On the Kor\'anyi Spherical maximal function on Heisenberg groups

We prove $L^p\to L^q$ estimates for the local maximal operator associated with dilates of the K\'oranyi sphere in Heisenberg groups. These estimates are sharp up to endpoints and imply new bounds on sparse domination for the corresponding global maximal operator. We also prove sharp $L^p\to L^q$ estimates for spherical means over the Kor\'anyi sphere, which can be used to improve the sparse domination bounds for the associated lacunary maximal operator.


Introduction
Let H n = R 2n × R be the Heisenberg group of real Euclidean dimension 2n + 1.We shall use the notation x = (x, x), y = (y, ȳ), with x, y ∈ R 2n and x, ȳ ∈ R, to denote elements of H n .The group law is given by x • y = (x + y, x + ȳ + 1 2 x Jy), where J := 0 I n −I n 0 is the 2n × 2n standard symplectic matrix (and I n is the n × n identity matrix).Further, the Korányi norm of an element x is defined to be 4 .This norm is homogeneous of degree one with respect to the natural parabolic dilation structure δ t ((x, x)) := (tx, t 2 x) on H n .Let S K be the sphere centred at the origin, of radius one with respect to the Korányi norm.There exists a unique Radon measure µ on S K induced by the Haar measure on H n .For t > 0, the averaging operator associated with the µ t , the t-dilate of this measure, is given by (1.1) and let Af (x, t) := A t f (x).
In 1981, Cowling [9] considered the global Korányi maximal function He showed that M is bounded on L p (H n ) for all p > 2n+1 2n and n ≥ 2.More recently, in [10], Ganguly and Thangavelu considered the lacunary variant on H n for n ≥ 2 and proved that M lac is bounded on L p for all 1 < p < ∞.They did so by establishing a (p, q )-sparse domination result for M lac .Closely following Lacey's approach in the Euclidean case [14], they made use of an induction argument relying on L p → L q estimates for spherical means over the unit Korányi sphere A 1 f (x) = f * µ(x), a problem which is also of interest in its own right.The objective is to find the best possible value of q in such an estimate.In [10] it is proved that A 1 maps L p (H n ) to L q (H n ) provided that ( 1 p , 1 q ) belongs to the interior of the triangle with corners (0, 0), (1, 1), ( 2n 2n+1 , 1 2n+1 ), as well as the line joining the points (0, 0) and (1, 1).In the following theorem, we provide L p → L q bounds that are sharp.
In light of the recent L p → L q estimates for the local maximal operator associated with codimension two spheres in the Heisenberg groups [2,23] (see the following remarks), it is natural to seek similar estimates for M .Such an estimate would also imply sparse bounds for the global maximal operator M. Our main theorem contains L p → L q estimates for M which are sharp up to endpoints.Theorem 1.2.Let n ≥ 2. Let R be the closed quadrilateral with corners There has been considerable progress in the recent years on the problem of establishing L p improving properties of localized maximal functions associated with surfaces, both in the Euclidean and Heisenberg settings.A few remarks are in order to shed light on the chief features of our problem which make it different from some related bodies of work.
The region R in Theorem 1.2, for n = 2.
Remarks 1.3.(i) Non-vanishing Rotational Curvature: A different maximal function on the Heisenberg group, considered in [2,3,[18][19][20]23], is associated to averages over codimension two spheres contained in a dilation invariant subspace of H n .In contrast, the averaging operator A t in (1.1) is associated to a surface of codimension one which respects the non-isotropic dilation structure on H n .Further, unlike the codimension two case, the oscillatory integral operators associated with A t possess non-vanishing rotational curvature at fixed time t (this fact is also implicit in [21] and proven directly in [25]).In this sense, A t could be considered to be a faithful analog of the Euclidean spherical average in the Heisenberg setting.
(ii) Vanishing of Cinematic Curvature: Unlike the Euclidean spherical averaging operator, the rank of the cinematic curvature matrix (7.8) associated with A is zero at the north and south poles of the Korányi sphere.To deal with the issue of flatness (that is, the absence of cinematic curvature in the sense of [26]) at the poles, we need to employ a scaling argument.The first step in such an argument is to dyadically decompose the relevant kernel based on the distance from the poles, followed by rescaling each piece to a region where the curvature does not vanish.This idea was used by Iosevich [13] to establish the L p boundedness of global maximal operators associated to families of flat, finite type curves in R 2 .It was also used in [16] and more recently [15] to prove L p → L q estimates for local maximal functions along some finite type curves in R 2 and hypersurfaces in R 3 .However, unlike [13,15,16], our problem is based in a non-Euclidean setting.The Heisenberg group structure calls for a new type of scaling argument.
Comparison with the Euclidean convolution structure.The Heisenberg group structure plays a crucial role in our analysis.It shows itself in the presence of the bilinear term involving the symplectic matrix J in the defining equation of the Korányi sphere centred at x and of radius t, given by (1.2) |x − y| 4 + |x − ȳ + 1  2 x Jy| 2 = t 4 .To illustrate its significance, we contrast our situation with that of the Korányi sphere in R 2n+1 when the translations are Euclidean and not given by the Heisenberg law.The defining equation of such a sphere centred at x and of radius t is given by (1.3) |x − y| 4 + |x − ȳ| 2 = t 4 .3 (in grey) depicts the region of almost sharp L p → L q estimates for the Korányi spherical maximal operator with Heisenberg group structure (given by Theorem 1.2).Since the necessary conditions for this operator (see §9) also apply to the Korányi spherical maximal operator described with respect to Euclidean convolution, the region corresponding Figure 3.A comparison between the regions of L p → L q boundedness for the Korányi spherical maximal operator corresponding to Heisenberg convolution (in grey) versus the Euclidean convolution (in blue), for n = 2.
