ess\,sup}}\limits _{y \in \mathbb {S}} \int _{\varrho (x,y) \ge r} |K_{F(\Box _b) (I-A_r)}(x,y)| \,d\mu (x) < \infty , \end{aligned}$$

(3)

where $\mu$ is the standard hypersurface measure on $\mathbb {S}$, $\varrho$ is the control distance for $\Box _b$, $K_T$ denotes the integral kernel of an operator T and $\{A_r\}_{r>0}$ is some “approximate identity” (as $r \downarrow 0$) satisfying, among other things, the uniform bound

$$\begin{aligned} \sup _{r>0} \Vert A_r\Vert _{1 \rightarrow 1} < \infty . \end{aligned}$$

(4)

In other works on spectral multipliers, this approximate identity is constructed as a function of the operator $\mathfrak {L}$ under consideration, such as the “heat propagator” $A_r = \exp (-r^2 \mathfrak {L})$ (see, e.g., [19]) or $A_r = {\varPhi }(r \sqrt{\mathfrak {L}})$ for a suitable Schwartz function ${\varPhi }$ with ${\varPhi }(0)=1$ (cf. [10]). However such choices of $A_r$ are forbidden in the case $\mathfrak {L}= \Box _b$ and $j\in \{0,n-1\}$, because the $L^1$-unboundedness of the Szegő projection is incompatible with (4).

We are then led to looking for an approximate identity $A_r$ outside the functional calculus of $\Box _b$, yet related to it, so as to be able to prove (3). Here comes a key observation: in the case $j=0$, the operator $\Box _b$ belongs to the joint functional calculus of two commuting differential operators on $\mathbb {S}$, namely, a sublaplacian L and a unit vector field T. Unlike $\Box _b$, the sublaplacian L does satisfy Gaussian-type heat kernel estimates, so $A_r = \exp (-r^2 L)$ satisfies (4) (indeed a sharp multiplier theorem for the sublaplacian L was proved in [11]). In fact, more is true: the T-derivatives of the heat kernel of L satisfy Gaussian-type estimates too. Based on these estimates, we can prove a spectral multiplier theorem of Mihlin–Hörmander type for the joint functional calculus of L and iT, which in turn (in combination with the weighted Plancherel-type estimates from [7]) allows us to derive (3) with $A_r = \exp (-r^2 L)$, whenever $\Vert F\Vert _{L^2_{s,\mathrm {sloc}}} < \infty$ for some $s>d/2$.

Spectral multiplier theorems for systems of commuting operators are not new in the literature. Actually, the classical Mihlin–Hörmander theorem for Fourier multipliers on $\mathbb {R}^d$ can be thought of as a spectral multiplier theorem for the joint functional calculus of the partial derivatives on $\mathbb {R}^d$. However, in settings other than $\mathbb {R}^d$, most of the available results (see, e.g., [8, 17, 30, 54, 55, 64, 66]) are of “Marcinkiewicz type”, i.e., they impose on the multiplier function a condition that is invariant under multiparameter rescaling, and the correspondingly obtained estimates appear not to be suitable to prove a weak type (1, 1) bound. Exceptions to this are the results of [58] and [49, §4], where a one-parameter rescaling of the multiplier is considered and weak type (1,1) estimates are obtained; however these results are not directly applicable to the system (L, iT) on the sphere, because [58] only applies to a product setting, whereas [49, §4] applies to left-invariant homogeneous operators on a homogeneous Lie group (this would be enough to deal with the Heisenberg group, but not the sphere).

For this reason, in Sect. 6 we develop an abstract version of [49, §4] in the context of doubling metric measure spaces, which includes the main result of [58] as a particular case and may be of independent interest. When applied to a single operator $\mathfrak {L}$, the result of Sect. 6 essentially reduces to the main result of [33], where only polynomial decay (of arbitrarily high order) is required on the heat kernel of $\mathfrak {L}$, in place of the usual Gaussian-type exponential decay. Because of the general character of the argument, we have tried to put minimal assumptions on the system of commuting operators, in order to obtain a statement that encompasses many different situations; we refer to Sect. 6 for an extensive discussion and examples. The resulting multivariate spectral multiplier theorem of Mihlin–Hörmander type (Theorem 6.1) is sufficiently strong to serve as a base for our sharp Theorem 1.2 and we expect that other similar applications may be found in the future.

Some general remarks about notation are in order. The letter C and variants such as $C_p$ will denote a finite positive quantity that may change from place to place. For any two nonnegative quantities A, B, we also write $A \lesssim B$ instead of $A \le C B$; moreover $A \sim B$ is the same as “$A \lesssim B$ and $B \lesssim A$”. We denote by $\mathbb {1}_{U}$ the characteristic function of a set U.

2 Unitary group action and joint spectral decomposition

This and the next two sections are devoted to the proof of Theorem 1.2. Indeed we need only to discuss the case $j=0$, i.e., the case of the Kohn Laplacian $\Box _b$ acting on (scalar-valued) functions on the sphere. In fact, by means of Geller’s Hodge star operator ([32, p. 5]; see also [7, Remark 4.6]), it is easily seen that the case $j=n-1$ in Theorem 1.2 can be reduced to the case $j=0$.

The Kohn Laplacian $\Box _b$ on $\mathbb {S}$ is invariant under the action of the unitary group $\mathrm {U}(n)$. It is therefore natural to exploit the representation theory of $\mathrm {U}(n)$ for the analysis of $\Box _b$, as was done in great detail in [26]. Here we just recall the main results that will be of use later.

As it is well-known (see, e.g., [11, 26]), the decomposition into irreducible representations of the natural representation of $\mathrm {U}(n)$ on $L^2(\mathbb {S})$ is multiplicity-free and is given by

$$\begin{aligned} L^2(\mathbb {S}) = \bigoplus _{p,q \ge 0} \mathcal {H}_{pq}, \end{aligned}$$

where $\mathcal {H}_{pq}$ is the space of (p, q)-bihomogeneous complex spherical harmonics (denoted by ${\varPhi }_{pq0}$ in [26]). By Schur’s Lemma, all $\mathrm {U}(n)$-equivariant operators R on $L^2(\mathbb {S})$ preserve this decomposition and are scalar when restricted to each $\mathcal {H}_{pq}$, i.e., $R|_{\mathcal {H}_{pq}} = \lambda _{pq}^R \mathop {\mathrm {id}}_{\mathcal {H}_{pq}}$ for some $\lambda _{pq}^R \in \mathbb {C}$. In particular, all such operators commute.

Let the sublaplacian L and the unit vector field T on $\mathbb {S}$ be defined as in [11] and set $U=-2(n-1)iT$. The operators $\Box _b$, L and U are $\mathrm {U}(n)$-equivariant, so they have a joint spectral decomposition in terms of complex spherical harmonics.

Proposition 2.1

For all $p,q \in \mathbb {N}$,

$$\begin{aligned} \dim \mathcal {H}_{pq}&= \frac{p+q+n-1}{n-1} \left( {\begin{array}{c}p+n-2\\ n-2\end{array}}\right) \left( {\begin{array}{c}q+n-2\\ n-2\end{array}}\right) \end{aligned}$$

(5)

$$\begin{aligned} \lambda _{pq}^L&= 4pq + 2(n -1)(p + q), \end{aligned}$$

(6)

$$\begin{aligned} \lambda _{pq}^U&= 2(n-1)(p-q), \end{aligned}$$

(7)

$$\begin{aligned} \lambda _{pq}^{\Box _b}&= 2q(p + n-1). \end{aligned}$$

(8)

In particular

$$\begin{aligned} 2\Box _b= L-U, \end{aligned}$$

(9)

and moreover, for all $p,q\in \mathbb {N}$,

$$\begin{aligned} \lambda _{pq}^{\Box _b} \ne 0 \quad \Rightarrow \quad \lambda _{pq}^{\Box _b} \le \lambda _{pq}^L \le (n+1) \lambda _{pq}^{\Box _b}. \end{aligned}$$

(10)

Proof

(5) is in [11, Corollary 2.6], (6) and (7) are in [11, Section 4], while (8) is in [26, Theorem 6]. From these expressions we immediately obtain (10) and

$$\begin{aligned} 2\lambda _{pq}^{\Box _b} = \lambda _{pq}^L - \lambda _{pq}^U, \end{aligned}$$

whence (9) follows. $\square$

3 Heat kernel estimates and a nonsharp multiplier theorem

Let $\mu$ be the standard hypersurface measure on $\mathbb {S}\subseteq \mathbb {C}^n$ and $\varrho$ denote the control distance for $\Box _b$. We refer to [7, Section 3] for precise definitions and discussion of the main properties of $\mu$ and $\varrho$. Here we just recall that $\varrho$ is $\mathrm {U}(n)$-invariant and

$$\begin{aligned} \varrho (z,w) \sim |1-\langle z,w\rangle |^{1/2}, \end{aligned}$$

(11)

where $\langle \cdot , \cdot \rangle$ denotes the standard Hermitian inner product on $\mathbb {C}^n$. Moreover, if V(r) denotes the $\mu$-measure of any $\varrho$-ball of radius $r \in [0,\infty )$, then

$$\begin{aligned} V(r) \sim \min \{1,r^Q\}, \end{aligned}$$

(12)

where $Q = 2n$. In the language of Sect. 6, this tells us that $(\mathbb {S},\varrho ,\mu )$ is a doubling metric measure space of homogeneous dimension Q and displacement parameter 0.

Let us introduce some weighted mixed Lebesgue norms defined on Borel functions $K : \mathbb {S}\times \mathbb {S}\rightarrow \mathbb {C}$, that will be repeatedly used: for all $p \in [1,\infty ]$, $r \in (0,\infty )$, and $\beta \in \mathbb {R}$,

$$\begin{aligned} |||K|||_{p,\beta ,r} = \mathop {\hbox {ess\,sup}}\limits \limits _y V(r)^{1/p'} \Vert K(\cdot ,y) \, (1+\varrho (\cdot ,y)/r)^\beta \Vert _{L^p(\mathbb {S})}. \end{aligned}$$

Here $p'$ denotes the conjugate exponent $p/(p-1)$ to p.

As already mentioned in the introduction, a key ingredient of our proof is the fact that the heat kernel of L, together with its U-derivatives, satisfies Gaussian-type estimates.

Proposition 3.1

There exists $b>0$ such that, for all $k \in \mathbb {N}$, there exists $C_k > 0$ such that, for all $t > 0$ and $x,y \in \mathbb {S}$,

$$\begin{aligned} |K_{U^k e^{-tL}}(x,y)| \le C_k \, t^{-k} \, V(t^{1/2})^{-1} \exp (-b \,\varrho (x,y)^2/t). \end{aligned}$$

Proof

These estimates are well known, at least for for $t \le 1$ (see, e.g., [40, Theorem 3] or [39, §4]). On the other hand, since $\mathbb {S}$ is compact and $\varrho$ is bounded, the estimates for $t \ge 1$ follow from the uniform bound

$$\begin{aligned} |||K_{U^k e^{-tL}} |||_{\infty ,0,t^{1/2}} \le C_k \, t^{-k}, \end{aligned}$$

which in turn is easily proved by $L^2$-spectral theory (cf. [11, p. 630] or [7, proof of Theorem 1.1]):

$$\begin{aligned} \begin{aligned} |||K_{U^k e^{-tL}} |||_{\infty ,0,t^{1/2}}&= V(t^{1/2}) \, \Vert U^k e^{-tL} \Vert _{1 \rightarrow \infty } \\&\le V(t^{1/2}) \, \Vert U^{k} e^{-tL/2} \Vert _{2 \rightarrow \infty } \, \Vert e^{-tL/2} \Vert _{1 \rightarrow 2} \end{aligned} \end{aligned}$$

and, for all $h \in \mathbb {N}$,

$$\begin{aligned} \begin{aligned} \Vert U^{h} e^{-tL/2} \Vert _{1 \rightarrow 2}^2&= \Vert U^{h} e^{-tL/2} \Vert _{2 \rightarrow \infty }^2\\&= \sum _{p,q \in \mathbb {N}} (\lambda _{pq}^U)^{2h} \, e^{-t \lambda _{pq}^L} \dim \mathcal {H}_{pq} / \mu (\mathbb {S}) \\&\le t^{-2h} \sum _{p,q \in \mathbb {N}} (t\lambda _{pq}^L)^{2h} \, e^{-t \lambda _{pq}^L} \dim \mathcal {H}_{pq} / \mu (\mathbb {S}) \\&\lesssim t^{-2h} \sum _{\ell \in \mathbb {N}} (t \ell ^2)^{2h} e^{-t\ell ^2} \ell ^{Q-1} \\&\lesssim \, t^{-2h} \, V(t^{1/2})^{-1} \end{aligned} \end{aligned}$$

by Proposition 2.1, Eq. (12) and [11, eq. (21)]. $\square$

Thanks to the above bounds on the heat kernel of L and its U-derivative, we can apply Theorem 6.1 to the system (L, U) and isotropic dilations $\epsilon _r(\lambda ) = r^2 \lambda$ on $\mathbb {R}^2$ (cf. Example 6.4).

Corollary 3.2

For all $\beta \ge 0$, all $\epsilon ,R > 0$, and all $G : \mathbb {R}^2 \rightarrow \mathbb {C}$ with ${{\mathrm{supp}}}G \subseteq [-R^2,R^2] \times [-R^2,R^2]$,

$$\begin{aligned} |||K_{G(L,U)}|||_{2,\beta ,R^{-1}} \le C_{\beta ,\epsilon } \, \Vert G(R^2\,\cdot ) \Vert _{L^\infty _{\beta +\epsilon }(\mathbb {R}^2)}. \end{aligned}$$

Proof

This is an instance of Theorem 6.1(ii) applied to the system (L, U).$\square$

From these estimates we can now derive a nonsharp spectral multiplier theorem for $\Box _b$. Define, for all $t>0$, the operator $A_t$ by

$$\begin{aligned} A_t = \exp (-t^2 L). \end{aligned}$$

Corollary 3.3

The following estimates hold.

• (i) For all $\beta \ge 0$, all $\epsilon ,R,t > 0$, and all $F : \mathbb {R}\rightarrow \mathbb {C}$ with ${{\mathrm{supp}}}F \subseteq [R/16,R]$,

  $$\begin{aligned} |||K_{F(\sqrt{\Box }_b) \,(1-A_t)} |||_{2,\beta ,R^{-1}} \le C_{\beta ,\epsilon } \, \Vert F(R\, \cdot ) \Vert _{L_{\beta +\epsilon }^\infty (\mathbb {R})} \, \min \{1,(Rt)^2\}. \end{aligned}$$

• (ii) For all $\beta \ge 0$, all $\epsilon ,R,r,t > 0$, and all $F : \mathbb {R}\rightarrow \mathbb {C}$ with ${{\mathrm{supp}}}F \subseteq [R/16,R]$,

  $$\begin{aligned}&\mathop {\hbox {ess\,sup}}\limits _y \int _{\varrho (x,y) \ge r} |K_{F(\sqrt{\Box }_b) \,(1-A_t)}(x,y)| \,d\mu (x) \\&\quad \le C_{\beta ,\epsilon }\, \Vert F(R \,\cdot ) \Vert _{L_{\beta +Q/2+\epsilon }^\infty (\mathbb {R})} \, \min \{1,(Rt)^2\} \, (1+Rr)^{-\beta }. \end{aligned}$$

• (iii) For all $\epsilon > 0$ and $F : \mathbb {R}\rightarrow \mathbb {C}$,

  $$\begin{aligned} \Vert F(\sqrt{\Box _b})\Vert _{L^1 \rightarrow L^{1,\infty }} \le C_{\epsilon } \, \Vert F\Vert _{L^\infty _{Q/2+\epsilon ,\mathrm {sloc}}} . \end{aligned}$$

Proof

• (i) Let

  $$\begin{aligned} G(\lambda _1,\lambda _2) = F(\sqrt{(\lambda _1-\lambda _2)_+/2}) \, (1-\exp (-t^2 \lambda _1)) \, \eta ((\lambda _1-\lambda _2)/(2\lambda _1)), \end{aligned}$$

  where $\eta \in C^\infty _c((0,2))$ and $\eta = 1$ on $[1/(n+1),1]$. By (9) and (10), it is easily seen that

  $$\begin{aligned} G(L,U) = F(\sqrt{\Box _b}) \, (1-A_t) \end{aligned}$$

  and moreover, for all $s \in [0,\infty )$,

  $$\begin{aligned} \Vert G(R^2 \,\cdot ) \Vert _{L^\infty _s(\mathbb {R}^2)} \le C_s \, \Vert F(R \,\cdot ) \Vert _{L^\infty _s(\mathbb {R})} \, \min \{ 1, (Rt)^2\}, \end{aligned}$$

  hence (i) follows by Corollary 3.2.