to the sharp L p → L q estimates for the Euclidean version is contained in R. In fact, this containment is proper, as can be seen by means of a standard Knapp example adapted to a homogeneous Euclidean hypersurface of degree four, which yields the sharpness of the edge P 3 P 4 in the interior of R. The quadrilateral P = P 1 P 2 P 3 P 4 (in blue) depicts the region in which scaling type arguments can be used to obtain a positive result for the Korányi maximal operator in the Euclidean setting.Its coordinates in R 2n+1 are given by P 1 = (0, 0), . The estimate at P 2 is the same as the L p bound contained in [8,Theorem 3.1] for global maximal operators associated to averages over Euclidean hypersurfaces which are graphs of smooth homogeneous functions with non-vanishing Hessian away from the origin, with the degree of homogeneity being at most twice the dimension of the surface.The estimates corresponding to P 3 and P 4 can be obtained by following the same arguments as in the proof of Theorem 1.2 (but in a more straightforward manner).Eschewing detailed calculations, which shall appear in the author's Ph.D. thesis [27], we now briefly discuss the reasons behind these differences between the Heisenberg and Euclidean settings.
In the Heisenberg case, the term x Jy changes the geometry of the Korányi sphere as it is translated away from the origin and is responsible for the non-vanishing of the rotational curvature, even at the poles.As a result, Hörmander's classical L 2 theory [30, Ch.IX.1] can be used to establish a sharp fixed time L 2 estimate.This in turn implies the positive result in Theorem 1.1 after interpolation with easily obtainable estimates involving the L 1 and L ∞ spaces.Using standard Sobolev embedding, the estimates at the vertices Q 1 , Q 2 , Q 3 of the quadrilateral R also follow.They are sharp up to endpoints and analogous to the corresponding estimates for the standard sphere in R 2n+1 defined using the Euclidean norm.This is in sharp contrast to the Korányi sphere with Euclidean translational structure, where the rotational curvature vanishes at the poles and a scaling argument (along the lines of [13]) is needed.Further, the fixed time L 2 estimate obtained using scaling is worse than the same estimate for the standard Euclidean sphere, and consequently, so are the corresponding diagonal and off-diagonal estimates.
As mentioned already, the Heisenberg estimate at Q 4 is indeed affected by the flatness at the poles, as is the corresponding estimate at P 4 for the Korányi sphere in the Euclidean case.In both cases, after suitable decomposition and scaling, we need to prove a fixed time L 2 estimate for each rescaled piece separately.However, in order to establish these estimates in the Heisenberg case, we need to introduce further localization and make use of an almost L 2 orthogonality that exists between the different localized pieces (see §7. 1.1).This is due to the surprising feature that the non-isotropic scaling in the "input variable" y is different from the one which shows up in the "output variable" x, with x, y in (1.2).This phenomenon is absent in the Euclidean version and occurs, again, due to the presence of the bilinear term x Jy in the second term in (1.2) (see §4, Remark 4.1).
Quite pleasantly, this "imbalanced" scaling argument yields a better estimate at Q 4 in the Heisenberg case than the estimate at P 4 for the Korányi sphere with Euclidean convolution structure.The former estimate is consistent with a new Knapp-type counterexample §9.5 which establishes the sharpness of the edge Q 3 Q 4 in the Heisenberg case.As mentioned already, the edge P 3 P 4 for the Korányi sphere in the Euclidean setting is also sharp, which can be seen by using a standard Knapp example adapted to a Euclidean degree four homogeneous surface.
The estimates at Q 4 (and P 4 ) also rely on an L ∞ bound on the kernel of an associated oscillatory integral operator.To do so, we shall seek to make use of estimates for standard oscillatory integrals of Carleson-Sjölin-Hörmander type, in particular a variant of Stein's theorem [29] formulated in [17] which relies on the maximal possible number of nonvanishing curvatures for a cone in the fibers of the canonical relation (see (7.9)).
The above discussion was focused on the estimates around the poles.Both the curvature conditions also hold at the equator and in the intermediate region of the Korányi sphere, enabling us to apply Hörmander's L 2 estimate and the aforementioned Stein's theorem directly.However, their verification is still technically involved in the Heisenberg case due to the additional bilinear term.
Plan of the paper.The paper is organized as follows: • §2 contains a localization argument for the function f , and a partition of unity argument which streamlines our analysis into three regimes, based on whether the kernel of the Fourier integral operator being considered is supported around the poles, at the equator or in an intermediate region.We also describe the corresponding parametrization of the Korányi sphere (explicitly around the poles and implicitly in the other two regimes).• In §3 we list the basic estimates and reduce them to corresponding ones for oscillatory integral operators.
• In §4 we use a scaling argument to understand the behaviour of our operator around the poles (or the north pole to be more specific, but the behaviour around the south pole is exactly the same due to symmetry).• §5 describes a Stein-Tomas type argument to reduce the estimate at Q 4 to an L 2 estimate and an L ∞ bound on the kernel of an oscillatory integral operator.• §6 contains a few technical results about matrices and radial functions which will be crucial to calculations verifying the non-vanishing of the rotation curvature and the additional curvature condition (7.9).• In §7, we verify these two conditions around the poles (for the rescaled operators).§7.1.1 also contains an almost orthogonality argument crucial to the proof of the fixed time L 2 estimate for the rescaled operators.• In §8, we verify the curvature conditions at the equator and in the intermediate region, thus finishing the proofs of the positive results contained in Theorems 1.1 and 1.2.• In §9 we provide counterexamples to show the sharpness of these theorems.
• In §10 we briefly discuss their implications on sparse bounds for the lacunary and global maximal functions.• §11 contains a technical lemma about the invariance of both the curvature conditions under the Heisenberg group law.
Notation.Partial derivatives and tangent vectors along the coordinate direction e j will be denoted by the subscript j for j ∈ {1, 2, . . ., 2n + 1}.For a function f : R 2n → R 2n , f shall be used to denote its Hessian which is a 2n × 2n matrix.By A B we shall mean that A ≤ C • B, where C is a positive constant and A ≈ B shall signify that A B and B A. A D B shall mean that A ≤ C • B with the positive constant C depending on the parameter D.
Acknowledgement.The author is grateful to her advisor, Andreas Seeger, for suggesting this problem and his invaluable advice at various stages of the project.She is also thankful to the Hausdorff Research Institute of Mathematics and the organizers of the trimester program "Harmonic Analysis and Analytic Number Theory" for a productive stay in the summer of 2021.Research supported in part by NSF grants DMS 1764295 and DMS 2054220.

Preliminaries
2.1.Localization.Without loss of generality, we can limit our consideration to functions f supported in a small subset of a thin neighborhood of the Korányi sphere of radius one centred at the origin.
To see this we use the group translation to tile Moreover, for t ∈ [1, 2], the t-dilates of the measure µ are supported in {w ∈ H n : |w| ≤ 2, | w| ≤ 4}, hence in the union of B k with |k j | ≤ 2 for j ≤ 2n and |k 2n+1 | ≤ 6. Denote this set of indices by J. Then where I(n) is a set of indices ñ with |n j −ñ j | ≤ C(J, n) for j = 1, . . ., 2n+1.This consideration of spatial orthogonality allows us to reduce to the case of functions f supported in the union of a finite number of Heisenberg tiles B n which cover the unit Korányi sphere and its t dilates for t ∈ [1,2].We can further partition these tiles (resp.the interval [1,2]) into a finite number of sub-tiles (resp.sub-interval) of small enough size and consider the averaging operator associated to each such piece separately, summing up the estimates at the end.This reduces matters to the case when f is supported in a small subset of a thin neighborhood of the unit Korányi sphere, of length at most 2 −400n 100n in each coordinate direction.As a consequence, we can also assume that A t f is supported in a small neighborhood of the origin of length at most 2 −400n 10n in each direction.We shall frequently make use of both Taylor approximation and the implicit function theorem to express the phase function of oscillatory integral operators.Both these theorems require the corresponding amplitude to be supported in a set of constant but sufficiently small size.By choosing a suitable smooth partition of unity, we can express A t as a finite sum of operators, each of which is localized to a small region of the Korányi sphere.It then suffices to prove the required estimates for each such operator independently.We now explicitly describe such a localization.
Given a point w ∈ S K , let Q(w) := { w ∈ S K : |w − w| < 2 −400n }.Since the Korányi sphere S K is a compact submanifold of H n , there exists a finite set of points {w ν } ν ⊆ S K such that the collection {Q ν := Q(w ν )} ν forms an open cover of S K .Let ν η ν be a smooth partition of unity subordinate to {Q ν } ν .Define µ ν := µη ν so that µ = ν µ ν .
We can rewrite (1.1) as Using a change of variables y = x − tω, ȳ = x − t 2 ω − t 2 x Jω, we can express Jy + e 2n+1 + 4t 3 e 2n+2 .To parametrize the Korányi sphere as the graph of a smooth function locally, we need to express one of the coordinates of y as a function of the other y−coordinates, x and t.The coordinate we choose, and hence the parametrization, will depend upon the localization neighborhood Q ν .
Since the Korányi sphere is symmetric about the equator, we can limit our attention to its northern hemisphere.We will choose different parametrizations near the equator and around the north pole.Unlike in the case of the Euclidean sphere, we shall see that the curvature properties depend upon the neighborhood Q ν being considered.Heuristically speaking, due to the non-isotropic dilation structure of the Korányi sphere, the region around the equator exhibits similar curvature properties as the Euclidean sphere.However, around the north pole, the Korányi sphere behaves like the surface ȳ = 1 + |y| 4 , with the (cinematic) curvature vanishing at the pole.Other than these two extreme cases, the intermediate region needs to be carefully considered as well.
Remark 2.1.The above discussion only applies to the cinematic curvature condition (see (7.9)).The rotational curvature does not see these different parametrizations, and remains non-vanishing throughout.This has been observed before in the context of the Korányi sphere in [25] and is also to be expected in view of the L p bounds on the global maximal operator in [9].
Let J denote the indexing set of the cover {Q ν } ν .Keeping the above discussion in mind, we partition J into three disjoint sets J = J Eq ∪ J IM ∪ J NP , where J NP = {ν : e 2n+1 ∈ Q ν } J Eq = {ν : there exists x = (x, x) ∈ Q ν with x = 0}, Near the North Pole.For ν ∈ J NP and for all x−y t , x−ȳ+ 1 and we can write Using Taylor's remainder formula, we can express Near the Equator.Observe that for A ∈ O(2n), the action (x, x, t, y, ȳ) → (Ax, x, t, Ay, ȳ) leaves the first and third terms in (2.1) unchanged, and replaces J by J = A JA in the second term.Since J is still a skew-symmetric matrix with J 2 = −I 2n , as long as our analysis is uniform in the set of all matrices satisfying these two properties, using horizontal rotations, it suffices to consider only those ν ∈ J Eq such that the corresponding neighborhood Q ν contains e 1 .Let J Eq denote the collection of all such ν.