• (ii) This follows from (i), since

  $$\begin{aligned}&\mathop {\hbox {ess\,sup}}\limits _y \int _{\varrho (x,y) \ge r} |K_{F(\sqrt{\Box }_b) \,(1-A_t)}(x,y)| \,d\mu (x) \\&\quad \le (1+Rr)^{-\beta } |||K_{F(\sqrt{\Box }_b) \,(1-A_t)} |||_{1,\beta ,R^{-1}} \end{aligned}$$

  and

  $$\begin{aligned} |||K_{F(\sqrt{\Box }_b) \,(1-A_t)} |||_{1,\beta ,R^{-1}} \le C_{\beta ,\epsilon } \, |||K_{F(\sqrt{\Box }_b) \,(1-A_t)} |||_{2,\beta +Q/2+\epsilon /2,R^{-1}} \end{aligned}$$

  by Hölder’s inequality (44).

• (iii) Suppose first that $F(0)=0$. Note that, by Proposition 3.1, the operators $A_t$ satisfy the “Poisson-type bounds” of [18, eq. (2) and (3)]. A simple adaptation of the argument in [19, Proof of Theorem 3.1] (see also the proof of Theorem 6.1(vi) below), exploiting (ii) in place of [19, eq. (4.18)], shows that $F(\sqrt{\Box _b})$ is of weak type (1, 1), with

  $$\begin{aligned} \Vert F(\sqrt{\Box _b})\Vert _{L^1 \rightarrow L^{1,\infty }} \le C_{\epsilon } \, \Vert F \Vert _{L^\infty _{Q/2+\epsilon ,\mathrm {sloc}}}. \end{aligned}$$

  In particular, if we take $F = \mathbb {1}_{(0,\infty )}$, then we recover the weak type (1,1) of the Szegő projection $\mathbb {1}_{\{0\}}(\Box _b) = I - \mathbb {1}_{(0,\infty )}(\Box _b)$ on the unit sphere [43].

  For a general F, it is then sufficient to split

  $$\begin{aligned} F(\sqrt{\Box _b}) = F(0) \, \mathbb {1}_{\{0\}}(\Box _b) + (F \mathbb {1}_{\mathbb {R}\setminus \{0\}})(\sqrt{\Box _b}) \end{aligned}$$

  and apply what we have just proved to the two summands. $\square$

4 Sharpening the result

Here we show how the weighted Plancherel estimates proved in [7] can be used to sharpen the multiplier theorem given by Corollary 3.3(iii) and obtain Theorem 1.2. As in [7, §5], define the weight $\varpi : \mathbb {S}\times \mathbb {S}\rightarrow [0,\infty )$ by

$$\begin{aligned} \varpi (z,w) = |1-|\langle z,w\rangle |^2|^{1/2}. \end{aligned}$$

Here are some basic properties of $\varpi$ that will be of use.

Lemma 4.1

For all $r>0$ and $\alpha ,\beta \ge 0$ such that $\alpha +\beta > Q$ and $\alpha < 2n-2$, and for all $w \in \mathbb {S}$,

$$\begin{aligned} \int _{\mathbb {S}} (1+\varrho (z,w)/r)^{-\beta } (1+\varpi (z,w)/r)^{-\alpha } \,d\mu (z) \le C_{\alpha ,\beta } V(r). \end{aligned}$$

(13)

Moreover

$$\begin{aligned} \int _{\mathbb {S}} (1+\varpi (z,w)/r)^{-\alpha } \,d\mu (z) \le C_{\alpha } \, \min \{1,r^{\alpha }\}. \end{aligned}$$

(14)

for all $r > 0$, all $\alpha \in [0,2n-2)$ and all $w \in \mathbb {S}$, and

$$\begin{aligned} \varpi (z,w) \le C \varrho (z,w) \end{aligned}$$

(15)

for all $z,w \in \mathbb {S}$.

Proof

By $\mathrm {U}(n)$-invariance of $\varpi$ and $\varrho$, we may assume that $w = (1,0,\dots ,0)$. If $z = (z_1,z_2,\dots ,z_n)=(z_1,z')$, then

$$\begin{aligned} \varpi (z,w) = |1-|z_1|^2|^{1/2} = |z'| \end{aligned}$$

and

$$\begin{aligned} \varrho (z,w) \sim |1-z_1|^{1/2} \gtrsim |z'| + |\mathfrak {I}z_1|^{1/2} \end{aligned}$$

(see [7, Proposition 3.1]). This gives (15) immediately, and (14) follows because

$$\begin{aligned} \int _{\mathbb {S}} \varpi (z,w)^{-\alpha } \,d\mu (z) < \infty \end{aligned}$$

for $\alpha < 2n-2$.

As for (13), the case $r \ge 1$ is trivial because $\mathbb {S}$ is compact, so we may assume that $r \le 1$. Since $\beta > Q-\alpha = (2n-2-\alpha ) + 2$ and $2n-2-\alpha > 0$, we can decompose $\beta = \beta _1+\beta _2$ so that $\beta _1 > 2n-2-\alpha$ and $\beta _2 > 2$, hence

$$\begin{aligned} \begin{aligned} \int _{\mathbb {S}}&(1+\varrho (z,w)/r)^{-\beta } (1+\varpi (z,w)/r)^{-\alpha } \,d\mu (z) \\&\le C_{\alpha ,\beta } \int _{\mathbb {S}} (1+(|z'|+|\mathfrak {I}z_1|^{1/2})/r)^{-\beta } (1+|z'|/r)^{-\alpha } \,d\mu (z) \\&\le C_{\alpha ,\beta } \int _{\mathbb {S}} (1+|\mathfrak {I}z_1|^{1/2}/r)^{-\beta _2} (1+|z'|/r)^{-\alpha -\beta _1} \,d\mu (z) \\&\le C_{\alpha ,\beta } \int _\mathbb {R}(1+|u|^{1/2}/r)^{-\beta _2} \,du \int _{\mathbb {R}^{2n-2}} (1+|v|/r)^{-\alpha -\beta _1} \,dv\\&= C_{\alpha ,\beta } \,r^{Q}, \end{aligned} \end{aligned}$$

and we are done. $\square$

Let us introduce the following bi-weighted mixed Lebesgue norms defined on Borel functions $K : \mathbb {S}\times \mathbb {S}\rightarrow \mathbb {C}$: for all $p \in [1,\infty ]$, $r \in (0,\infty )$, and $\alpha ,\beta \in \mathbb {R}$,

$$\begin{aligned} |||K|||_{p,\beta ,\alpha ,r} = \mathop {\hbox {ess\,sup}}\limits _y V(r)^{1/p'} \Vert K(\cdot ,y) \, (1+\varrho (\cdot ,y)/r)^\beta \, (1+\varpi (\cdot ,y)/r)^\alpha \Vert _{L^p(\mathbb {S})}. \end{aligned}$$

As in [7, §2] (see also [10, 19]), for all $N \in \mathbb {N}\setminus \{0\}$ and $F : \mathbb {R}\rightarrow \mathbb {C}$ supported in [0, 1], let the norm $\Vert F \Vert _{N,2}$ be defined by

$$\begin{aligned} \Vert F \Vert _{N,2} = \left( \frac{1}{N} \sum _{i=1}^N \sup _{\lambda \in \left[ \frac{i-1}{N},\frac{i}{N}\right] } |F(\lambda )|^2 \right) ^{1/2}. \end{aligned}$$

Proposition 4.2

For all $\alpha \in [0,1/2)$, all $N \in \mathbb {N}\setminus \{0\}$, all $t > 0$, and all $F : \mathbb {R}\rightarrow \mathbb {C}$ vanishing off (0, N),

$$\begin{aligned} |||K_{F(\sqrt{\Box }_b)} |||_{2,0,\alpha ,N^{-1}}&\le C_{\alpha } \, \Vert F(N\,\cdot ) \Vert _{N,2} , \end{aligned}$$

(16)

$$\begin{aligned} |||K_{F(\sqrt{\Box }_b) \,(1-A_t)} |||_{2,0,\alpha ,N^{-1}}&\le C_{\alpha } \, \Vert F(N\,\cdot ) \Vert _{N,2} \min \{1,(Nt)^2\}. \end{aligned}$$

(17)

Proof

We prove only (17), the other estimate being similar and easier. By Proposition 2.1 and [7, Lemma 4.3], $K_{F(\sqrt{\Box }_b) \,(1-A_t)}$ is a “kernel polynomial” in the sense of [7, §5], which satisfies the assumptions of [7, Proposition 5.3]. Hence, for all $\theta \in [0,1)$,

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {S}} |K_{F(\sqrt{\Box }_b) \,(1-A_t)}(x,y)|^2 \, \varpi (x,y)^\theta \,d\mu (x) \\&\le C_\theta \, N^{Q-1-\theta } \sum _{j=2}^N \max \{ |F(\sqrt{\lambda _{pq}^{\Box _b}}) (1-\exp (-t^2 \lambda _{pq}^L))|^2 :(j-1)^2 \le \lambda _{pq}^{\Box _b} \le j^2 \} \\&\le C_\theta \, N^{Q-1-\theta } (1-e^{-(n+1) t^2 N^2}) \, \sum _{j=2}^N \max \{ |F(\sqrt{\lambda _{pq}^{\Box _b}})|^2 :(j-1)^2 \le \lambda _{pq}^{\Box _b} \le j^2 \}, \end{aligned} \end{aligned}$$

where we have used (10). As in [7, proof of Theorem 1.1], we then deduce that, for all $\theta \in [0,1)$,

$$\begin{aligned}&\int _{\mathbb {S}} |K_{F(\sqrt{\Box }_b) \,(1-A_t)}(x,y)|^2 \, \varpi (x,y)^\theta \,d\mu (x) \\&\qquad \le C_\theta \, N^{Q-\theta } \, \Vert F(N\,\cdot )\Vert _{N,2}^2 \, \max \{1,(Nt)^2\}^2. \end{aligned}$$

Since $V(N^{-1}) \sim N^{-Q}$ for $N \in \mathbb {N}\setminus \{0\}$, the conclusion follows by combining the two instances of the above estimate corresponding to $\theta =0$ and $\theta = 2\alpha$. $\square$

Fix a nonnegative function $\xi \in C^\infty _c((-1/16,1/16))$ such that

$$\begin{aligned} \int _\mathbb {R}\xi (t) \,dt = 1 \qquad \text {and}\qquad \int _\mathbb {R}t^k \,\xi (t) \,dt = 0 \text { for }k=1,\dots ,2Q. \end{aligned}$$

(18)

As in [10, 19], we first deal with the operator $(F * \xi )(\sqrt{\Box _b})$ corresponding to a smoothened version of the multiplier F.

Proposition 4.3

The following estimates hold.

• (i) For all $\beta \ge 0$, all $\alpha \in [0,1/2)$, all $N \in \mathbb {N}\setminus \{0\}$, all $\epsilon ,t > 0$, and all $F : \mathbb {R}\rightarrow \mathbb {C}$ with ${{\mathrm{supp}}}F \subseteq [N/4,3N/4]$,

  $$\begin{aligned} |||K_{(F*\xi )(\sqrt{\Box }_b) \,(1-A_t)} |||_{2,\beta ,\alpha ,N^{-1}} \le C_{\alpha ,\beta ,\epsilon } \, \Vert F(N\,\cdot ) \Vert _{L_{\beta +\epsilon }^2(\mathbb {R})} \, \min \{1,(Nt)^2\}. \end{aligned}$$

• (ii) For all $\beta \ge 0$, all $N \in \mathbb {N}\setminus \{0\}$, all $\epsilon ,r,t > 0$, and all $F : \mathbb {R}\rightarrow \mathbb {C}$ with ${{\mathrm{supp}}}F \subseteq [N/4,3N/4]$,

  $$\begin{aligned}&\mathop {\hbox {ess\,sup}}\limits _y \int _{\varrho (x,y) \ge r} |K_{(F*\xi )(\sqrt{\Box }_b) \,(1-A_t)}(x,y)| \,d\mu (x) \\&\quad \le C_{\beta ,\epsilon }\, \Vert F(N\,\cdot ) \Vert _{L_{\beta +d/2+\epsilon }^2(\mathbb {R})} \min \{1,(Nt)^2\} (1+Nr)^{-\beta }. \end{aligned}$$

• (iii) For all $\epsilon > 0$ and $F : \mathbb {R}\rightarrow \mathbb {C}$ with ${{\mathrm{supp}}}F \subseteq [1/2,\infty )$,

  $$\begin{aligned} \Vert (F*\xi )(\sqrt{\Box _b})\Vert _{L^1 \rightarrow L^{1,\infty }} \le C_{\epsilon } \, \Vert F \Vert _{L^2_{d/2+\epsilon ,\mathrm {sloc}}}. \end{aligned}$$

Proof

• (i) By Corollary 3.3(i) and Young’s inequality,

  $$\begin{aligned} |||K_{(F*\xi )(\sqrt{\Box }_b) \,(1-A_t)} |||_{2,\beta ,N^{-1}} \le C_{\beta ,\epsilon } \Vert F(N\,\cdot ) \Vert _{L_{\beta +\epsilon }^\infty (\mathbb {R})} \, \min \{1,(Nt)^2\}. \end{aligned}$$

  for all $\beta \ge 0$, $N \in \mathbb {N}\setminus \{0\}$, $\epsilon ,t > 0$, and $F : \mathbb {R}\rightarrow \mathbb {C}$ with ${{\mathrm{supp}}}F \subseteq [N/8,7N/8]$; in particular, by Sobolev embedding and the inequality (15),

  $$\begin{aligned} |||K_{(F*\xi )(\sqrt{\Box }_b) \,(1-A_t)} |||_{2,\beta ,\alpha ,N^{-1}} \le C_{\beta ,\alpha ,\epsilon } \Vert F(N\,\cdot ) \Vert _{L_{\beta +\alpha +1/2+\epsilon }^2(\mathbb {R})} \, \min \{1,(Nt)^2\}. \end{aligned}$$

  for all $\alpha ,\beta \ge 0$, $N \in \mathbb {N}\setminus \{0\}$, $\epsilon ,t > 0$, and $F : \mathbb {R}\rightarrow \mathbb {C}$ with ${{\mathrm{supp}}}F \subseteq [N/8,7N/8]$. On the other hand, by Proposition 4.2 and [19, eq. (4.9)],

  $$\begin{aligned} |||K_{(F*\xi )(\sqrt{\Box }_b) \,(1-A_t)} |||_{2,0,\alpha ,N^{-1}} \le C_\alpha \Vert F(N \,\cdot ) \Vert _{2} \, \min \{1,(Nt)^2\} \end{aligned}$$

  for all $\alpha \in [0,1/2)$, $N \in \mathbb {N}\setminus \{0\}$, $t > 0$, and $F : \mathbb {R}\rightarrow \mathbb {C}$ with ${{\mathrm{supp}}}F \subseteq [N/8,7N/8]$. Interpolation of the last two inequalities (cf., for instance, [52, proof of Lemma 1.2] or [51, proof of Proposition 13]) gives the conclusion.

• (ii) Note that

  $$\begin{aligned}&\mathop {\hbox {ess\,sup}}\limits _y \int _{\varrho (x,y) \ge r} |K_{(F*\xi )(\sqrt{\Box }_b) \,(1-A_t)}(x,y)| \,d\mu (x) \\&\quad \le |||K_{(F*\xi )(\sqrt{\Box }_b) \,(1-A_t)} |||_{1,\beta ,0,N^{-1}} (1+Nr)^{-\beta }. \end{aligned}$$

  Moreover, for all nonnegative $\alpha \in (1/2-\epsilon /2,1/2)$, since $d/2+\epsilon /2+\alpha > d/2+1/2=Q/2$, Hölder’s inequality and (13) give that

  $$\begin{aligned} |||K_{(F*\xi )(\sqrt{\Box }_b) \,(1-A_t)} |||_{1,\beta ,0,N^{-1}} \le C_{\alpha ,\beta ,\epsilon } \, |||K_{(F*\xi )(\sqrt{\Box }_b) \,(1-A_t)} |||_{2,\beta +d/2+\epsilon /2,\alpha ,N^{-1}}. \end{aligned}$$

  The conclusion follows by (i).

• (iii) This follows from (ii) in the same way as Corollary 3.3(iii) follows from Corollary 3.3(ii). $\square$

It remains to deal with $(F-F*\xi )(\sqrt{\Box _b})$; however this part satisfies even better estimates.

Proposition 4.4

For all $\epsilon > 0$ and $F : \mathbb {R}\rightarrow \mathbb {C}$ with ${{\mathrm{supp}}}F \subseteq [1/2,\infty )$,

$$\begin{aligned} \Vert (F-F*\xi )(\sqrt{\Box _b})\Vert _{L^1 \rightarrow L^1} \le C_{\epsilon } \, \Vert F \Vert _{L^2_{d/2+\epsilon ,\mathrm {sloc}}}. \end{aligned}$$

Proof

We follow the argument in [10, proof of Theorem 3.6] (cf. also [19, proof of Theorem 3.2]).

Set $\epsilon ' = \min \{1/4,\epsilon \}$ and choose $\alpha \in [0,1/2)$ such that $d/2+\epsilon '+\alpha > Q/2$. Choose $\eta \in C^\infty _c((0,\infty ))$ such that ${{\mathrm{supp}}}\eta \subseteq [1/4,3/4]$ and $\sum _{k \in \mathbb {Z}} \eta _k = 1$ on $(0,\infty )$, where $\eta _k = \eta (2^{-k} \cdot )$. In particular, if $F_k = \eta _k F$, then $F = \sum _{k \in \mathbb {N}} F_k$. Set $\xi _k = 2^k \xi (2^k \cdot )$. By (33), Hölder’s inequality, (14), (12), the “Plancherel estimate” (16), and [19, Proposition 4.6] (which applies because of (18)), for all $k \in \mathbb {N}$,

$$\begin{aligned} \begin{aligned} \Vert (F_k -F_k*\xi )(\sqrt{\Box _b})&\Vert _{L^1 \rightarrow L^1} \\&= |||(F_k-F_k*\xi )(\sqrt{\Box _b}) |||_{1,0,0,2^{-k}} \\&\le C_\epsilon \, 2^{-k\alpha } \, V(2^{-k})^{-1/2} \, |||(F_k-F_k*\xi )(\sqrt{\Box _b}) |||_{2,0,\alpha ,2^{-k}} \\&\le C_\epsilon \, 2^{k(Q/2-\alpha )} \, \Vert F_k(2^k \cdot ) - F_k(2^k \cdot ) * \xi _k \Vert _{2^k,2} \\&\le C_\epsilon \, 2^{k(Q/2-\alpha -d/2-\epsilon ')} \, \Vert F_k(2^k \cdot ) \Vert _{L^2_{d/2+\epsilon '}} \\&\le C_\epsilon \, 2^{k(Q/2-\alpha -d/2-\epsilon ')} \, \Vert F \Vert _{L^2_{d/2+\epsilon ,\mathrm {sloc}}}. \end{aligned} \end{aligned}$$

The conclusion follows by summing over $k \in \mathbb {N}$. $\square$

Proof of Theorem 1.2

By combining Propositions 4.3(iii) and 4.4,

$$\begin{aligned} \Vert F(\sqrt{\Box _b})\Vert _{L^1 \rightarrow L^{1,\infty }} \lesssim \Vert F \Vert _{L^2_{d/2+\epsilon ,\mathrm {sloc}}} \end{aligned}$$

for all $\epsilon > 0$ and $F : \mathbb {R}\rightarrow \mathbb {C}$ with ${{\mathrm{supp}}}F \subseteq [1/2,\infty )$. On the other hand, since the eigenvalues $\lambda _{pq}^{\Box _b}$ of $\Box _b$ are nonnegative integers by (8) and $\mathbb {1}_{\{0\}}(\Box _b)$ is of weak type (1, 1) (see Corollary 3.3(iii)),

$$\begin{aligned} \Vert F(\sqrt{\Box _b})\Vert _{L^1 \rightarrow L^{1,\infty }} = \Vert F(0) \, \mathbb {1}_{\{0\}}(\Box _b)\Vert _{L^1 \rightarrow L^{1,\infty }} \lesssim |F(0)| \end{aligned}$$

for all $F : \mathbb {R}\rightarrow \mathbb {C}$ with ${{\mathrm{supp}}}F \subseteq (-\infty ,1)$. Hence a partition of unity argument yields

$$\begin{aligned} \Vert F(\sqrt{\Box _b})\Vert _{L^1 \rightarrow L^{1,\infty }} \le C_{\epsilon } \, \Vert F \Vert _{L^2_{d/2+\epsilon ,\mathrm {sloc}}}. \end{aligned}$$

Interpolation of this bound with the trivial estimate

$$\begin{aligned} \Vert F(\sqrt{\Box _b})\Vert _{L^2 \rightarrow L^{2}} \le \Vert F\Vert _{\infty } \le C_{\epsilon } \, \Vert F \Vert _{L^2_{d/2+\epsilon ,\mathrm {sloc}}} \end{aligned}$$

gives $L^p$-boundedness of $F(\sqrt{\Box _b})$ for $p \in (1,2)$; the case $p \in (2,\infty )$ follows by applying this result to $\overline{F}$ in place of F. $\square$

5 Transplantation to the Heisenberg group and sharpness

In this section we prove Corollary 1.3 and Theorem 1.4 via transplantation. Our approach is an extension of the technique of [42], which is based on perturbation theory for self-adjoint operators. In [42] scale-invariant $L^p$-bounds for the functional calculus of a self-adjoint differential operator D on a d-manifold M are transplanted to analogous bounds for the homogeneous constant-coefficient differential operator $D_0$ on $\mathbb {R}^d$ corresponding to the principal symbol of D (with respect to a choice of coordinates) at an arbitrary point of M.

The results of [42] do not apply to our situation, because the Kohn Laplacian on the Heisenberg group (that would play the role of $D_0$ above) is not a constant-coefficient operator. Indeed the method of “freezing the coefficients” is not appropriate for the analysis of operators such as the Kohn Laplacian on a CR manifold (see, e.g., [3, 28, 59]). For this reason we introduce in Definition 5.1 below a generalization of this method, based on a system of (possibly nonisotropic) dilations. Correspondingly, we prove a general transplantation result (Theorem 5.2) for an arbitrary self-adjoint differential operator acting on sections of a vector bundle over a smooth manifold. Finally we apply the general result to the Kohn Laplacian.

Manifolds are assumed to be smooth and second-countable. Vector bundles over manifolds are assumed to be smooth as well. By a hermitian vector bundle we mean a complex vector bundle with a (smooth) hermitian metric. A smooth measure on a manifold is a positive Borel measure whose density with respect to the Lebesgue measure in all coordinate charts is smooth and nowhere vanishing.

If $\mathcal {E}$ is a vector bundle over M and $U \subseteq M$, we denote by $\mathcal {E}|_U$ the vector bundle over U obtained by restriction. By $\mathcal {T}_d^k$ we denote the trivial bundle over $\mathbb {R}^d$ with fibre $\mathbb {C}^k$, equipped with the standard hermitian metric. Sections of $\mathcal {T}_d^k$ will be identified with $\mathbb {C}^k$-valued functions.

By a system of dilations $(\delta _R)_{R>0}$ on $\mathbb {R}^d$ we mean a family of linear automorphisms of $\mathbb {R}^d$ of the form $\delta _R = \exp ((\log R) A)$ for some positive self-adjoint linear endomorphism A of $\mathbb {R}^d$; note that $\det \delta _R = R^Q$, where $Q = {{\mathrm{tr}}}A$ is the “homogeneous dimension” associated with the system of dilations.

Definition 5.1

Let M be a d-manifold, equipped with a smooth measure $\mu$. Let $\mathcal {E}$ be a hermitian vector bundle of rank k over M. Let $D : C^\infty (\mathcal {E}) \rightarrow C^\infty (\mathcal {E})$ be a differential operator. We say that a differential operator $D_0 : C^\infty (\mathcal {T}^k_d) \rightarrow C^\infty (\mathcal {T}^k_d)$ is a local model for D at the point $x \in M$ of order $\gamma \in \mathbb {R}$ with respect to a system of dilations $(\delta _R)_{R>0}$ on $\mathbb {R}^d$ if there exist a coordinate chart $\phi : U \rightarrow V \subseteq \mathbb {R}^d$ of M centred at x and an orthonormal frame $X = (X_l)_{l=1}^k$ for $\mathcal {E}|_U$ such that, if we define

• (a) ${\varPsi } : C^\infty (\mathcal {E}|_U) \rightarrow C^\infty (\mathcal {T}^k_d|_V)$ by

  $$\begin{aligned} {\varPsi } \left( \sum _{l=1}^k f_l X_l \right) = ((f_l \circ \phi ^{-1}) \, a^{1/2})_{l=1}^k, \end{aligned}$$

  (19)

  where $a \in C^\infty (V)$ is the density with respect to the Lebesgue measure of the push-forward of $\mu$ via $\phi$, and

• (b) $D_R : C^\infty (\mathcal {T}^k_d|_{\delta ^{-1}_R(V)}) \rightarrow C^\infty (\mathcal {T}^k_d|_{\delta ^{-1}_R(V)})$, for all $R > 0$, as the differential operator given by

  $$\begin{aligned} D_R f = R^{\gamma } ({\varPsi } D {\varPsi }^{-1}(f \circ \delta ^{-1}_{R})) \circ \delta _R, \end{aligned}$$

then, for all $f \in C^\infty _c(\mathcal {T}^k_d)$,

$$\begin{aligned} \lim _{R \downarrow 0} D_{R} f = D_0 f \qquad \text {in }L^2. \end{aligned}$$

(20)

Note that the domains $C^\infty (\mathcal {T}^k_d|_{\delta ^{-1}_R(V)})$ and $C^\infty (\mathcal {T}^k_d)$ of the differential operators $D_R$ and $D_0$ may be different. However, if we identify $C^\infty _c(\mathcal {T}^k_d|_{\delta ^{-1}_R(V)})$ with a subspace of $C^\infty _c(\mathcal {T}^k_d)$, then, for all $f \in C^\infty _c(\mathcal {T}^k_d)$, we have that ${{\mathrm{supp}}}f \subseteq \delta ^{-1}_R(V)$ for all sufficiently small $R> 0$, so $D_R f$ is well defined as an element of $C^\infty _c(\mathcal {T}^k_d)$ for such small R and the condition (20) makes sense.

Similar considerations show that the coordinate chart and the local orthonormal frame in Definition 5.1 could be replaced with their restrictions to any smaller open neighbourhood of x, without changing the limit operator $D_0$ in (20). Hence two differential operators with same germ at x have the same local model.

The definition (19) of the map ${\varPsi }$ involves the density a of the measure on M, in such a way that ${\varPsi }|_{C^\infty _c(\mathcal {E}|_U)}$ extends to an isometry $L^2(\mathcal {E}|_U) \rightarrow L^2(\mathcal {T}^k_d|_V)$. This corresponds to the fact that our functional calculi, based on the spectral theorem, are initially defined on $L^2$ and are equivariant with respect to $L^2$-isometries. Similarly, we require $L^2$-convergence in (20). However, in applications (cf. proof of Proposition 5.4), it may happen that the convergence in (20) holds in a stronger sense and that the limit operator $D_0$ is independent of the positive smooth function a in (19).

The system of dilations $(\delta _R)_{R>0}$ is crucial in determining the local model. If we take isotropic dilations $\delta _R(v) = Rv$ and let $\gamma$ be the order of D as a differential operator, then the local model $D_0$ is nothing else than the principal part of the constant-coefficient differential operator obtained by freezing the coefficients of D at x in the chosen coordinates. Hence Theorem 5.2 extends some results of [42]. On the other hand, our applications involve nonisotropic dilations.

In what follows, we denote by $C_0(\mathbb {R})$ the space of complex-valued continuous functions on $\mathbb {R}$ vanishing at infinity. Strong convergence of operators is always understood in the sense of the strong $L^2$ operator topology.

Theorem 5.2

Let M be a d-manifold, equipped with a smooth measure. Let $\mathcal {E}$ be a hermitian vector bundle of rank k over M. Let $D : C^\infty (\mathcal {E}) \rightarrow C^\infty (\mathcal {E})$ be a formally self-adjoint differential operator. Suppose that D has a local model $D_0 : C^\infty (\mathcal {T}^k_d) \rightarrow C^\infty (\mathcal {T}^k_d)$ at some point $x \in M$ of order $\gamma \in \mathbb {R}$ with respect to some dilations $(\delta _R)_{R>0}$ on $\mathbb {R}^d$. Assume that $D_0$ is essentially self-adjoint on $C^\infty _c(\mathcal {T}^k_d)$ and denote its unique self-adjoint extension by $D_0$ as well. Let $\tilde{D}$ be any self-adjoint extension of D. Then, for all $F \in C_0(\mathbb {R})$ and all neighbourhoods $U \subseteq M$ of x,

$$\begin{aligned} \Vert F(D_0) \Vert _{L^1(\mathcal {T}^k_d) \rightarrow L^{1,\infty }(\mathcal {T}^k_d)} \le \liminf _{r\downarrow 0} \Vert P_U F(r^\gamma \tilde{D}) P_U \Vert _{L^1(\mathcal {E}) \rightarrow L^{1,\infty }(\mathcal {E})} \end{aligned}$$

(21)

and, for all $p \in [1,\infty ]$,

$$\begin{aligned} \Vert F(D_0) \Vert _{L^p(\mathcal {T}^k_d) \rightarrow L^p(\mathcal {T}^k_d)} \le \liminf _{r\downarrow 0} \Vert P_U F(r^\gamma \tilde{D}) P_U \Vert _{L^p(\mathcal {E}) \rightarrow L^p(\mathcal {E})}, \end{aligned}$$

(22)

where $P_U$ is the operator of multiplication by $\mathbb {1}_U$.

Proof

The right-hand sides of (21) and (22) do not increase if we replace U by a smaller neighbourhood of x. Hence we may assume that U is open and is the domain of a coordinate chart $\phi : U \rightarrow V$ centred at x and of an orthonormal frame X for $\mathcal {E}|_U$ such that the conditions of Definition 5.1 are satisfied. We may also assume that the topological boundary $\partial U$ has zero measure in M and that V is bounded in $\mathbb {R}^d$. Let the map ${\varPsi } : C^\infty (\mathcal {E}|_U) \rightarrow C^\infty (\mathcal {T}^k_d|_V)$ and the differential operators $D_R$ ($R > 0$) be as in Definition 5.1.

Take an open set $U_* \supseteq U$ of full measure in M and such that $\phi$ and X can be extended to a coordinate chart $\phi _* : U_* \rightarrow V_* \subseteq \mathbb {R}^d$ and an orthonormal frame $X_*$ for $\mathcal {E}|_{U_*}$. To construct such a set $U_*$, take a countable open cover $\{U_n\}_{n\in \mathbb {N}}$ of M such that $U_0=U$, each $U_n$ carries local coordinates for M and a local frame for $\mathcal {E}$, and each $\partial U_n$ has zero measure in M, and then define $U_* = \bigcup _{n \in \mathbb {N}}(U_n \setminus \overline{\bigcup _{m<n} U_m})$.

We can then define ${\varPsi }_* : C^\infty (\mathcal {E}|_{U_*}) \rightarrow C^\infty (\mathcal {T}^k_d|_{V_*})$ in the same way as ${\varPsi }$ in (19), with $\phi _*$ and $X_*$ in place of $\phi$ and X. In particular ${\varPsi }_*|_{C^\infty _c(\mathcal {E}|_{U_*})}$ extends to an isometry $\tilde{\varPsi } : L^2(\mathcal {E}) \rightarrow L^2(\mathcal {T}^k_d|_{V_*})$.

Correspondingly we extend the differential operators $D_R$ to (densely defined) self-adjoint operators $\tilde{D}_R$ on $L^2(\mathcal {T}^k_d)$ as follows:

$$\begin{aligned} \tilde{D}_R f = R^\gamma (\tilde{D}_1 (f \circ \delta ^{-1}_R)) \circ \delta _R, \end{aligned}$$

where $\tilde{D}_1$ is the self-adjoint operator on $L^2(\mathcal {T}^k_d) = L^2(\mathcal {T}^k_d|_{V_*}) \oplus L^2(\mathcal {T}^k_d|_{\mathbb {R}^d \setminus V_*})$ given by

$$\begin{aligned} \tilde{D}_1 = \left( \begin{array}{c|c} \tilde{\varPsi } \tilde{D} \tilde{\varPsi }^{-1} &{} 0 \\ \hline 0 &{} 0 \end{array}\right) . \end{aligned}$$

In particular $\tilde{D}_R f = D_R f$ for all $f \in C^\infty _c(\mathcal {T}^k_d|_{\delta _R^{-1}(V)})$, and

$$\begin{aligned} \lim _{R \downarrow 0} \tilde{D}_{R} f = D_0 f \end{aligned}$$

in $L^2$ for all $f \in C^\infty _c(\mathcal {T}^k_d)$.

Since $D_0$ is essentially self-adjoint on $C^\infty _c(\mathcal {T}^k_d)$, by [41, §VIII.1.1, Corollary 1.6] we conclude that $\tilde{D}_{R}$ converges strongly in the generalized sense to $D_0$ as $R \downarrow 0$ (as defined in [41, §VIII.1.1, p. 429]). Consequently, by [41, §VIII.1.1, Corollary 1.4], the resolvents $(\tilde{D}_{R} - \zeta )^{-1}$ converge strongly to $(D_0 - \zeta )^{-1}$ as $R \downarrow 0$ for all $\zeta \in \mathbb {C}\setminus \mathbb {R}$. An application of the Stone–Weierstraß theorem to the class of functions $F \in C_0(\mathbb {R})$ such that

$$\begin{aligned} F(\tilde{D}_{R}) \rightarrow F(D_0) \quad \text {and}\quad \overline{F}(\tilde{D}_{R}) \rightarrow \overline{F}(D_0) \qquad \text {strongly as }R \downarrow 0 \end{aligned}$$

(23)

shows that (23) holds for all $F \in C_0(\mathbb {R})$.