Main Estimates
Keeping the above parametrizations in mind, we are led to consider three types of generalized Radon transforms, associated to incidence relations given by the equations (2.6), (2.7) and (2.8), with kernels supported in Q ν for ν in J NP , J Eq and J IM .We choose ν NP , ν Eq , ν IM in J NP , J Eq and J IM respectively, and fix them for the rest of the paper.For ∈ {NP, Eq, IM}, we define µ := µ ν and η := η ν .
Recall that we have reduced consideration to functions f with small support around a thin neighborhood of the Korányi sphere, so that A t f is supported around a small neighborhood of the origin.We can choose a smooth function χ 1 supported in a slightly larger neighborhood of the origin so that x Jy t 2  .
We now define smooth cut-off functions χNP (x, t, y) Observe that x and y are small in the supports of all three functions, y 1 is small in the support of χNP and ȳ is small in the support of χ Eq .Further, due to our assumptions on the supports of η NP , η Eq , η IM , it follows that G(x, t, y), H 1 (x, t, y , ȳ) and H(x, t, y) are of size about one in the support of χNP , χ Eq and χ IM respectively.
We can thus write (3.1) When considering the last integral above (at the north pole), it will be convenient to introduce a shear transformation in the x-variables (depending smoothly on t) By this change of variables and χ NP (x, t, y) = χNP (x, x + t 2 , y).Note that while the shear transform leaves the Lebesgue space estimates unchanged, the smooth cut off χ NP is now supported in the set where both x and G(x, t, ȳ) are of size about one, while x and y are small.The right hand sides of (3.1) and (3.2) represent operators with Schwartz kernel K of three types depending on the neighborhood of localization: In the Intermediate Region: By applying the Fourier transform to δ 0 , we can express where the phase function around the north pole, the equator and in the intermediate region is given by ψ NP (x, t, y, θ) := θ(G(x, t, y) − ȳ), ψ Eq (x, t, y, θ) := θ(H 1 (x, t, y , ȳ) − y 1 ) and ψ IM (x, t, y, θ) := θ( H(x, t, y) − ȳ) respectively; the smooth cut-off χ can be χ NP , χ Eq or χ IM , and ỹ can be y or (y , ȳ) depending on the support of the kernel.Then K is well defined as an oscillatory integral distribution.
Next, we perform a dyadic decomposition in the frequency variable θ.Let ζ 0 be a smooth radial function on R with compact support in {|θ| < 1} such that ζ 0 (θ) = 1 for |θ| ≤ 1/2.We set For k ≥ 0, ψ of any of the three forms above and corresponding χ, we define where ỹ can be y or (y , ȳ) depending on the support of the kernel.Further, let Following are the main estimates for M k , which hold uniformly for all three possible forms of the phase function ψ in the support of the corresponding smooth cut off χ.
(iii) Set q 5 := 2(n+2) n .For every > 0, there exists C > 0 such that Proof of Theorem 1.2, given Proposition 3.1.It suffices to show the required bounds for Mf (x) := k≥0 M k f .We shall show that M is of restricted weak type (p, q) for ( 1 p , 1 q ) ∈ {Q 2 , Q 3 }.Observe that for n ≥ 2, we have 1 − 2n+1 2 < 0. To deduce the required restricted weak type estimates for M at Q 2 , Q 3 , we recall the Bourgain interpolation argument ( [5], [6]): Suppose we are given sublinear operators T k so that for k ≥ 1, for some p 0 , q 0 , p 1 , q 1 ∈ [1, ∞], a 0 , a 1 > 0. Then the operator k≥1 T k is of restricted weak type (p, q), where ) and ϑ = a 0 a 0 +a 1 ∈ (0, 1).The restricted weak type estimate for M at Q 2 = ( 2n 2n+1 , 2n 2n+1 ) now follows from (3.4).Similarly, the restricted weak type bound at Q 3 = ( 2n 2n+1 , 1 2n+1 ) follows from (3.5).Interpolation between the estimates at , due to the loss of a 2 k factor on the right hand side of (3.6), we only have , standard interpolation between (3.6) and the case q = ∞ of (3.5) implies that Interpolating again between the estimates at Q 1 , Q 2 , Q 3 and Q 4, yields the L p → L q boundedness of M for ( 1 p , 1 q ) in the interior of the quadrilateral formed by those vertices.Letting → 0, we conclude the same for ( 1p , 1 q ) in the interior of R. Since bounds for M imply bounds for M , this concludes the proof of part (i) of Theorem 1.2.

3.1.
Reduction to estimates on the averaging operator.We consider a discrete dyadic cover of the interval [1,2].Given a non-negative integer k, let Z k denote the set of left endpoints of all dyadic intervals of the form (ν2 −k , (ν + 1)2 −k ) (with ν ∈ Z) which intersect [1,2], endowed with the counting measure.It is clear that #(Z k ) = 2 k .An application of the fundamental theorem of calculus gives the following pointwise bound Taking the L q norm on both sides, using the triangle inequality, then dominating the supremum by the q (Z k ) norm and using the Minkowski inequality for the second term, we obtain By a change of variables and using homegeneity (in θ) for k ≥ 1, we have ) and b Eq , b IM defined analogously (ỹ might stand for y or (y , ȳ) depending on the support).In particular, |θ| ∈ [1/2, 2] in the support of all three types of symbols.
Using homegeneity, it is not hard to see that the operator 2 −k d dt A k t is a linear combination of expressions involving operators which obey the same Lebesgue space estimates as the operator A k t .Thus parts (i) and (ii) of Proposition 3.1 are a consequence of the following fixed time estimates.
with the implicit constant independent of t.