On the other hand, by construction,

$$\begin{aligned} F(\tilde{D}_1) f = \tilde{\varPsi } F(\tilde{D}) \tilde{\varPsi }^{-1} f \end{aligned}$$

for all $f \in L^2(\mathcal {T}^k_d|_V)$, hence

$$\begin{aligned} F(\tilde{D}_R) f = (\tilde{\varPsi } F(R^{\gamma } \tilde{D}) \tilde{\varPsi }^{-1} (f \circ \delta ^{-1}_{R})) \circ \delta _R \end{aligned}$$

for all $f \in L^2(\mathcal {T}^k_d|_{\delta _R^{-1}(V)})$.

We now prove (21). Since $F(D_0)$ is $L^2$-bounded, for all $f \in L^1 \cap L^2(\mathcal {T}^k_d)$ and $\alpha > 0$, if $f_n \in C^\infty _c(\mathcal {T}^k_d)$ and $f_n \rightarrow f$ in $L^1 \cap L^2$, then, by Markov’s inequality,

$$\begin{aligned} \mu _0\Bigl (\Bigl \{ |F(D_0)f|> \alpha \Bigr \}\Bigr ) \le \sup _{r>0} \inf _{\epsilon>0} \liminf _{n \rightarrow \infty } \mu _0\Bigl (B_r \cap \Bigl \{ |F(D_0) f_n| > \alpha /(1+\epsilon ) \Bigr \}\Bigr ) \end{aligned}$$

where $\mu _0$ is the Lebesgue measure and $B_r$ is the closed ball of radius r in $\mathbb {R}^d$ centred at the origin. Hence it is sufficient to prove that, for all compact sets $K \subseteq H$, all $f \in C^\infty _c(\mathcal {T}^k_d)$, and all $\alpha > 0$,

$$\begin{aligned} \mu _0\Bigl (K \cap \Bigl \{ |F(D_0)f| > \alpha \Bigr \}\Bigr ) \le \kappa _F \Vert f \Vert _1 / \alpha \end{aligned}$$

where $\kappa _F = \liminf _{R \downarrow 0} \kappa _{F,R}$ and $\kappa _{F,R} = \Vert P_U F(R^{\gamma } \tilde{D}) P_U \Vert _{L^1(\mathcal {E}) \rightarrow L^{1,\infty }(\mathcal {E})}$. On the other hand, by (23) and Markov’s inequality,

$$\begin{aligned} \mu _0\Bigl (K \cap \Bigl \{ |F(D_0)f|> \alpha \Bigr \}\Bigr ) \le \inf _{\epsilon>0} \liminf _{R\downarrow 0} \mu _0\left( K \cap \left\{ |F(\tilde{D}_R)f| > \alpha /(1+\epsilon ) \right\} \right) . \end{aligned}$$

Let R be sufficiently small that $\tilde{K} := {{\mathrm{supp}}}f \cup K \subseteq \delta _R^{-1}(V)$. Let $\mu$ be the measure on M. Let $a \in C^\infty (V)$ be as in Definition 5.1, and set $A_R = \max _{\delta _R(\tilde{K})} a^{1/2}$ and $B_R = \max _{\delta _R(\tilde{K})} a^{-1/2}$. Let Q be the homogeneous dimension associated with the dilations $\delta _R$. Then

$$\begin{aligned} \begin{aligned} \mu _0&\left( K \cap \left\{ |F(\tilde{D}_R)f|> \alpha \right\} \right) \\&= R^{-Q} \mu _0\left( \delta _R(K) \cap \left\{ |\tilde{\varPsi } F(R^{\gamma }\tilde{D})\tilde{\varPsi }^{-1}(f\circ \delta ^{-1}_R)|> \alpha \right\} \right) \\&\le B_R^2 \, R^{-Q} \mu \left( U \cap \left\{ |F(R^{\gamma }\tilde{D}){\varPsi }^{-1}(f\circ \delta ^{-1}_R)| > \alpha /A_R \right\} \right) \\&\le B_R^2 \,A_R \, R^{-Q} \,\kappa _{F,R} \, \Vert {\varPsi }^{-1}(f \circ \delta _R^{-1}) \Vert _{L^1(\mathcal {E})} / \alpha \\&\le B_R^2 \, A^2_R \, R^{-Q} \,\kappa _{F,R} \, \Vert f \circ \delta _R^{-1} \Vert _{L^1(\mathcal {T}^k_d)} / \alpha \\&= B_R^2 \, A^2_R \,\kappa _{F,R} \, \Vert f \Vert _{L^1(\mathcal {T}^k_d)} / \alpha \end{aligned} \end{aligned}$$

and in particular, since $\lim _{R \downarrow 0} A_R B_R = 1$,

$$\begin{aligned} \liminf _{R \downarrow 0} \mu _0\left( K \cap \left\{ |F(\tilde{D}_R)f| > \alpha \right\} \right) \le \kappa _F \, \Vert f\Vert _{L^1(\mathcal {T}^k_d)}/\alpha . \end{aligned}$$

The proof of (22) is similar and follows the lines of the proof of [42, Theorem 2], keeping track of the constants. $\square$

Corollary 5.3

With the notation and hypotheses of Theorem 5.2, assume moreover that $\tilde{D}$ is nonnegative. Then $D_0$ is nonnegative and

$$\begin{aligned} \varsigma (D_0) \le \varsigma (\tilde{D}) \qquad \text {and}\qquad \varsigma _-(D_0) \le \varsigma _-(\tilde{D}). \end{aligned}$$

Proof

We may assume that $\tilde{D} \ne 0$, otherwise $D_0=0$ by (20) and the result is trivial. Let ${\varSigma }$ be the $L^2$-spectrum of $\tilde{D}$. Then, from the definition of $\varsigma$ and the fact that $\Vert F(\tilde{D})\Vert _{L^2 \rightarrow L^2} = \sup _{\lambda \in {\varSigma }} |F(\lambda )|$ for all continuous F, it follows easily that $\varsigma (\tilde{D}) \ge 1/2$ (see, e.g., [62, §2.6.2, proof of Theorem 1] or [47, Proposition 2.3.15]).

Take $s > \varsigma (\tilde{D})$ and let $\kappa _s \in (0,\infty )$ be such that

$$\begin{aligned} \Vert F(\tilde{D}) \Vert _{L^2 \rightarrow L^2} + \Vert F(\tilde{D}) \Vert _{L^1 \rightarrow L^{1,\infty }} \le \kappa _s \, \Vert F\Vert _{L^2_{s,\mathrm {sloc}}}. \end{aligned}$$

(24)

for all bounded Borel functions $F : \mathbb {R}\rightarrow \mathbb {C}$. We show now that a similar inequality holds with $D_0$ in place of $\tilde{D}$.

Take any F such that $\Vert F\Vert _{L^2_{s,\mathrm {sloc}}} < \infty$. Since $s>1/2$, by Sobolev embedding F is continuous and bounded on $(0,\infty )$. Arguing as in the proof of Corollary 3.3(iii), it is not restrictive to assume that $F(0) = 0$. Let $\eta _n \in C_c^\infty ((0,\infty ))$, for $n \in \mathbb {Z}$, be as in the proof of Proposition 4.4 and define $F_N = \sum _{|n|\le N} F \eta _n$. Theorem 5.2 and (24) then give that

$$\begin{aligned} \Vert F_N(D_0) \Vert _{L^2 \rightarrow L^{2}} + \Vert F_N(D_0) \Vert _{L^1 \rightarrow L^{1,\infty }} \le \kappa _s \, \Vert F_N \Vert _{L^2_{s,\mathrm {sloc}}} \le C_s \, \kappa _s \, \Vert F\Vert _{L^2_{s,\mathrm {sloc}}}. \end{aligned}$$

for all $N \in \mathbb {N}$. On the other hand, $F_N(D_0) \rightarrow F(D_0)$ strongly as $N \rightarrow \infty$. Consequently

$$\begin{aligned} \Vert F(D_0) \Vert _{L^2 \rightarrow L^{2}} + \Vert F(D_0) \Vert _{L^1 \rightarrow L^{1,\infty }} \le C_s \, \kappa _s \, \Vert F \Vert _{L^2_{s,\mathrm {sloc}}}. \end{aligned}$$

This proves that $\varsigma (\tilde{D}) \ge \varsigma (D_0)$. The other inequality is proved analogously. $\square$

We now apply these results to the Kohn Laplacian. To this purpose we exploit the analysis of [3].

Proposition 5.4

Let M be a CR manifold of hypersurface type and dimension $2n-1$, equipped with a compatible hermitian metric. Let $\Box _b$ be the Kohn Laplacian acting on section of the bundle ${\varLambda }^{0,j} M$ of (0, j)-forms, where $j \in \{0,\dots ,n-1\}$. Let $\mathcal {J} = \{ J \subseteq \{1,\dots ,n-1\} :|J|=j\}$ and let $\mathcal {T}^\mathcal {J}$ be the trivial bundle over $\mathbb {R}^{2n-1}$ with fibre $\mathbb {C}^\mathcal {J}$. Let $y \in M$ and let $\lambda _1,\dots ,\lambda _{n-1}$ be the eigenvalues of the Levi form of M at p. Define vector fields on $\mathbb {R}^{2n-1}$ by

$$\begin{aligned} U_0 = \frac{\partial }{\partial u_0}, \quad U_k = \frac{\partial }{\partial u_k} + 2 \lambda _k \, u_{n-1+k} \frac{\partial }{\partial u_0}, \quad U_{n-1+k} = \frac{\partial }{\partial u_{n-1+k}} - 2 \lambda _k \, u_{k} \frac{\partial }{\partial u_0} \end{aligned}$$

for $k =1,\dots ,n-1$. Let $\Box _b^y : C^\infty (\mathcal {T}^{\mathcal J}) \rightarrow C^\infty (\mathcal {T}^{\mathcal J})$ be the differential operator defined by

$$\begin{aligned} \Box _b^y (f_J)_{J \in \mathcal {J}} = (\Box _b^{y,J} f_J)_{J \in \mathcal {J}}, \end{aligned}$$

(25)

where

$$\begin{aligned} {-}4 \, \Box _b^{y,J} = \sum _{k=1}^{2n-2} U_k^2 +4 i \left( \sum _{k \in J} \lambda _k - \sum _{k \notin J} \lambda _k\right) U_0. \end{aligned}$$

(26)

Then $\Box _b^y$ is a local model for $\Box _b$ at p of order 2 with respect to the dilations

$$\begin{aligned} \delta _R(u_0,u_1,\dots ,u_{2n-2}) = (R^2 u_0,R u_1,\dots ,R u_{2n-2}) \end{aligned}$$

on $\mathbb {R}^{2n-1}$.

Proof

This follows from the results of [3, Chapter 4]. Indeed, if the complex vector fields $Z_k$ and $Z_k^y$ are defined as in [3, eqs. (21.1) and (21.5)], and a is any smooth function on the domain of the $Z_k$, then it is easily seen that, for all $f \in C^\infty _c(\mathbb {R}^{2n-1})$,

$$\begin{aligned} R (Z_k ( f \circ \delta _R^{-1})) \circ \delta _R&\rightarrow Z_k^y f \\ (a (f \circ \delta _R^{-1})) \circ \delta _R&\rightarrow a(0) f \end{aligned}$$

as $R \downarrow 0$, in the LF-space topology of $C^\infty _c(\mathbb {R}^{2n-1})$. Consequently, by composition, for all nowhere zero smooth functions a and for all $f \in C^\infty _c(\mathbb {R}^{2n-1})$,

$$\begin{aligned} R^2 (a^{-1} Z_k \overline{Z}_l ((af) \circ \delta _R^{-1}) \circ \delta _R \rightarrow Z_k^y \overline{Z}_l^y f \end{aligned}$$

as $R \downarrow 0$, in the topology of $C^\infty _c(\mathbb {R}^{2n-1})$, and also in $L^2$. From this it is not difficult to see that the operator $\Box _b^y$ given by [3, eq. (21.10)] is a local model, in the sense of our Definition 5.1, of the operator $\Box _b$ given by [3, eq. (20.43)]. In order to conclude, it is sufficient to observe that [3, eq. (22.1)], which corresponds to (25) and (26) above, is obtained from [3, eq. (21.10)] by means of the coordinate changes [3, eqs. (21.14) and (21.21)], which commute with the dilations $\delta _R$. $\square$

Note that the vector fields $U_0,U_1,\dots ,U_{2n-2}$ of Proposition 5.4 are left-invariant with respect to the nilpotent Lie group law on $\mathbb {R}^{2n-1}$ defined by

$$\begin{aligned}&(u_0,u_1,\dots ,u_{2n-2}) \cdot (u'_0,u'_1,\dots ,u'_{2n-2}) \nonumber \\&\qquad = \Biggl (u_0+u_0'+ 2\sum _{k=1}^{n-1} \lambda _k (u_k' u_{n-1+k} - u_k u_{n-1+k}'),u_1+u_1',\dots ,u_{2n-2}+u_{2n-2}'\Biggr )\nonumber \\ \end{aligned}$$

(27)

(cf. [3, Definition (1.14)]). Hence the operators $\Box _b^{y,J}$ and $\Box _b^y$ are essentially self-adjoint (see, e.g., [56] or [48, Proposition 3.2]).

It is now immediate that Theorems 1.1 and 1.2 may be transplanted to the Heisenberg group.

Proof of Corollary 1.3

The sphere $\mathbb {S}$ is a strictly pseudoconvex CR manifold with a Levi metric [59, Definition 1.5], so the eigenvalues of the Levi form at each point are all equal to 1. Hence, if we apply Proposition 5.4 to the Kohn Laplacian $\Box _b^\mathbb {S}$ on the sphere, we obtain as local model (at any point) an operator that corresponds, in suitable coordinates and trivializations, to the Kohn Laplacian $\Box _b^H$ on the Heisenberg group with the standard strictly pseudoconvex structure (see [28, §§4–5]). Consequently $\varsigma (\Box _b^H) \le \varsigma (\Box _b^\mathbb {S})$ by Corollary 5.3. $\square$

As for Theorem 1.4, the following variation of a result of [50] will be of use.

Proposition 5.5

Let G be a 2-step stratified group of topological dimension d, $\mathfrak {L}$ a homogeneous sublaplacian thereon and V a left-invariant vector field in the second layer of G, such that $\mathfrak {L}\ge iV$. Then

$$\begin{aligned} \varsigma _-(\mathfrak {L}-iV) \ge d/2. \end{aligned}$$

Proof

The proof follows the lines of the argument in [50, Section 2], where the case $V =0$ is treated. Here we just list the main steps and modifications needed.

As in [50, Section 2], let Q be the homogeneous dimension of G, $\mathfrak {g}_2 \cong \mathbb {R}^{d_2}$ be the second layer of the Lie algebra of G and $\mathbf {U}$ be the corresponding vector of central derivatives. Then $-i V = \beta \cdot \mathbf {U}$ for some $\beta \in \mathfrak {g}_2$. Similarly as in [50, eq. (10)], define ${\varOmega }_{\beta ,t}^{\chi ,\theta }$ as the convolution kernel of $m_t^\chi (\mathfrak {L}+\beta \cdot \mathbf {U}) \, \theta (t \mathbf {U})$.

By arguing as in [50, Proposition 5], one obtains an expression for ${\varOmega }_{\beta ,t}^{\chi ,\theta }$ analogous to the right-hand side of [50, eq. (11)], where $I^\theta$ is replaced by

$$\begin{aligned}&I^\theta _\beta (t,s,r,y,v) = \int _{\mathfrak {g}_2^*} \exp (i t {\varPhi }(y,v,\mu )-is{\varSigma }(y,v)-irR(s,r,y,\mu )) \\&\qquad \times \exp (i(1-sr) \langle \mu , \beta \rangle ) \, B(sr,\mu ) \,\theta (\mu ) \,d\mu . \end{aligned}$$

Hence the argument of [50, Proposition 7] gives that

$$\begin{aligned} t^{Q-d/2} \, {\varOmega }_{\beta ,t}^{\chi ,\theta }(2ty,t^2 v) = e^{i\pi d_1/4} e^{it{\varPsi }_\beta (y,v)} A^{\chi ,\theta }(y,v) + O(t^{-1}), \end{aligned}$$

where ${\varPsi }_\beta (y,v) = {\varPsi }(y,v) + \beta \cdot \mu ^c(y,v)$.

One can then repeat the proof of [50, Theorem 8] to obtain that

$$\begin{aligned} \Vert m_t^\chi (\mathfrak {L}+\beta \cdot \mathbf {U}) \Vert _{p\rightarrow p} \ge C_{p,\chi ,\beta } \, t^{d(1/p-1/2)}, \qquad p \in [1,2], \, t \ge 1. \end{aligned}$$

Comparison of this estimate with [50, eq. (9)] gives the conclusion. $\square$

Proof of Theorem 1.4

As in Proposition 5.4, let $\lambda _1,\dots ,\lambda _{n-1}$ be the eigenvalues of the Levi form at a point $y \in M$. Since M is not Levi-flat, we can choose $y \in M$ so that not all $\lambda _j$ are zero. Hence the group law (27) defines a 2-step stratified structure on $\mathbb {R}^{2n-1}$, for which $\mathfrak {L}= -\sum _{k=1}^{2n-2} U_k^2$ is a homogeneous sublaplacian and $U_0$ is a vector field in the second layer. Consequently, by Corollary 5.3 and Proposition 5.5,

$$\begin{aligned} \varsigma _-(\Box _b) \ge \varsigma _-(\Box _b^y) = \max _{J \in \mathcal {J}} \varsigma _-(\Box _b^{y,J}) \ge (2n-1)/2 \end{aligned}$$

and we are done. $\square$

6 An abstract multivariable multiplier theorem

In this section we prove a multiplier theorem for commuting operators in the setting of a doubling metric measure space X. More precisely, we will consider a system of strongly commuting, possibly unbounded self-adjoint operators $U_1,\dots ,U_n$ on $L^2(X)$. Here “strongly commuting” means that such operators have a joint spectral resolution E on $\mathbb {R}^n$, so

$$\begin{aligned} U_j = \int _{\mathbb {R}^n} \lambda _j \,dE(\lambda ) \end{aligned}$$

for $j=1,\dots ,n$, and a joint functional calculus is defined by

$$\begin{aligned} F(U_1,\dots ,U_n) = \int _{\mathbb {R}^n} F(\lambda ) \,dE(\lambda ) \end{aligned}$$

for all Borel functions $F : \mathbb {R}^n \rightarrow \mathbb {C}$. In the following we aim at giving a sufficient condition for the weak type (1, 1) and $L^p$-boundedness of an operator $F(U_1,\dots ,U_n)$ in terms of a smoothness condition of Mihlin–Hörmander type on the multiplier F:

$$\begin{aligned} \sup _{r \ge 0} \Vert (F \circ \epsilon _r) \, \chi \Vert _{L^\infty _s(\mathbb {R}^n)} < \infty . \end{aligned}$$

(28)

for some $s\ge 0$ sufficiently large. Here $(\epsilon _r)_{r>0}$ is a system of dilations on $\mathbb {R}^n$, $\epsilon _0 : \mathbb {R}^n \rightarrow \mathbb {R}^n$ is the constant function 0, and $\chi \in C^\infty _c(\mathbb {R}^n \setminus \{0\})$ is a cutoff supported on an annulus.

A model result for us is the classical Mihlin–Hörmander theorem for Fourier multipliers on $\mathbb {R}^n$, thought of as a multiplier theorem for the joint functional calculus of the partial differential operators $U_j = -i\partial _j$ on $\mathbb {R}^n$. In this case isotropic dilations $\epsilon _r(\lambda ) = r\lambda$ are considered and a condition (28) of order $s>n/2$ is required.

In the general setting described above, clearly some assumptions on the operators $U_1,\dots ,U_n$ are necessary, other than self-adjointness and strong commutativity, in order to prove $L^p$-bounds for $p \ne 2$. This problem has been studied extensively in the case $n=1$, especially for a single nonnegative self-adjoint operator $\mathfrak {L}= U_1$; in this case the choice of dilations $\epsilon _r$ is irrelevant and (28) reduces to (2). Several assumptions on $\mathfrak {L}$ have been considered in the literature (see, e.g., [4, 10, 19, 20, 33, 45, 46] and references therein), which usually involve estimates for the heat propagator $e^{-t\mathfrak {L}}$ or the wave propagator $\cos (t\sqrt{\mathfrak {L}})$ associated with $\mathfrak {L}$. When bounds on the heat propagator (such as Gaussian-type bounds) are assumed, a classical proof strategy (which appears to originate in the study of group-invariant differential operators on Lie groups, see [1, 9, 12, 13, 29, 37, 38, 52] and references therein) consists in a “change of variable”, i.e., writing $F(\mathfrak {L}) = G(e^{-t\mathfrak {L}})$ for $G(s) = F(-t^{-1}\log s)$, and then using the particularly favourable bounds on the heat propagator $e^{-t\mathfrak {L}}$ to obtain $L^p$-boundedness of $G(e^{-t\mathfrak {L}})$ whenever G is sufficiently smooth and supported away from 0; since the “change of variables” is given by a smooth function, smoothness properties of G can be reduced to smoothness properties of F.

For a system $U_1,\dots ,U_n$ of several commuting operators, there seems not to be an obvious standard generalization of heat propagator bounds. In [58] the case of direct products is considered, where $X = X_1 \times \dots \times X_n$ and each $U_j$ operates on a different factor $X_j$ of the product; there a multiplier theorem for the system $U_1,\dots ,U_n$ is proved by assuming bounds for each heat propagator $e^{-tU_j}$ on $X_j$. However there are many systems of commuting operators that are not in “direct product form”. Numerous examples come from the setting of Lie groups and homogeneous spaces, such as the systems of commuting differential operators associated to Gelfand pairs [25, 31, 35, 65]. In these cases, one can usually find an operator $\mathfrak {L}= P(U_1,\dots ,U_n)$ for some polynomial P such that good bounds hold for the heat propagator $e^{-t\mathfrak {L}}$, as well as for the “$U_j$-derivatives” $U_j e^{-t\mathfrak {L}}$ of the propagator. In [48, 49] such systems of commuting group-invariant differential operators on Lie groups are studied and, in the case of homogeneous operators on a homogeneous Lie group, a multiplier theorem of Mihlin–Hörmander type is proved. The basic idea in the proof is again to use a change of variables such as $F(U_1,\dots ,U_n) = G(e^{-t\mathfrak {L}},U_1 e^{-t\mathfrak {L}},\dots ,U_n e^{-t\mathfrak {L}})$ in order to reduce the functional calculus of the original (unbounded) operators to the functional calculus of operators satisfying good bounds.

What follows can be considered as an extension of the multiplier theorem of [49, §4] to the setting of abstract operators on a doubling metric measure space. To this purpose, here we take as an assumption the existence of a “change of variables” with suitable smoothness and invertibility properties and such that the operators resulting from this change satisfy suitably good bounds. In order for our result to encompass the various different cases mentioned above, we do not prescribe a specific form for this “change of variables”, and we state fairly minimal hypotheses that are enough for the argument to work.

So far we have been vague on the “good bounds” to be required on the operators resulting from the “change of variables”. In the case $n=1$, a typical assumption is given by Gaussian-type heat kernel bounds, i.e., superexponential decay in space of the integral kernel of the heat propagator. However in [33] it is shown that polynomial decay of arbitrarily large order is sufficient. Following [33], an analogous polynomial decay assumption is stated in our result below. This assumption is sufficient to prove the weak type (1, 1) of $F(U_1,\dots ,U_n)$ under a condition of the form (28) of order $s>Q/2$, where Q is the “homogeneous dimension” of X (see definitions below).

Note that in the smoothness condition (28) an $L^\infty$ Sobolev norm is used. It would be interesting to investigate whether sharper results involving other Sobolev norms are possible. In the case $n=1$, this problem is extensively discussed in [19], where further assumptions on the operator, such as “Plancherel-type estimates”, are used to replace $L^\infty _s$ with $L^q_s$ for some s. A similar approach is used in [49] in the case of systems of group-invariant operators on Lie groups. For the sake of brevity, we will not pursue this here.

Let $(X,\varrho ,\mu )$ be a doubling metric measure space of homogeneous dimension Q and displacement parameter N. In other words, $(X,\varrho )$ is a metric space and $\mu$ is a positive regular Borel measure on X, such that all balls $B(x,r) = \{ y \in X :\varrho (x,y) < r\}$ have nonzero finite measure $V(x,r) = \mu (B(x,r))$, and moreover there exist $c',c'' > 0$ such that, for all $x,y \in X$, $\lambda ,r \in [0,\infty )$,

$$\begin{aligned} V(x,\lambda r)&\le c' (1+\lambda )^Q \, V(x,r), \end{aligned}$$

(29)

$$\begin{aligned} V(y, r)&\le c'' (1+\varrho (x,y)/r)^N \, V(x,r). \end{aligned}$$

(30)

The constants $Q,N,c',c''$ contained in the inequalities (29), (30) will be referred to as the “structure constants” of the doubling space $(X,\varrho ,\mu )$.

A basic consequence of the doubling property (29) is the following integrability property: for all $s>Q$ and $y \in X$,

$$\begin{aligned} \int _X (1+\varrho (x,y)/r)^{-s} \,d\mu (x) \le C_s \, V(y,r). \end{aligned}$$

(31)

In the following we will consider bounded linear operators T between Lebesgue spaces on X with an integral kernel, that is, a locally integrable function $K_T : X \times X \rightarrow \mathbb {C}$ such that

$$\begin{aligned} Tf(x) = \int _X K_T(x,y) \, f(y) \,d\mu (y) \end{aligned}$$

(32)

for all $f \in C_c(X)$ and $\mu$-almost all $x \in X$. Not all the operators of interest have an integral kernel, so we also consider a weaker notion: in case the function $K_T$ is just locally integrable on $\{ (x,y) \in X \times X :x \ne y\}$ and (32) holds for $\mu$-a.e. $x \notin {{\mathrm{supp}}}f$, then we say that $K_T$ is the off-diagonal kernel of T.

In dealing with integral kernels, we will make repeated use of certain mixed weighted Lebesgue norms on functions $K : X \times X \rightarrow \mathbb {C}$, defined as follows: for all $p \in [1,\infty ]$, $s \in [0,\infty )$, $r \in (0,\infty )$,

$$\begin{aligned} |||K|||_{p,s,r} = \mathop {\hbox {ess\,sup}}\limits _y V(y,r)^{1/p'} \Vert K(\cdot ,y) \, (1+\varrho (\cdot ,y)/r)^s \Vert _{L^p(X)} \end{aligned}$$

where $p'=p/(p-1)$ is the conjugate exponent. We will also write $|||K |||_{p,s}$ in place of $|||K |||_{p,s,1}$. It is worth noting that, if the operator T has integral kernel $K_T$, then

$$\begin{aligned} \Vert T\Vert _{1 \rightarrow 1} = |||K_T |||_{1,0,r} \end{aligned}$$

(33)

for all $r \in (0,\infty )$.

Theorem 6.1

Let $U_1,\dots ,U_n$ be strongly commuting, possibly unbounded self-adjoint operators on $L^2(X)$, with joint spectrum ${\varSigma } \subseteq \mathbb {R}^n$. Let $\gamma _1,\dots ,\gamma _n \in (0,\infty )$ and define dilations $\epsilon _r$ on $\mathbb {R}^n$ by $\epsilon _r(\lambda _1,\dots ,\lambda _n) = (r^{\gamma _1} \lambda _1,\dots ,r^{\gamma _n} \lambda _n)$ for all $r \in (0,\infty )$. Let ${\varSigma }_* = \overline{\bigcup _{r>0} \epsilon _r({\varSigma })}$. Assume that, for some $m \ge n$, there exists a continuous map ${\varPsi } = ({\varPsi }_1,\dots ,{\varPsi }_m) : \mathbb {R}^n \rightarrow \mathbb {R}^m$ such that:

• (A) ${\varPsi }(0) \ne 0$ and there exist an open neighbourhood ${\varOmega }$ of ${\varPsi }(0)$ in $\mathbb {R}^m$ and a smooth map ${\varPhi } : {\varOmega } \rightarrow \mathbb {R}^n$ such that ${\varPhi } \circ {\varPsi }$ is the identity on ${\varPsi }^{-1}({\varOmega }) \cap {\varSigma }_*$;

• (B) for all $j=1,\dots ,m$ and $r>0$, the operator ${\varPsi }_j \circ \epsilon _r(U_1,\dots ,U_n)$ has an integral kernel and

  $$\begin{aligned} \sup _{r>0} |||K_{{\varPsi }_j \circ \epsilon _r(U_1,\dots ,U_n)} |||_{2,a,r} <\infty \end{aligned}$$

  (34)

  for all $a \in [0,\infty )$ and $j=1,\dots ,m$.

Then the following estimates hold.

• (i) For all bounded Borel functions $F : \mathbb {R}^{n} \rightarrow \mathbb {C}$ supported in $[-1,1]^{n}$, the operator $F \circ \epsilon _r(U_1, \dots , U_n)$ has an integral kernel for all $r \in (0,\infty )$ and

  $$\begin{aligned} \sup _{r>0} |||K_{F \circ \epsilon _r(U_1, \dots , U_n)} |||_{2,0,r} \le C \Vert F\Vert _{\infty }. \end{aligned}$$

• (ii) For all $p \in [1,\infty ]$, $b,s \in [0,\infty )$ with $s > b+Q(1/p-1/2)_+ + N(1/2-1/p)_+$ and all bounded Borel functions $F : \mathbb {R}^{n} \rightarrow \mathbb {C}$ supported in $[-1,1]^{n}$,

  $$\begin{aligned} \sup _{r>0} |||K_{F \circ \epsilon _r(U_1, \dots , U_n)} |||_{p,b,r} \le C_{p,s,b} \Vert F\Vert _{L^\infty _s}. \end{aligned}$$

• (iii) For all $F : \mathbb {R}^n \rightarrow \mathbb {C}$ in the Schwartz class, for all $a\in [0,\infty )$ and $p \in [1,\infty ]$,

  $$\begin{aligned} \sup _{r>0} |||K_{F \circ \epsilon _r(U_1,\dots ,U_n)} |||_{p,a,r} < \infty . \end{aligned}$$

• (iv) For all $s > Q/2$ and all bounded Borel functions $F : \mathbb {R}^{n} \rightarrow \mathbb {C}$ supported in $[-1,1]^{n}$,

  $$\begin{aligned} \sup _{r>0} \Vert F \circ \epsilon _r(U_1, \dots , U_n) \Vert _{1 \rightarrow 1} \le C_s \Vert F\Vert _{L^\infty _s}. \end{aligned}$$

• (v) Let E be the joint spectral resolution of $U_1,\dots ,U_n$. Then $E(\{0\})$ is bounded on $L^p(X)$ for all $p \in [1,\infty ]$. Moreover $E(\{0\})=0$ if $\mu (X) = \infty$.

• (vi) For all $s > Q/2$ and all bounded Borel functions $F : \mathbb {R}^{n} \rightarrow \mathbb {C}$, if

  $$\begin{aligned} \sup _{r>0} \Vert (F \circ \epsilon _r) \, \chi \Vert _{L^\infty _s} < \infty \end{aligned}$$

  for some cutoff $\chi \in C^\infty _c(\mathbb {R}^{n} \setminus \{0\})$ with $\bigcup _{r>0} \epsilon _r(\{\chi \ne 0\}) = \mathbb {R}^{n} \setminus \{0\}$, then the operator $F(U_1,\dots ,U_n)$ is of weak type (1, 1) and bounded on $L^p(X)$ for all $p \in (1,\infty )$, and moreover

  $$\begin{aligned} \Vert F(U_1,\dots ,U_n)\Vert _{L^1 \rightarrow L^{1,\infty }} \le C_{\chi ,s} \sup _{r\ge 0} \, \Vert (F \circ \epsilon _r) \, \chi \Vert _{L^\infty _s}. \end{aligned}$$

Here are some examples of applications of the above theorem.

Example 6.2

Let $X = \mathbb {R}^d$ with the Euclidean metric and Lebesgue measure, so $Q = d$. Then Theorem 6.1 can be applied with $n=d$, $\gamma _j=1$ and $U_j=-i\partial _j$ for $j=1,\dots ,d$, and ${\varPsi } : \mathbb {R}^d \rightarrow \mathbb {R}^{d+1}$ given by

$$\begin{aligned} {\varPsi }(\lambda _1,\dots ,\lambda _d) = ( e^{-(\lambda _1^2+\dots +\lambda _d^2)}, \lambda _1 \, e^{-(\lambda _1^2+\dots +\lambda _d^2)}, \dots , \lambda _d \, e^{-(\lambda _1^2+\dots +\lambda _d^2)}). \end{aligned}$$

Note that $F(-i\partial _1,\dots ,-i\partial _d)$ is the Fourier multiplier operator on $\mathbb {R}^d$ corresponding to the multiplier F. Hence Theorem 6.1(vi) in this case essentially reduces to the classical Mihlin–Hörmander theorem, with a smoothness condition of order $s>d/2$ on the multiplier. It is known that the threshold d / 2 is sharp in this case, so the condition $s> Q/2$ in Theorem 6.1(vi) cannot be weakened in general. By suitably adjusting the metric on $\mathbb {R}^d$ [34], one could also take arbitrary $\gamma _j \in [1,\infty )$, thus obtaining nonisotropic versions of the Mihlin–Hörmander theorem on $\mathbb {R}^d$ (cf. [23, 44]).