(ii) For 2 ≤ q ≤ ∞, with the implicit constant independent of t.
The above fixed time estimates are sufficient to establish the positive result in Theorem 1.1.
To prove the estimate at q 5 , taking an L q 5 ,∞ norm on both sides of (3.7), we conclude that Thus part (iii) of Proposition 3.1 is a consequence of the following estimate.
n .We have For all k ≥ 1, we have the following estimates In view of the above it suffices in what follows to consider the case of large k.The bounds (3.9), (3.10) then follow by an interpolation argument using (3.12), (3.13) and the fixed time 3.2.Reduction to oscillatory integral estimates.To prove (3.14) we use oscillatory integral operators of the form where Φ can be of the form Φ NP (x, t, y) = ȳG(x, t, y), Φ Eq (x, t, y) = y 1 H 1 (x, t, y , ȳ), and Φ IM (x, t, y) = ȳ H(x, t, y), with G, H 1 and H as defined in equations (3.3), (2.7) and (2.8).The corresponding amplitude b is compactly supported in R 2n+1 × R × R 2n−1 × R, and can be of the form b NP , b Eq or b IM as defined in (3.8) (with the role of θ played by ȳ or y 1 , depending on the situation).
To reduce matters to oscillatory integral operators of the above form, we define The integrals on the right can be interpreted as scaled Fourier transforms in the ȳ and y 1 variables respectively.After renaming w to ȳ and ȳ to θ, we have around the north pole and in the intermediate regions.Similarly, renaming w 1 to y 1 and y 1 to θ, we obtain around the equator.By Plancherel's theorem with the implicit constant uniform in t ∈ [1,2].
The proof will be given using Hörmander's standard L 2 estimate [12] (also see [30, Ch.IX.1]) combined with almost-orthogonality arguments.By the same argument, the L 2 → L q 5 ,∞ bound (3.11) is reduced to the following estimate.

A Scaling Argument at the Poles
In this section, we focus on the case when the kernel of the operator T k is supported around the poles (more specifically, around the north pole but the same argument works for the south pole due to symmetry about the equator).In fact, the L p improving property at Q 4 is completely determined by how the operator behaves around the poles.We shall decompose the kernel of the operator T k based on the distance from the north pole and use a stretching argument to precisely understand this behaviour.
We define Due to condition (2.5) on R, we have In fact, g can be expressed as g (w) = u (|w|), where u is a smooth function in one variable satisfying the condition We thus have and for a fixed t Remark 4.1.Observe that the input variable y is scaled by 2 − , while the output variable x is scaled by a factor of 2 −3 .This is due to the presence of the bilinear term involving the matrix J in the phase function.This phenomenon was not present in [13], where a scaling type type argument was used to establish the L p boundedness of global maximal operators associated to families of flat, finite type curves in R 2 (also see [16]).
We introduce an operator T k, t defined by e i2 k−4 Φ (x,t,y) a (x, t, y)f (y, ȳ)dy and let The following lemma will be useful for T T * type arguments in the upcoming sections.Lemma 4.2.Let a, b, c ∈ Z and for i = 1, 2, let U i , U i be operators defined on R 2n+1 such that Then for 1 ≤ p, q ≤ ∞, Proof.As a consequence of (4.7), we have Taking L q norms on both sides and changing variables on the right hand side, we get f L p , this yields the desired estimate.
To prove (3.17), we shall also need a fixed time L 2 estimate for each scaled piece T k, t .Proposition 4.3.Let T k, t be as defined in (4.5).For all f ∈ L 2 (R 2n+1 ) and t ∈ [1, 2], with the implicit constant uniform in t ∈ [1,2].
The above proposition shall be proven in §8.
) .If we sum this up for 0 ≤ ≤ k 4 , we would obtain the fixed time L 2 estimate (3.16) for T k around the poles but up to a logarithmic loss in 2 k .However, as we shall see in §8.3, the rotational curvature remains non-vanishing at the poles and thus (3.16) can be directly obtained without incurring any logarithmic loss for the estimates at Q 1 , Q 2 and Q 3 .

A Stein-Tomas Argument
In this section, we shall reduce the estimate (3.17), for the operators T k (around the equator and in the intermediate region) and T k, (around the poles), to an L q 5 ,1 → L q 5 ,∞ estimate for the respective operator precomposed with its adjoint via the T T * technique.Using a Stein-Tomas type argument, the said estimate will then be obtained by interpolating between an L 2 → L 2 and an L 1 → L ∞ estimate.We first give the details when the kernel of T k is supported around the poles (more precisely, the north pole), where the argument needs to be run separately for each piece T k, .5.1.Around the North Pole.Since T k f = k 4 =0 T k, f , to prove (3.17), it suffices to show for each 0 ≤ ≤ k 4 .Using a T T * argument (and the fact that the dual space of L q ,1 is L q,∞ ), the above estimate is a consequence of (5.1) We define the operator S k, acting on functions g : Then (5.1) follows from For m ≥ 0 and We observe . Thus, using the Bourgain interpolation trick for S k, = m≥0 S k, m , (5.3) is a consequence of (5.5) To show (5.5), using the Cauchy-Schwarz inequality, we have where we have used Corollary 4.4 and the fact that #(Z m k (t)) ∼ 2 m for all t ∈ Z k .
To prove (5.6), we need the following estimate on the kernel of the operator T k, t (T k, t ) * .