Example 6.3

Suppose that L is a nonnegative self-adjoint operator on $L^2(X)$ satisfying, for some $h \in (0,\infty )$, the following heat kernel estimate: for all $a \in [0,\infty )$,

$$\begin{aligned} \sup _{t>0} |||K_{e^{-tL}} |||_{2,a,t^{1/h}} < \infty . \end{aligned}$$

(35)

Then Theorem 6.1 can be applied with $n=1$, $U_1 = L$, $\gamma _1=h$ and ${\varPsi }(\lambda ) = e^{-\lambda }$. In the case $h>1$, the estimate (35) can be obtained by Hölder’s inequality from the following pointwise Gaussian-type heat kernel estimate: for all $t>0$ and $x,y \in X$,

$$\begin{aligned} |K_{e^{-t L}}(x,y)| \le C \, \, V(y,t^{1/h})^{-1} \exp \left( -b \left( \varrho (x,y)/t^{1/h}\right) ^{h/(h-1)}\right) . \end{aligned}$$

(36)

This is one of the usual assumptions in abstract spectral multiplier theorems for a single operator L (cf., e.g., [19, Assumption 2.2], where $h \ge 2$ is required).

Example 6.4

Let L and h be as in Example 6.3. Suppose moreover that D is a self-adjoint operator on $L^2(X)$, that commutes strongly with L and satisfies, for some $k \in (0,\infty )$, the following estimate: for all $a \in [0,\infty )$,

$$\begin{aligned} \sup _{t>0} t^{k/h} |||K_{D e^{-tL}} |||_{2,a,t^{1/h}} < \infty . \end{aligned}$$

(37)

Then Theorem 6.1 can be applied with $n=2$, $(U_1,U_2) = (L,D)$, $(\gamma _1,\gamma _2)=(h,k)$ and ${\varPsi }(\lambda _1,\lambda _2) = (e^{-\lambda _1},\lambda _2 e^{-\lambda _1})$. As before, in the case $h>1$, the estimate (37) can be obtained from the following Gaussian-type estimate for the “D-derivative” of the heat kernel of L: for all $t>0$ and $x,y \in X$,

$$\begin{aligned} |K_{D e^{-t L}}(x,y)| \le C \, t^{-k/h} \, V(y,t^{1/h})^{-1} \exp (-b (\varrho (x,y)/t^{1/h})^{h/(h-1)}). \end{aligned}$$

(38)

See, e.g., [2, 22, 61, 63] for examples of differential operators L, D satisfying (36) and (38); see also [47,48,49] for examples of commuting operators.

Example 6.5

Suppose that X is the product $X_1 \times X_2$ of two doubling metric measure spaces of homogeneous dimensions $Q_1$ and $Q_2$, so $Q = Q_1 + Q_2$. For $j=1,2$ let $L_j$ be a nonnegative self-adjoint operator on $L^2(X_j)$ satisfying the analogue of (35): for some $h_j \in (0,\infty )$ and all $a \in [0,\infty )$,

$$\begin{aligned} \sup _{t>0} |||K_{e^{-tL_j}} |||_{2,a,t^{1/h_j}} < \infty . \end{aligned}$$

(39)

Let $\tilde{L}_1 = L_1 \otimes I$ and $\tilde{L}_2 = I \otimes L_2$ be the corresponding operators on $L^2(X_1 \times X_2)$. Then Theorem 6.1 can be applied with $n=2$, $(U_1,U_2) = (\tilde{L}_1,\tilde{L}_2)$, $(\gamma _1,\gamma _2) = (h_1,h_2)$, ${\varPsi }(\lambda _1,\lambda _2) = (e^{-2\lambda _1-\lambda _2}, e^{-\lambda _1-2\lambda _2})$, since

$$\begin{aligned} K_{{\varPsi }_1 \circ \epsilon _r(\tilde{L}_1,\tilde{L}_2)}(x,y)&= K_{e^{-2 r^{h_1} L_1}}(x_1,y_1) \, K_{e^{-r^{h_2} L_2}}(x_2,y_2),\\ K_{{\varPsi }_2 \circ \epsilon _r(\tilde{L}_1,\tilde{L}_2)}(x,y)&= K_{e^{-r^{h_1} L_1}}(x_1,y_1) \, K_{e^{-2 r^{h_2} L_2}}(x_2,y_2) \end{aligned}$$

and

$$\begin{aligned} (1+\varrho (x,y)/r)^a \le (1+\varrho _1(x_1,y_1)/r)^a (1+\varrho _2(x_2,y_2)/r)^a. \end{aligned}$$

This gives an alternative proof and a generalization of the main result of [58], where Gaussian-type estimates like (36) are required for each $L_j$, and only the case $h_1 = h_2$ is considered.

Remark 6.6

In the case the map ${\varPsi }$ in Theorem 6.1 is smooth, one could replace the assumption (A) with the following:

• (A’) ${\varPsi }(0) \ne 0$, $d{\varPsi }(0)$ is injective, ${\varPsi }^{-1}({\varPsi }(0)) = \{0\}$ and ${\varPsi }^{-1}({\varGamma })$ is compact for some compact neighbourhood ${\varGamma }$ of ${\varPsi }(0)$ in $\mathbb {R}^m$.

Indeed, under the assumption (A’), the existence of the smooth local inverse ${\varPhi }$ as in (A) can be obtained from the constant rank theorem. On the other hand, smoothness of ${\varPsi }$ is never used in the proof of Theorem 6.1. In fact, one could even weaken the smoothness assumption on the local inverse ${\varPhi }$ of ${\varPsi }$ as follows:

• (A”) ${\varPsi }(0) \ne 0$ and there exist an open neighbourhood ${\varOmega }$ of ${\varPsi }(0)$ in $\mathbb {R}^m$ and a continuous map ${\varPhi } : {\varOmega } \rightarrow \mathbb {R}^n$ which is smooth on ${\varOmega } \setminus \{{\varPsi }(0)\}$ and such that ${\varPhi } \circ {\varPsi }$ is the identity on ${\varPsi }^{-1}({\varOmega })$.

However, under this weaker assumption, one would obtain weaker versions of items (ii), (iii), (iv) of Theorem 6.1, where F is constant in some neighbourhood of 0. A further generalization would be to consider several maps ${\varPsi }$ instead of a single one and assume that, for all $a \in [0,\infty )$, the kernel estimate (34) is satisfied by some ${\varPsi }$ that may depend on a. In the case $n=1$, this idea has been exploited in the study of the $L^p$ functional calculus for certain pseudodifferential operators L, where better and better estimates are available for the kernel of $e^{-tL^M}$ as $M \in \mathbb {N}$ grows [21]. For the sake of brevity, we will not pursue this here.

As mentioned above, one of the main ideas in the proof of Theorem 6.1 is to use the map ${\varPsi }$ as a “change of variables”, in order to replace the (possibly unbounded) operators $U_1,\dots ,U_n$ with the bounded operators ${\varPsi }_j \circ \epsilon _r(U_1,\dots ,U_n)$, $j=1,\dots ,m$. By means of this change of variables, the proof of Theorem 6.1 is essentially reduced to the following result, which is a multivariate extension of a result of [33].

For notational convenience, set

$$\begin{aligned} \exp _0(\lambda ) = e^\lambda -1. \end{aligned}$$

(40)

Proposition 6.7

Let $M_1,\dots ,M_m$ be pairwise commuting, self-adjoint bounded operators on $L^2(X)$, admitting integral kernels $K_1,\dots ,K_m$.

• (i) Suppose that, for some $\kappa \ge 0$ and $a > 0$,

  $$\begin{aligned} |||K_j|||_{2,0} \le \kappa , \qquad |||K_j|||_{1,a} \le \kappa \end{aligned}$$

  (41)

  for $j=1,\dots ,m$. Then, for all $h \in \mathbb {Z}^m$, the operator

  $$\begin{aligned} \exp _0(i (h_1 M_1 + \dots + h_m M_m)) \end{aligned}$$

  has an integral kernel $E_h$ satisfying

  $$\begin{aligned} |||E_h|||_{1,b} \le C_{\kappa ,a,b} \, |h|_1^{\gamma _{a,b}} \end{aligned}$$

  for all $b \in \left[ 0,a\right)$. Here $|h|_1 = |h_1| + \dots + |h_m|$ and the constants in the previous inequality depend only on the specified parameters and on the structure constants of $(X,\varrho ,\mu )$; in particular

  $$\begin{aligned} \gamma _{a,b} = 2^{\lfloor (b + Q/2)/(a-b) \rfloor } (1 + (b+Q/2)(1 + 1/(a-b))). \end{aligned}$$

  (42)

• (ii) Suppose moreover that, for some $b \in [0,a)$,

  $$\begin{aligned} |||K_j|||_{2,b} \le \kappa \end{aligned}$$

  for $j=1,\dots ,m$. Then, for all $h \in \mathbb {Z}^m$,

  $$\begin{aligned} |||E_h|||_{2,b} \le C_{\kappa ,a,b} \, |h|_1^{\gamma _{a,b} + 1}. \end{aligned}$$

  (43)

• (iii) Under the previous assumptions, if $F \in L^2_s(\mathbb {R}^m)$ for some

  $$\begin{aligned} s > \gamma _{a,b} + 1 + m/2 \end{aligned}$$

  and $F(0) = 0$, then the operator $F(M_1,\dots ,M_m)$ has an integral kernel satisfying

  $$\begin{aligned} |||K_{F(M_1,\dots ,M_m)} |||_{2,b} \le C_{m,\kappa ,a,b,s} \Vert F\Vert _{L^2_s}. \end{aligned}$$

Remark 6.8

Proposition 6.7 is stated in terms of the norms $|||\cdot |||_{p,s}$, instead of the more general $|||\cdot |||_{p,s,r}$. However the norm $|||\cdot |||_{p,s,r}$ can be considered as the norm $|||\cdot |||_{p,s}$ defined in terms of the rescaled distance $\varrho /r$. Moreover it is easily seen that the “rescaled space” $(X,\varrho /r,\mu )$ satisfies the same estimates (29) and (30) as the original space $(X,\varrho ,\mu )$, with the same structure constants. For this reason, it is not restrictive to omit the “scale parameter” r in Proposition 6.7 and other statements where just a single scale is used.

Before entering the proofs of the above statements, let us briefly recall a few basic facts about integral kernels. If an operator has integral kernel K, then its adjoint corresponds to the integral kernel $K^*$ given by

$$\begin{aligned} K^*(x,y) = \overline{K(y,x)}. \end{aligned}$$

Composition of operators corresponds (under suitable integrability conditions on the kernels) to the following “convolution” of integral kernels:

$$\begin{aligned} K_1 * K_2(x,y) = \int _X K_1(x,z) \, K_2(z,y) \,d\mu (z). \end{aligned}$$

The following lemma collects a few useful inequalities, among which is an extension of Young’s inequality for convolution.

Lemma 6.9

Let H, K be the integral kernels of the operators S, T respectively.

• (i) For all $a,b \in [0,\infty )$ and $1 \le p < q \le \infty$,

  $$\begin{aligned} |||K|||_{p,a} \le C_{p,q,a,b} \, |||K|||_{q,b} \qquad \text {if }b > a+ Q(1/p - 1/q). \end{aligned}$$

  (44)

• (ii) For all $a \in [0,\infty )$ and $1 \le p,q,r \le \infty$ with $1/p+1/q=1+1/r$,

  $$\begin{aligned} |||H * K|||_{r,a} \le (c'')^{1/p'} \, |||H|||_{p,a}^{p/r} \, |||H^*|||_{p,a+N/p'}^{p/q'} \, |||K|||_{q,a+N/p'}. \end{aligned}$$

  (45)

  The constants $c''$ and N in (45) are the same as in (30).

• (iii) For all $p \in [1,\infty ]$,

  $$\begin{aligned} |||H * K |||_{p,0} \le \Vert S\Vert _{p \rightarrow p} \, |||K|||_{p,0}. \end{aligned}$$

  (46)

Proof

• (i) This follows immediately from Hölder’s inequality and (31).

• (ii) Note that $1/r' = 1/p'+1/q'$. Define

  $$\begin{aligned} \tilde{H}_1(x,y)&= H(x,y) \, V(y,1)^{1/p'} (1+\varrho (x,y))^a,\\ \tilde{H}_2(x,y)&= H(x,y) \, V(x,1)^{1/p'} (1+\varrho (x,y))^{a+N/p'},\\ \tilde{K}(y,z)&= K(y,z) \, V(z,1)^{1/q'} (1+\varrho (y,z))^{a+N/p'}. \end{aligned}$$

  Then, by (30) and the inequality $(1+\varrho (x,z)) \le (1+\varrho (x,y)) \, (1+\varrho (y,z))$,

  $$\begin{aligned} \begin{aligned}&V(z,1)^{1/r'} |H(x,y) K(y,z)| (1+\varrho (x,z))^{a} \\&\qquad \le (c'')^{1/p'} | \tilde{H}_1(x,y)|^{p/r} |\tilde{H}_2(x,y)|^{p/q'} |\tilde{K}(y,z)|^{q/r} |\tilde{K}(y,z)|^{q/p'}. \end{aligned} \end{aligned}$$

  Integration in y and Hölder’s inequality yield

  $$\begin{aligned} \begin{aligned} V(z,1)^{1/r'}&|H*K(x,z)| (1+\varrho (x,z))^{a} \\&\le (c'')^{1/p'} \left( \int _X |\tilde{H}_1(x,y)|^{p} |\tilde{K}(y,z)|^{q} \,d\mu (y) \right) ^{1/r} \\&\qquad \times \left( \int _X |\tilde{H}_2(x,y)|^{p} \,d\mu (y) \right) ^{1/q'} \left( \int _X |\tilde{K}(y,z)|^{q} \,d\mu (y) \right) ^{1/p'}. \end{aligned} \end{aligned}$$

  By raising both sides to the power r, integrating in x and taking the essential supremum in z, the inequality (45) follows.

• (iii) This follows easily from the fact that $H*K(\cdot ,y) = S(K(\cdot ,y))$. ${\square }$

We are now ready to prove Theorem 6.1 and Proposition 6.7. Let us see first how Theorem 6.1 is derived from Proposition 6.7.

Proof of Theorem 6.1

By replacing ${\varPsi }$ with ${\varPsi } \circ \epsilon _r$ for sufficiently small $r>0$ and possibly permuting the components of ${\varPsi }$, we may assume the following:

• there exists $\psi \in C_c(\mathbb {R}^n)$ such that $\psi \cdot {\varPsi }_1 = 1$ on $[-1,1]^n$;

• there exist an open neighbourhood ${\varOmega }$ of ${\varPsi }([-1,1]^n)$ in $\mathbb {R}^m$ and a smooth map ${\varPhi } : {\varOmega } \rightarrow \mathbb {R}^n$ such that ${\varPhi } \circ {\varPsi }$ is the identity on ${\varPsi }^{-1}({\varOmega }) \cap \tilde{\varSigma }$.

• (i) If $F : \mathbb {R}^n \rightarrow \mathbb {C}$ is supported in $[-1,1]^n$, then $F = (F \psi ) {\varPsi }_1$, hence

  $$\begin{aligned} \begin{aligned} |||K_{F \circ \epsilon _r(U_1,\dots ,U_n)}|||_{2,0,r}&\le \Vert (F \psi ) \circ \epsilon _r(U_1,\dots ,U_n) \Vert _{2 \rightarrow 2} \, |||K_{{\varPsi }_1 \circ \epsilon _r(U_1,\dots ,U_n)} |||_{2,0,r} \\&\le \Vert F\Vert _\infty \, \Vert \psi \Vert _\infty \, |||K_{{\varPsi }_1 \circ \epsilon _r(U_1,\dots ,U_n)} |||_{2,0,r} \end{aligned} \end{aligned}$$

  for all $r>0$, and the conclusion follows by (34).

• (ii) We consider first the case $p=2$, i.e., we want to prove the inequality

  $$\begin{aligned} |||K_{F \circ \epsilon _r(U_1, \dots , U_n)} |||_{2,b,r} \le C_{s,b} \, \Vert F\Vert _{L^\infty _s}. \end{aligned}$$

  (47)

  for all $r > 0$, all $F : \mathbb {R}^n \rightarrow \mathbb {C}$ supported in $[-1,1]^n$, and all $s > b \ge 0$. Suppose first that $s > b + Q/2 + 2 + m/2$. Then we can find $a > 0$ sufficiently large that $s > \gamma _{a,b} + 1 + m/2$, where $\gamma _{a,b}$ is given by (42). By (34) and Hölder’s inequality (44), there exists $\kappa \in [0,\infty )$ such that

  $$\begin{aligned} |||K_{{\varPsi }_j \circ \epsilon _r(U_1,\dots ,U_n)}|||_{1,a,r} \le \kappa , \qquad |||K_{{\varPsi }_j \circ \epsilon _r(U_1,\dots ,U_n)}|||_{2,b,r} \le \kappa \end{aligned}$$

  for all $r > 0$ and $j = 1,\dots ,m$. Take some $\eta \in C^\infty _c(\mathbb {R}^m)$ supported in ${\varOmega }$ and identically 1 on ${\varPsi }([-1,1]^n)$. Then, for all $F : \mathbb {R}^n \rightarrow \mathbb {C}$ supported in $[-1,1]^n$,

  $$\begin{aligned} F = ((F \circ {\varPhi }) \, \eta ) \circ {\varPsi } \qquad \text {on } {\varSigma }_\epsilon , \end{aligned}$$

  so, for all $r > 0$,

  $$\begin{aligned} F \circ \epsilon _r(U_1,\dots ,U_n) = ((F \circ {\varPhi }) \, \eta ) ({\varPsi } \circ \epsilon _{r}(U_1,\dots ,U_n)). \end{aligned}$$

  By Proposition 6.7(iii) applied to the rescaled space $(X,\varrho /r,\mu )$ and the operators ${\varPsi }_j \circ \epsilon _{r}(U_1,\dots ,U_n)$, we then obtain that

  $$\begin{aligned} |||K_{F \circ \epsilon _r(U_1,\dots ,U_n)}|||_{2,b,r} \le C_{s,b} \Vert (F \circ {\varPhi }) \, \eta \Vert _{L^2_s} \le C_{s,b} \Vert F\Vert _{L^\infty _s}, \end{aligned}$$

  since $\eta$ is smooth and compactly supported and ${\varPhi }$ is smooth on ${{\mathrm{supp}}}\eta$. This gives (47) for $b \ge 0$, $s > b + Q/2 + 2 + m/2$. On the other hand, (47) also holds when $s=b=0$, by (i). The full range $s>b$ is then obtained by interpolation (cf., e.g., the proofs of [52, Lemma 1.2], [19, Lemma 4.3], [49, Theorem 2.7]). So the proof of the case $p=2$ is concluded. Note now that the case $p < 2$ follows from the case $p=2$ by Hölder’s inequality (44). As for $p>2$, take any real-valued $\xi \in C^\infty _c(\mathbb {R}^n)$ such that $\xi = 1$ on $[-1,1]^n$, and define q by $1/q=1/2+1/p$. Then $q<2$ and, from what we have just proved, it follows that

  $$\begin{aligned} \sup _{r>0} |||K_{\xi \circ \epsilon _r(U_1,\dots ,U_n)} |||_{q,a,r} < \infty \end{aligned}$$

  (48)

  for all $a \in [0,\infty )$. Moreover $F = \xi \cdot F$ for all $F : \mathbb {R}^n \rightarrow \mathbb {C}$ supported in $[-1,1]^n$; hence, by Young’s inequality (45) and the estimates (47) and (48),

  $$\begin{aligned} \begin{aligned} |||K_{F \circ \epsilon _r(U_1,\dots ,U_n)}&|||_{p,b,r} \\&\le C_p |||K_{\xi \circ \epsilon _r(U_1,\dots ,U_n)} \, |||_{q,b+N/q',r} |||K_{F \circ \epsilon _r(U_1,\dots ,U_n)} |||_{2,b+N/q',r} \\&\le C_{p,b} \Vert F\Vert _{L^\infty _s} \end{aligned} \end{aligned}$$

  for all $r \in (0,\infty )$ and $b,s \in [0,\infty )$ with $s > b+N/q' = b+N(1/2-1/p)$.

• (iii) Let $a \in [0,\infty )$ and $p\in [1,\infty ]$, and set

  $$\begin{aligned} s = a+Q(1/p-1/2)_+ + N(1/2-1/p)_+ +1. \end{aligned}$$

  Let $\phi _0 \in C_c(\mathbb {R}^n)$ and $\phi \in C_c(\mathbb {R}^n \setminus \{0\})$ be supported in $[-1,1]^n$ and such that $\phi _0 + \sum _{k > 0} \phi _k = 1$, where $\phi _k = \phi \circ \epsilon _{2^{-k}}$ for all $k > 0$. Then, by (ii), for all $r>0$,

  $$\begin{aligned} |||K_{(F \phi _0) \circ \epsilon _r(U_1,\dots ,U_n)}|||_{p,a,r} \le C_{p,a} \Vert F \phi _0 \Vert _{L^\infty _{s}}, \end{aligned}$$

  and moreover, for all $k \in \mathbb {N}\setminus \{0\}$,

  $$\begin{aligned} \begin{aligned} |||K_{(F \phi _k) \circ \epsilon _r(U_1,\dots ,U_n)}|||_{p,a,r}&\le C_p \, 2^{kQ/p'} \, |||K_{(F \phi _k) \circ \epsilon _r(U_1,\dots ,U_n)}|||_{p,a,2^{-k} r} \\&\le C_{p,a} \, 2^{kQ/p'} \, \Vert (F \circ \epsilon _{2^k}) \phi \Vert _{L^\infty _{s}}, \end{aligned} \end{aligned}$$

  by (29). Since it is easily seen that

  $$\begin{aligned} \Vert F \phi _0 \Vert _{L^\infty _{s}} + \sum _{k>0} 2^{kQ/p'} \, \Vert (F \circ \epsilon _{2^k}) \phi \Vert _{L^\infty _{s}} \end{aligned}$$

  is controlled by some Schwartz norm of F, the conclusion follows.

• (iv) This follows immediately from (ii) and (33).

• (v) Note that $E(\{0\}) = \mathbb {1}_{\{0\}}(U_1,\dots ,U_n)$. Hence, by (i),

  $$\begin{aligned} |||K_{E(\{0\})}|||_{2,0,r} \le C \end{aligned}$$

  uniformly in $r > 0$, so

  $$\begin{aligned} \int _X |K_{E(\{0\})}(x,y)|^2 \,d\mu (y) \le C V(y,r)^{-1} \end{aligned}$$

  for a.e. $y \in X$ and all $r>0$. If $\mu (X) = \infty$, then the right-hand side tends to zero as $r \rightarrow \infty$, hence $K_{E(\{0\})} = 0$ a.e. and consequently $E(\{0\}) = 0$. If instead $\mu (X) < \infty$, then by (29) we can find $r > 0$ such that $B(y,r) = X$ for all $y \in X$. Therefore, by Hölder’s inequality,

  $$\begin{aligned} |||K_{E(\{0\})}^*|||_{1,0} = |||K_{E(\{0\})}|||_{1,0} \le |||K_{E(\{0\})}|||_{2,0,r} \le C, \end{aligned}$$

  hence $\Vert E(\{0\})\Vert _{p\rightarrow p} \le C < \infty$ for $1 \le p \le \infty$ by (33) and interpolation.

• (vi) Note that $F = F(0) \mathbb {1}_{\{0\}} + F \mathbb {1}_{\mathbb {R}^n \setminus \{0\}}$, so

  $$\begin{aligned} F(U_1,\dots ,U_n) = F(0) E(\{0\}) + (F \mathbb {1}_{\mathbb {R}^n \setminus \{0\}})(U_1,\dots ,U_n). \end{aligned}$$

  Hence, because of (v), it is not restrictive to assume in the following that $F(0)=0$. In particular

  $$\begin{aligned} \Vert F(U_1,\dots ,U_n)\Vert _{2 \rightarrow 2} \le \Vert F\Vert _\infty \le C_{\chi ,s} \sup _{R>0} \Vert (F \circ \epsilon _R) \,\chi \Vert _{L^\infty _s}. \end{aligned}$$

  Let $\omega \in C^\infty _c(\mathbb {R}^n \setminus \{0\})$ be such that ${{\mathrm{supp}}}\omega \subseteq [-1,1]^n$ and $\sum _{k \in \mathbb {Z}} \omega _k(\lambda ) = 1$ for $\lambda \ne 0$, where $\omega _k = \omega \circ \epsilon _{2^{-k}}$. Take moreover a cutoff $\eta \in C^\infty _c(\mathbb {R}^n)$ which is identically 1 on $[-1,1]^n$. Hence, by (ii), the operators $A_t = \eta \circ \epsilon _t(U_1,\dots ,U_n)$ satisfy

  $$\begin{aligned} \sup _{t>0} |||K_{A_t} |||_{\infty ,b,t} < \infty \end{aligned}$$

  for all $b \in [0,\infty )$, and in particular they satisfy the “Poisson-type bounds” of [18, eq. (2) and (3)]. Fix $a \in (Q/2,s)$ and an integer $M > s$. For $r,t > 0$, $k \in \mathbb {Z}$ and a.e. $y \in X$, we then have

  $$\begin{aligned}&\int _{X \setminus B(y,r)} |K_{(F \omega _k)(U_1,\dots ,U_n) \, (I-A_t)}(x,y)|\, d\mu (x) \\&\qquad \le \left( \int _{X \setminus B(y,r)} \frac{(1 + 2^k \varrho (x,y))^{-2a}}{V(y,2^{-k})} \, d\mu (x) \right) ^{1/2} \, |||K_{(F \omega _k (1-\eta \circ \epsilon _t))(U_1,\dots ,U_n)} |||_{2,a,2^{-k}} \\&\qquad \le C_{\beta } (1+2^k r)^{Q/2 - a} \Vert (F \circ \epsilon _{2^k})\, \omega \Vert _{L^\infty _s} \Vert (1 - \eta ) \circ \epsilon _{2^k t} \Vert _{C^M({{\mathrm{supp}}}\omega )} \end{aligned}$$

  by Hölder’s inequality, (29) and (ii). Note now that the quantity $\Vert (1 - \eta ) \circ \epsilon _{t} \Vert _{C^M({{\mathrm{supp}}}\omega )}$ is continuous in t, vanishes for $t \le 1$ and is constant for t large, therefore is bounded uniformly in $t > 0$. We then conclude that the series $\sum _{k \in \mathbb {Z}} K_{(F \omega _k (1-\eta \circ \epsilon _t))(U_1,\dots ,U_n)}$ converges, off the diagonal of $X \times X$, to a function $K_t$ satisfying

  $$\begin{aligned} \begin{aligned} \mathop {\hbox {ess\,sup}}\limits _{y \in X} \int _{X \setminus B(y,r)} |K_t(x,y)|\, d\mu (x)&\le C_s \sum _{k :2^k> 1/t} (1+2^k r)^{Q/2-a} \Vert (F \circ \epsilon _{2^k}) \,\omega \Vert _{L^\infty _s} \\&\le C_{\chi ,s} (r/t)^{Q/2-a} \sup _{R>0} \Vert (F \circ \epsilon _R) \, \chi \Vert _{L^\infty _s}, \end{aligned} \end{aligned}$$

  and it is easily checked that $K_t$ is the off-diagonal kernel of

  $$\begin{aligned} (F (1 - \eta \circ \epsilon _t))(U_1,\dots ,U_n) = F(U_1,\dots ,U_n) \,(1-A_t). \end{aligned}$$

  If we take $r = t$ in the previous inequality, then [18, Theorem 1] implies that $F(U_1,\dots ,U_n)$ is of weak type (1, 1) and bounded on $L^p$ for $1 < p \le 2$, with norm controlled by $\sup _{R>0} \Vert (F \circ \epsilon _R) \, \chi \Vert _{L^\infty _s}$. For $2< p < \infty$, it is sufficient to apply the result just obtained to the function $\overline{F}$.$\square$

We are left with the proof of Proposition 6.7. The proof, which follows the lines of [33, §2], will require some preliminary considerations and lemmas.

By Young’s inequality (45), for all $a \in [0,\infty )$, the norm $|||\cdot |||_{1,a}$ is submultiplicative, i.e.,

$$\begin{aligned} |||K_1 * K_2 |||_{1,a} \le |||K_1 |||_{1,a} \, |||K_2 |||_{1,a}; \end{aligned}$$

consequently, the space of kernels

$$\begin{aligned} B_a^0 = \{K :|||K|||_{1,a}, |||K^*|||_{1,a} < \infty \}, \end{aligned}$$

equipped with the norm

$$\begin{aligned} \Vert K\Vert _{B_a} = \max \{ |||K|||_{1,a}, |||K^*|||_{1,a}\} \end{aligned}$$

(49)

and with the operations of convolution and involution, is a Banach $*$-algebra (with isometric involution).

Let us denote by $T_K$ the operator corresponding to the integral kernel K. The identity (33), together with interpolation, shows that the correspondence $K \mapsto T_K$ embeds $B_a^0$ continuously into the space $\mathcal {B}(L^p)$ of bounded operators on $L^p(X)$ for all $p \in [1,\infty ]$, with

$$\begin{aligned} \Vert K\Vert _{p \rightarrow p} := \Vert T_K\Vert _{p \rightarrow p} \le \Vert K\Vert _{B_a}. \end{aligned}$$

(50)

If the algebra $B_a^0$ has an identity element I, then we set $B_a = B_a^0$. Otherwise we formally introduce an identity I, i.e., we consider the unitization $B_a = \mathbb {C}I \oplus B_a^0$, which is a Banach $*$-algebra with norm

$$\begin{aligned} \Vert \lambda I + H \Vert _{B_a} = |\lambda | + \Vert H\Vert _{B_a} \end{aligned}$$

for every $\lambda \in \mathbb {C}$ and $H \in B_a^0$. Notice that, if we extend analogously the definition of the $|||\cdot |||_{1,a}$-norm by setting

$$\begin{aligned} |||\lambda I + H |||_{1,a} = |\lambda | + |||H|||_{1,a}, \end{aligned}$$

then the extension is still a submultiplicative norm, and (49) holds for every $K \in B_a$. Moreover the embedding into $\mathcal {B}(L^p)$ extends to the whole $B_a$, together with the inequality (50).

The introduction of the identity I makes the manipulation of kernels easier. For instance, for every $K \in B_a$, the exponential $e^K$ is defined via power series as an element of $B_a$, and $\Vert e^K\Vert _{B_a} \le e^{\Vert K\Vert _{B_a}}$; moreover the corresponding operator on $L^2(X)$ is nothing else than the exponential $e^{T_K}$. In the case K is a “genuine kernel”, i.e., $K \in B_a^0$, then $e^K \in I + B_a^0$ (as it is clear by inspection of the power series); therefore, by (40), $e^K - I \in B_a^0$ is the integral kernel of the operator $\exp _0(T_K)$.

Let $M_1,\dots ,M_m \in \mathcal {B}(L^2)$ be pairwise commuting, self-adjoint operators admitting integral kernels $K_1,\dots ,K_m$. According to the above discussion, under the hypothesis (41), for all $h \in \mathbb {Z}^m$, the operator $\exp _0(i (h_1 M_1 + \dots + h_m M_m))$ has integral kernel

$$\begin{aligned} E_h = A_1^{* h_1} * \dots * A_m^{* h_m} - I; \end{aligned}$$

here $A_j = e^{i K_j}$ and $A_j^{* h_j}$ is the iterated convolution of $|h_j|$ factors of the form $A_j$ or $A_j^*$, depending on the sign of $h_j$. Proposition 6.7(i) will then be proved by showing that, for all $b \in [0,a)$ and $h \in \mathbb {Z}^m \setminus \{0\}$,

$$\begin{aligned} \Vert A_1^{* h_1} * \dots * A_m^{* h_m}\Vert _{B_{b}} \le C_{\kappa ,a,b} |h|_1^{\gamma _{a,b}}. \end{aligned}$$

(51)

From now on, we will assume that $h \in \mathbb {N}^m$; the proof in the general case can be obtained by replacing some of the $A_j$ with $A_j^*$ in the argument below.

The idea is to decompose each $A_j$, or rather the kernel $A_j' = A_j - I$, into pieces supported at different distances from the diagonal. Namely, for some parameter $r \ge 1$ (which will be fixed later) we set

$$\begin{aligned} A_{j,k}(x,y) = {\left\{ \begin{array}{ll} A'_j(x,y) &{}\text {if }e^{k} r \le \varrho (x,y) < e^{k+1} r, \\ 0 &{}\text {otherwise,} \end{array}\right. } \end{aligned}$$

for $k > 0$, and $A_{j,0} = I + A'_{j,0}$, where

$$\begin{aligned} A'_{j,0}(x,y) = {\left\{ \begin{array}{ll} A'_j(x,y) &{}\text {if }\varrho (x,y) < er, \\ 0 &{}\text {otherwise.} \end{array}\right. } \end{aligned}$$

By dominated convergence, $A_j' = A'_{j,0} + \sum _{k > 0} A_{j,k}$ in $B_{a}^0$, and consequently $A_j = \sum _{k \ge 0} A_{j,k}$ in $B_{a}$. By so decomposing each factor in $A_1^{* h_1} * \dots * A_m^{* h_m}$, we obtain an infinite sum of products of the form

$$\begin{aligned} P_{\alpha ,\beta } = A_{\alpha _1,\beta _1} * \dots * A_{\alpha _n,\beta _n}, \end{aligned}$$

where $n = |h|_1$, $\alpha = (\alpha _1,\dots ,\alpha _n) \in \{1,\dots ,m\}^n$, $\beta = (\beta _1,\dots ,\beta _n) \in \mathbb {N}^n$; to simplify notation, we set $\mathcal {I}_n = \{1,\dots ,m\}^n \times \mathbb {N}^n$, and

$$\begin{aligned} |\beta |_H = \left| \{ u :\beta _u \ne 0 \}\right| . \end{aligned}$$

In order to estimate the $B_{b}$-norm of the products $P_{\alpha ,\beta }$, we use the fact that $\delta = a - b > 0$; hence, for all $k > 0$,

$$\begin{aligned} \Vert A_{j,k}\Vert _{2\rightarrow 2} \le \Vert A_{j,k}\Vert _{B_{b}} \le e^{-k\delta } r^{-\delta } \Vert A_j'\Vert _{B_{a}} \le e^{-k\delta } r^{-\delta } \tilde{\kappa }, \end{aligned}$$

(52)

where $\tilde{\kappa }= e^\kappa$. For $k = 0$ this does not work, however we can exploit cancellation from the $L^2$-theory: since $\Vert A_j\Vert _{2\rightarrow 2} = \Vert e^{iT_{K_j}}\Vert _{2 \rightarrow 2} \le 1$ by spectral theory,

$$\begin{aligned} \Vert A_{j,0}\Vert _{2 \rightarrow 2} \le \Vert A_j\Vert _{2\rightarrow 2} + \Vert A_j - A_{j,0}\Vert _{B_{b}} \le 1 + r^{-\delta } \tilde{\kappa }; \end{aligned}$$

(53)

moreover, by (41) and Young’s inequality (45),

$$\begin{aligned} |||A_{j,0}'|||_{2,0} \le |||A_{j}'|||_{2,0} \le \sum _{n>0} \left| \left| \left| \frac{(i K_j)^{*n}}{n!} \right| \right| \right| _{2,0} \le \sum _{n>0} \frac{\Vert K_j\Vert _{B_0}^{n-1} |||K_j|||_{2,0}}{n!} \le \tilde{\kappa }\end{aligned}$$

(54)

and the same holds for $(A_{j,0}')^*$.