Proposition 5.1.Let K k, t,t denote the kernel of T k, t (T k, t ) * , where T k, t and T k, t are as defined in (4.5) for t, t ∈ [1,2].Then for 0 ≤ < k 4 , (5.7) Postponing the proof of the above proposition to §7.2, we use it for now to prove (5.6).The above estimate implies that For m ≥ 4 , using the estimate above, we have On the other hand, for m ≤ 4 or = k 4 , we can use the trivial bound of 2 −2n (using the fact that This implies (5.6), and concludes the proof of (3.11).

5.2.
Around the Equator and in the Intermediate Region.The proof of estimate (3.17) when the operator T k t has kernel supported away from the north pole proceeds in the exact manner as above.The following replaces Proposition 5.1 in this setting.Proposition 5.2.Let K k t,t denote the kernel of T k t (T k t ) * , where T k t and T k t are as defined in (3.15) for t, t ∈ Z k , with Φ = Φ Eq or Φ = Φ IM .Then The above Proposition shall be proved in §8.1.2and §8.2.2.Below we show how it implies (3.17).Using the T T * technique, it suffices to prove that We define S k and S k m as in (5.2) and (5.4), with T k, t , T k, t replaced with T k t , T k t respectively.The above estimate then follows from which by Bourgain's interpolation trick is a consequence of the estimates (5.9) Estimate (5.9) follows like in the previous case, using the Cauchy-Schwarz inequality and the fixed time estimate (3.16) for T k with kernel supported around the equator or in an intermediate region.
(5.10) is a straightforward consequence of (5.8), for we have This concludes the proof of (3.17).
Thus, in order to prove the positive results in Theorems 1.1 and 1.2, we need to establish Propositions 4.3 and 5.1 around the poles (proven in §7.1 and §7.2 respectively), Proposition 5.2 around the equator and in the intermediate region (contained in §8.1.2and §8.2.2 respectively) and Proposition 3.4 in all three cases (proven in §8.1.1,§8.2.1 and §8.3).

Results about Radial Functions and Matrix Inverses
This section contains a few results about the Hessian of the radial function g and its interaction with a skew symmetric matrix J satisfying J 2 = −I d , which will aid us in the calculations to follow.Lemma 6.1.Let d ∈ N, u : R → R be a smooth function and let g : R d → R be given by g(w) = u(|w|).
Proof.The first part is a simple application of the chain rule, so we focus on the proof of the second part.Setting r = |w| and taking partial derivatives twice, we have ww .
Here I d denotes the identity matrix of order d.Taking determinants on both sides, we have The claim then implies the desired equality, for then it follows that det g (w) = u (r) r Using the induction hypothesis, we conclude that Next we have a formula for the inverses of matrices of the form J +g (w) where J is a skew-symmetric matrix of dimension d with J 2 = −I d and w ∈ R d .Lemma 6.2.Let J be a skew-symmetric matrix with J 2 = −I d and let w ∈ R d .For any σ, λ, κ, γ ∈ R, we have Proof.This can be verified directly, using the facts that J 2 = −I d , ww ww = |w| 2 ww and w Jw = 0 (due to J being skew-symmetric).
Here (7.1) Φ (x, t, y, ȳ) = y x + 1 2 x T Jy + t 2 g ( We seek to apply Hörmander's classical L 2 bound ([30, ch.IX.1]).For a fixed phase function Φ supported in R 2n+1 ×R 2n+1 , this bound guarantees the fixed time estimate in Proposition 4.3, provided that the rank of the (2n + 1) × (2n + 1) mixed Hessian matrix Φ (x,y) is equal to 2n + 1.The implicit constant in the inequality depends on the lower bound for the determinant of Φ (x,y) , the upper bound on finitely many derivatives of Φ and the amplitude a, and the size of the support of a.
In our case, we require uniform L 2 bounds for a family of operators with phase functions Φ and amplitudes a for 400n ≤ ≤ k 4 .From the definition of a , it is clear that for large , the derivatives of a upto a finite order are bounded from above by an absolute constant times the derivatives of a.
Coming to the phase functions, recall that g (in the definition of Φ above) is given by g (w) = g(w) + 2 −4 g (w), with g satisfying property (4.3), namely The above estimate is also true for higher order derivatives, perhaps with an absolute constant greater than 100n.Thus for large enough (say ≥ 400n), g and its higher order derivatives essentially behave like g and consequently Φ is a smooth perturbation of Φ(x, t, y, ȳ) = y x + x T Jy + t 2 g( −y t ) .However, the size of the x support of the amplitude a is no longer uniform in , since a is supported in the set where |x| 2 3 and |x| 2 4 .Hence we need to introduce spatial localization in the x variable.To this effect, let ρ be a smooth, non-negative, compactly supported function in R 2n+1 such that µ∈Z 2n+1 ρ(x − µ) = 1 for all x ∈ R 2n+1 .Define e i2 k−4 Φ (x,t,y) a (x, t, y)f (y, ȳ)dy.
Then we can write The kernel of each T k, t,µ is now supported in a set of constant size.Since Hörmander's estimate is stable under smooth perturbations, it can be applied to show that uniformly in µ and t, provided we establish that (7.2) rank (Φ ) (x,y) = 2n + 1 with det(Φ ) (x,y) bounded from below uniformly in , t and µ.Let (7.3) Ξ (x, t, y) := ∇ x,t Φ (x, t, y).
We calculate where Υ ( ) and satisfies The partial derivative with respect to t is not needed at the moment but will be helpful for calculations in the next subsection.Note that since g is a homogeneous polynomial of degree 4, we have Using the above in the last co-ordinate, and subscripts 1, 2, . . ., 2n, 2n + 1 to denote partial derivatives in y 1 , y 2 , . . ., y 2n and ȳ respectively, we have Thus, for a fixed t ∈ [1, 2], the mixed Hessian (Φ ) (x,y) is of the form Using (6.1), we get , γ = 0 and note that for ≥ 400n, max{σ, λ, κ, γ, |w|} ≤ 1, using (4.4) and the fact that |w| ≤ 1 2 .Further 7.1.1.Almost orthogonality.We have thus verified (7.2) and can conclude using Hörmander's L 2 estimate that (7.5) for each µ ∈ Z 2n+1 .To avoid cumbersome notation, we suppress the dependence on k, and t and denote T k, t,µ by T µ .In order to obtain the required L 2 estimate for T k, t = µ∈Z 2n+1 T µ , by the Cotlar-Stein lemma, it suffices to show that (7.6) sup for large enough N .Because of (7.5), it is enough to consider the cases when µ = ν.The kernel of and is non-zero only for |µ − ν| ≤ M for some constant M depending on ρ.In view of (7.5), this implies the desired estimate for T µ T * ν L 2 for a large enough constant c N .We now consider T µ T * ν whose kernel is given by Integrating by parts N times yields .
By symmetry we have the same estimate for R 2n+1 |K µν * (x, x)|dx and thus it follows by the Schur test that . The above estimate implies (7.6) with N replaced by N −2n, which is still good enough for large enough N (say N > 4n + 2).An application of the Cotlar-Stein lemma concludes the proof of Proposition 4.3.7.2.Proof of Proposition 5.1.Let Ξ = ∇ x,t Φ as in (7.3), N ∈ R 2n+2 be a unit vector, and let C N (x, t, y) be the (2n + 1) × (2n + 1) cinematic curvature matrix with respect to N given by (7.8) We would like to argue as in the proof of [17,Proposition 3.4] using a variant of Stein's theorem [29] according to which the kernel estimate (5.7) holds provided the rank of the mixed Hessian (Φ ) (x,t),y = 2n + 1 and the curvature condition (7.9) N, Ξ j = 0, j = 1, . . ., 2n + 1 =⇒ rank C = 2n is satisfied; i.e. the conic surface Σ x,t parametrized by y → Ξ (x, t, y) has the maximal number 2n of nonvanishing principal curvatures.The argument would be a consequence of the method of stationary phase applied to the oscillatory integral operator given by T k, t (T k, t ) * .However, the kernel estimate in [17] was proved for a single oscillatory integral operator with a fixed phase function Φ, whereas we need to apply it for the family of operators T k, t with < k 4 .The implicit constant in Stein's theorem depends on the upper bound on finitely many spatial derivatives of the phase function Φ and the amplitude a , the size of the y support of a and the lower bound for the determinant of the invertible 2n × 2n minor of C .
We have already observed in the previous subsection that the spatial derivatives of the amplitude a are bounded from above by a constant times the corresponding derivatives of a for all large .Further, arguing similarly as in the last subsection using property (4.3) of g , it can be seen that Φ and ∇ (x,t) Φ are smooth perturbations of Φ and ∇ (x,t) Φ, where Φ = y x + x T Jy + t 2 g( −y t ) .Moreover, the size of the y support of a is bounded in , and while the same is not true for the size of the x support, arguing as in the previous subsection using (7.7), we can still conclude that Since the estimate in [17] is stable under smooth perturbations, it suffices to show that rank C = 2n and the determinant of its invertible 2n × 2n minor is bounded from below by an absolute constant independent of .
The second part of Lemma 6.1 implies that which is uniformly bounded away from 0, since Hence, we conclude that g (w) is invertible.Let C (x, t, y) denote the 2n × 2n matrix with the (i, j)-th entry given by Ξ ij , N for 1 ≤ i, j ≤ 2n.Using (7.11), we conclude that C (x, t, y) is invertible as well, with its determinant bounded from below by an absolute constant independent of for ≥ 400n.
Further, observe that for i = 2n + 1, N, Ξ ij = N, ȳ−1 Ξ j = 0 for 1 ≤ j ≤ 2n + 1.Since the cinematic curvature matrix at (x, t, y) takes the form we conclude that it is of rank 2n.This finishes the proof of Proposition 5.1.
8. Proof of Propositions 3.4 and 5.2 In this section, we prove the fixed time L 2 estimate (3.4) for the operator T k t and the kernel estimate (5.8) for the kernel K k tt corresponding to the equatorial and intermediate regions.We shall also observe that the former estimate is stable in the intermediate region, thus obtaining the L 2 estimate for T k with kernel supported around the north pole as well.We shall use Hörmander's L 2 estimate [30, ch.IX.1] and the curvature condition in [17,Proposition 3.4] again to prove Propositions 3.16 and 5.2 respectively.Since we use them for a fixed oscillatory operator (and not a family of them as in the previous section), these results are directly applicable.

8.1.
Estimates around the Equator.Recall that the phase function of T k around the equator is given by Φ = Φ Eq = y 1 H 1 (x, t, y , ȳ) with .
Here F (x, t, y) is the defining function of the Korányi sphere centred at x and of radius t 8.1.1.Proof of Proposition 3.4 at the Equator.By Hörmander's L 2 estimate, it suffices to prove that the mixed Hessian Φ x,y is of rank 2n + 1 in the support of b, with the determinant bounded from below by an absolute constant independent of t.