Set $\tilde{b} = b + Q/2$.

Lemma 6.10

For all $n \in \mathbb {N}\setminus \{0\}$ and $(\alpha ,\beta ) \in \mathcal {I}_n$,

$$\begin{aligned} \Vert P_{\alpha ,\beta }\Vert _{B_{b}} \le C_{\kappa ,b} \, n \left( r \sum _{u=1}^n e^{\beta _u}\right) ^{\tilde{b}} e^{-\delta |\beta |_1} (r^{-\delta } \tilde{\kappa })^{|\beta |_H} (1+r^{-\delta } \tilde{\kappa })^n. \end{aligned}$$

(55)

In particular, if $n \le (r^{-\delta } \tilde{\kappa })^{-1}$, then

$$\begin{aligned} \Vert P_{\alpha ,\beta }\Vert _{B_{b}} \le C_{\kappa ,b} \, n^{1+\tilde{b}} \left( r e^{|\beta |_\infty }\right) ^{\tilde{b}} e^{-\delta |\beta |_1} (r^{-\delta } \tilde{\kappa })^{|\beta |_H}, \end{aligned}$$

(56)

where $|\beta |_\infty = \max \{\beta _1,\dots ,\beta _n\}$.

Proof

Note that (56) follows from (55) because $(1+r^{-\delta } \tilde{\kappa })^n \le (1+1/n)^n \le e$ whenever $n \le (r^{-\delta } \tilde{\kappa })^{-1}$. We are therefore reduced to proving (55).

By (52) and (53), a simpler estimate can be immediately obtained, for the $\mathcal {B}(L^2)$-norm: for all $n \in \mathbb {N}$ and $(\alpha ,\beta ) \in \mathcal {I}_n$,

$$\begin{aligned} \Vert P_{\alpha ,\beta }\Vert _{2 \rightarrow 2} \le e^{-\delta |\beta |_1} (r^{-\delta } \tilde{\kappa })^{|\beta |_H} (1+r^{-\delta } \tilde{\kappa })^n. \end{aligned}$$

From this, (46) and (54) we then get, for all $j\in \{1,\dots ,m\}$,

$$\begin{aligned} |||P_{\alpha ,\beta } * A'_{j,0} |||_{2,0} \le \tilde{\kappa }\, e^{-\delta |\beta |_1} (r^{-\delta } \tilde{\kappa })^{|\beta |_H} (1+r^{-\delta } \tilde{\kappa })^n. \end{aligned}$$

On the other hand, it is easily proved that $(P_{\alpha ,\beta } * A'_{j,0})(x,y)$ vanishes for $d(x,y) \ge R$, where $R = e(r + r \sum _{u=1}^n e^{\beta _u}) \ge 1$. Hence, by Hölder’s inequality and the doubling condition (29),

$$\begin{aligned} |||P_{\alpha ,\beta } * A'_{j,0}|||_{1,b} \le (1+R)^b |||P_{\alpha ,\beta } * A'_{j,0}|||_{1,0} \le C_b R^{\tilde{b}} |||P_{\alpha ,\beta } *A'_{j,0}|||_{2,0} \end{aligned}$$

and finally

$$\begin{aligned} |||P_{\alpha ,\beta } * A'_{j,0}|||_{1,b} \le C'_{\kappa ,b} \left( r + r \sum _{u=1}^n e^{\beta _u}\right) ^{\tilde{b}} e^{-\delta |\beta |_1} (r^{-\delta } \tilde{\kappa })^{|\beta |_H} (1+r^{-\delta } \tilde{\kappa })^n; \end{aligned}$$

(57)

here $C'_{\kappa ,b}$ depends only on $\kappa$, b and the constants in (29).

We now prove, for all $n \in \mathbb {N}\setminus \{0\}$ and $(\alpha ,\beta ) \in \mathcal {I}_n$, the inequality

$$\begin{aligned} |||P_{\alpha ,\beta }|||_{1,b} \le C_{\kappa ,b} \, n \left( r \sum _{u=1}^n e^{\beta _u}\right) ^{\tilde{b}} e^{-\delta |\beta |_1} (r^{-\delta } \tilde{\kappa })^{|\beta |_H} (1+r^{-\delta } \tilde{\kappa })^n, \end{aligned}$$

where $C_{\kappa ,b} = C'_{\kappa ,b} + 1$, by induction on n.

Set $\alpha ' = (\alpha _1,\dots ,\alpha _{n-1})$, $\beta ' = (\beta _1,\dots ,\beta _{n-1})$. If $\beta _n \ne 0$, then

$$\begin{aligned} |||P_{\alpha ,\beta }|||_{1,b} \le |||P_{\alpha ',\beta '}|||_{1,b} \, |||A_{\alpha _n,\beta _n}|||_{1,b}; \end{aligned}$$

since the first factor $|||P_{\alpha ',\beta '}|||_{1,b}$ is 1 when $n=1$ and is controlled by the inductive hypothesis when $n>1$, while the second factor $|||A_{\alpha _n,\beta _n}|||_{1,b}$ is controlled by (52), the conclusion follows. If instead $\beta _n = 0$, then

$$\begin{aligned} |||P_{\alpha ,\beta }|||_{1,b} \le |||P_{\alpha ',\beta '}|||_{1,b} + |||P_{\alpha ',\beta '} A'_{\alpha _n,0} |||_{1,b}; \end{aligned}$$

the required estimate then follows by majorizing $|||P_{\alpha ',\beta '}|||_{1,b}$ as before and $|||P_{\alpha ',\beta '} A'_{\alpha _n,0} |||_{1,b}$ by (57).

An analogous argument proves the same estimate for $|||P_{\alpha ,\beta }^*|||_{1,b}$ and the two estimates together give (56). $\square$

From now on, the argument becomes purely Banach-algebraic. In fact, the inequality (56) can be improved via a combinatorial technique. The following lemma should be compared with [33, Lemma (2.6)].

Lemma 6.11

Let $\nu \in \mathbb {N}\setminus \{0\}$. For all $n \in \mathbb {N}\setminus \{0\}$ and $\beta \in \mathbb {N}^n$, there exists $I \subseteq \{1,\dots ,n\}$ such that:

• (i) $|I| \le 2^{\nu -1} -1$;

• (ii) $\beta _j \ne 0$ for all $j \in I$;

• (iii) $\nu \sum _{j \in J} \beta _j \le |\beta |_1$ for all $J \subseteq \{1,\dots ,n\}$ with $|J| \le |I| + 1$ and $J \cap I = \emptyset$.

Proof

Modulo reordering and padding with zeros, we may assume that the sequence $\beta _1,\dots ,\beta _n$ is nonincreasing and $n \ge 2^{\nu }-1$. In particular

$$\begin{aligned} |\beta |_1 \ge \sum _{k=0}^{\nu -1} \sum _{j=2^k}^{2^{k+1}-1} \beta _j, \end{aligned}$$

hence there exists $k \le \nu -1$ such that

$$\begin{aligned} |\beta |_1 \ge \nu \sum _{j=2^k}^{2^{k+1}-1} \beta _j. \end{aligned}$$

Set $I = \{j \in \{1,\dots ,2^{k}-1\} :\beta _j \ne 0\}$. Clearly (i) and (ii) are satisfied. Moreover, since $\beta$ is nonincreasing, if $I \ne \{1,\dots ,2^{k}-1\}$, then, for every $J \subseteq \{1,\dots ,n\}$ which is disjoint from I, we have $\beta _j = 0$ for all $j \in J$, so that (iii) is trivially satisfied. In the case $I = \{1,\dots ,2^{k}-1\}$, instead, we have $J \subseteq \{2^k,\dots ,n\}$, hence, if $|J| \le |I| + 1 = 2^k$,

$$\begin{aligned} \nu \sum _{j \in J} \beta _j \le \nu \sum _{j = 2^k}^{2^{k+1}-1} \beta _j \le |\beta |_1, \end{aligned}$$

because $\beta$ is nonincreasing, and we are done. ${\square }$

Lemma 6.12

Let $\nu \in \mathbb {N}\setminus \{0\}$. For all $n \in \mathbb {N}$ with $1 \le n \le (r^{-\delta } \tilde{\kappa })^{-1}$, and for all $(\alpha ,\beta ) \in \mathcal {I}_n$,

$$\begin{aligned} \Vert P_{\alpha ,\beta }\Vert _{B_{b}} \le C_{\kappa ,b,\nu } \, r^{2^{\nu -1} \tilde{b}} n^{2^{\nu -1}(1+\tilde{b})} e^{(\tilde{b}/\nu - \delta )|\beta |_1} (r^{-\delta } \tilde{\kappa })^{|\beta |_H}. \end{aligned}$$

Proof

Let $I \subseteq \{1,\dots ,n\}$ be given by Lemma 6.11 applied to $\nu$, n, $\beta$; in particular $l = |I| \le 2^{\nu -1}-1$. Let $j_1,\dots ,j_l$ be an increasing enumeration of the elements of I, so that $\beta _{j_1},\dots ,\beta _{j_l}$ are nonzero, and set $j_0 = 0$, $j_{l+1} = n+1$. Then

$$\begin{aligned} P_{\alpha ,\beta } = P_{\alpha ^{(0)},\beta ^{(0)}} A_{\alpha _{j_1},\beta _{j_1}} P_{\alpha ^{(1)},\beta ^{(1)}} \dots A_{\alpha _{j_l},\beta _{j_l}} P_{\alpha ^{(l)},\beta ^{(l)}}, \end{aligned}$$

where $\alpha ^{(k)} = (\alpha _{j_{k-1} + 1},\dots ,\alpha _{j_k - 1})$, $\beta ^{(k)} = (\beta _{j_{k-1} + 1},\dots ,\beta _{j_k - 1})$, and

$$\begin{aligned} \nu \sum _{k=0}^{l} |\beta ^{(k)}|_\infty \le |\beta |_1. \end{aligned}$$

Moreover (56) gives, for the $\beta ^{(k)}$ which are nonempty,

$$\begin{aligned} \Vert P_{\alpha ^{(k)},\beta ^{(k)}}\Vert _{B_{b}} \le C_{\kappa ,b} \, n^{1+\tilde{b}} r^{\tilde{b}} e^{\tilde{b} |\beta ^{(k)}|_\infty } e^{-\delta |\beta ^{(k)}|_1} (r^{-\delta } \tilde{\kappa })^{|\beta ^{(k)}|_H}, \end{aligned}$$

whereas

$$\begin{aligned} \Vert A_{\alpha _{j_k},\beta _{j_k}}\Vert _{B_{b}} \le e^{-\beta _{j_k} \delta } r^{-\delta } \tilde{\kappa }\end{aligned}$$

by (52). Since

$$\begin{aligned} |\beta |_1= & {} |\beta ^{(0)}|_1 + \beta _{j_1} + |\beta ^{(1)}|_1 + \dots + \beta _{j_l} + |\beta ^{(l)}|_1,\\ |\beta |_H= & {} |\beta ^{(0)}|_H + 1 + |\beta ^{(1)}|_H + \dots + 1 + |\beta ^{(l)}|_H, \end{aligned}$$

the conclusion follows by multiplying these estimates together. $\square$

Proof of Proposition 6.7

• (i) From the previous discussion, we are reduced to proving (51) for $h \in \mathbb {N}^m \setminus \{0\}$. Having fixed such an h, we choose $r = (|h|_1 \tilde{\kappa })^{1/\delta }$ and $\nu = \nu _{a,b} := \lfloor \tilde{b}/\delta \rfloor + 1$, so that $\epsilon _{a,b} := \delta - \tilde{b} /\nu > 0$. Hence Lemma 6.12 gives, for $(\alpha ,\beta ) \in \mathcal {I}_{|h|_1}$,

  $$\begin{aligned} \Vert P_{\alpha ,\beta }\Vert _{B_{b}} \le C_{\kappa ,a,b} \, |h|_1^{\gamma _{a,b}} e^{-\epsilon _{a,b} |\beta |_1} |h|_1^{-|\beta |_H}. \end{aligned}$$

  Set now $\alpha = (1,\dots ,1,2,\dots ,2,\dots ,m,\dots ,m)$, with $h_1$ occurrences of 1, $h_2$ occurrences of 2, and so on. Then

  $$\begin{aligned} \begin{aligned} \Vert A_1^{* h_1} * \dots * A_m^{* h_m}\Vert _{B_{b}}&\le \sum _{\beta \in \mathbb {N}^{|h|_1}} \Vert P_{\alpha ,\beta }\Vert _{B_{b}} \\&\le C_{\kappa ,a,b} \, |h|_1^{\gamma _{a,b}} \sum _{\beta \in \mathbb {N}^{|h|_1}} e^{-\epsilon _{a,b} |\beta |_1} |h|_1^{-|\beta |_H} \\&= C_{\kappa ,a,b} \, |h|_1^{\gamma _{a,b}} \left( 1 + |h|_1^{-1} \sum _{k=1}^\infty e^{-\epsilon _{a,b} k}\right) ^{|h|_1} \\&\le C_{\kappa ,a,b} \, \exp \left( \sum _{k=1}^\infty e^{-\epsilon _{a,b} k}\right) \, |h|_1^{\gamma _{a,b}}, \end{aligned} \end{aligned}$$

  and we are done.

• (ii) We assume as before that $h_1,\dots ,h_m \ge 0$. First of all, by arguing as in (54), one immediately obtains that

  $$\begin{aligned} |||A_j-I|||_{2,b} \le \tilde{\kappa }\end{aligned}$$

  for $j=1,\dots ,m$. Now we proceed inductively on $|h|_1$. The case $|h|_1 = 0$ is trivial. If instead $h_j \ne 0$ for some j, then

  $$\begin{aligned} E_h = E_{h'} + (A_1^{* h'_1} * \dots * A_m^{* h'_m}) * (A_j - I), \end{aligned}$$

  where $h' = (h_1,\dots ,h_{j-1},h_j - 1, h_{j+1}, \dots ,h_m)$, hence

  $$\begin{aligned} |||E_h|||_{2,b} \le C_{\kappa ,a,b} (|h'|_1^{\gamma _{a,b} + 1} + |h'|_1^{\gamma _{a,b}}) \le C_{\kappa ,a,b} \, |h|_1^{\gamma _{a,b}+1} \end{aligned}$$

  by (51), Young’s inequality (45) and the inductive hypothesis.

• (iii) We may assume, modulo rescaling, that $\kappa = 1$, so in particular the joint spectrum of $M_1,\dots ,M_m$ is contained in $[-1,1]^m$. The Fourier series expansion

  $$\begin{aligned} F(\lambda ) = \sum _{h \in \mathbb {Z}^m} \hat{F}(h) \, e^{i h \cdot \lambda } = \sum _{0 \ne h \in \mathbb {Z}^m} \hat{F}(h) \, (e^{i h \cdot \lambda } - 1) \end{aligned}$$

  for $\lambda \in (-\pi ,\pi )$ (where the last equality is due to the fact that $F(0) = 0$, and the convergence is uniform because $s > m/2$) then implies that

  $$\begin{aligned} F(M_1,\dots ,M_m) = \sum _{0 \ne h \in \mathbb {Z}^m} \hat{F}(h) \, \exp _0(i (h_1 M_1 + \dots + h_m M_m)) \end{aligned}$$

  with convergence in $\mathcal {B}(L^2)$. On the other hand,

  $$\begin{aligned} \sum _{0 \ne h \in \mathbb {Z}^m} |\hat{F}(h)| \, |||E_h|||_{2,b} \le C_{a,b} \sum _{0 \ne h \in \mathbb {Z}^m} |\hat{F}(h)| \, |h|_1^{\gamma _{a,b} + 1} \le C_{a,b,m,s} \Vert F\Vert _{L^2_s} \end{aligned}$$

  by (43) and Hölder’s inequality, since $s > \gamma _{a,b} + 1 + m/2$, and we are done.$\square$