Taking partial derivatives with respect to y j and using the chain rule, we obtain for 2 ≤ j ≤ 2n + 1. Observe that the second term is a scalar multiple of Ξ 1 = ∇ (x,t) H for 2 ≤ j ≤ 2n + 1.Then it follows that the tangent space of Ξ at (x, t, y) is spanned by Ξ 1 , . . ., Ξ 2n+1 .We shall repeatedly use this kind of observation to simplify the expressions for tangent vectors and their derivatives.
Henceforth, I shall denote the 2n × 2n identity matrix I 2n .The above discussion implies that the mixed Hessian Φ (x,y) at x = 0 and y = 0 has the same determinant as the matrix (8.1) .
Here I y 1 is the 2n × 2n diagonal matrix with y 1 as the first diagonal entry and the rest being 1. Henceforth, we shall abbreviate H 1 (0, t, 0, ȳ) to H 1 .The determinant of the above matrix is equal to .
We use Lemma 6.2 to obtain 4|y| 2 I − ȳJ Thus the absolute value of the determinant of the matrix in (8.1) greater than or equal to Using Corollary 6.3 and the fact that |y 1 | ∼ 1, we conclude that and the implicit constant is independent of t ∈ [1,2].It follows that the same is true for Φ x,y or in other words, the mixed Hessian of Φ is of rank 2n + 1.This concludes the proof of Proposition 3.4 around the equator.
Let δ ij := δ 0 (i − j) where δ 0 denotes the Dirac measure at the origin in R. Recall that H 1 = H 1 (0, t, 0, ȳ) and thus |y| = |H 1 | when y = 0.For 2 ≤ i, j ≤ 2n + 1, we have Here we have used that y j = 0 for 2 ≤ j ≤ 2n and also the fact that the dot product of N with terms involving the first derivatives of ∇ (x,t) F is zero since they lie in the tangent space spanned by vectors Ξ 1 , . . ., Ξ 2n+1 .Using (8.4) and the properties J 2 = −I 2n and J = −J, we calculate For i = 2n + 1 and 2 ≤ j ≤ 2n, we have Let P : R 2n → R 2n−1 be the projection onto the orthogonal complement of e 1 .At the equator, when x = 0, y = 0 and ȳ = 0, we have H 1 = t and the cinematic curvature matrix with respect to N is given by Since |T | 1, the rank of the above matrix is 2n, provided [1,2].This verifies (8.3), and finishes the proof of Proposition 5.2 at the equator.8.2.Estimates in the Intermediate Region.Recall that the phase function of T k in the intermediate region is given by Φ = Φ IM = ȳ H(x, t, y) with .
Here we have used the same idea as in Remark 8.1 to get a simplified form for the tangent vectors.
We abbreviate H(0, t, y 1 e 1 ) = H.The mixed Hessian of Φ when x = 0 and y j = 0 for 2 ≤ j ≤ 2n has determinant equal to that of the matrix Using Schur's complement, the absolute value of the determinant of the above matrix is ȳ H where H y(Jy) .
Here we use the fact that y j = 0 for 2 ≤ j ≤ 2n and the dot product of N with terms involving the first derivatives of ∇ (x,t) F disappears since these terms lie in the space spanned by the tangent vectors.For 2 ≤ i ≤ 2n, using (8.5), we calculate For i = 1 and 2 ≤ j ≤ 2n, using the fact that ∂ ∂y 1 H(0, t, y 1 e 1 ) = − Thus, the rank of the above matrix is 2n, provided |D + λ −1 B B| is uniformly bounded from below by a positive constant.We have We claim that This can be seen by optimizing the function We end this section by briefly describing how the calculation in §8.2.1 also yields the fixed time L 2 estimate for T k at the poles.In §8.2.1, we showed that the determinant of Hessian of the the phase function is bounded from below by a constant times the determinant of the matrix X given by Here H is defined uniquely such that |y| 4 + | H| 2 = t 4 .At the poles, since |y| = 0 and H = t 2 , it follows that X = − t 2 2 J and ] and J satisfying the condition J 2 = −I.Thus the mixed Hessian of the phase function has rank 2n + 1 at the poles as well, which implies the fixed time estimate (3.16) for the corresponding T k via Hörmander's L 2 estimate.Observe that this argument works directly without using the Taylor's expansion (2.4) of the phase function.
Remark 8.4.We also note that arguing directly as above would not work when calculating the rank of the cinematic curvature matrix C around the poles.As is clear from (8.6), C has rank 0 when |y| = 0. Thus the scaling argument of §4 is required to understand the behaviour of the Kóranyi sphere around the poles.

Necessary Conditions
We provide five counter-examples, corresponding to each edge of the quadrilateral R in Theorem 1.2 and one for the point Q 2 , which show the necessity of all the conditions in Theorem 1.2.They are suitable modifications of the examples in [1,22,24] for the Euclidean case, which were in turn adapted from standard examples for spherical means and maximal functions.The first four counter-examples are similar to the corresponding ones in [23] for the maximal function associated to codimension two spheres on the Heisenberg groups.The fifth one is new and can be viewed as the replacement of a standard Knapp type example.It has the interesting feature that the dimensions of the set associated to the input test function are not equal to the dimensions of the set on which the output function (under the action of the operator) is large.This is quite unlike what is observed for the Euclidean Knapp examples.
We also briefly discuss the counter-examples which imply the necessary condition in Theorem 1.1 for the single average and which can be easily inferred from the counter-examples for the maximal operator.
Recall that for x = (x, x) ∈ H n , the Korányi norm of x is defined to be 9.1.The line connecting Q 1 and Q 2 .This is the necessary condition p ≤ q imposed by translation invariance and noncompactness of the group H n (see [11] for the analogous argument in the Euclidean case).9.2.The line connecting Q 2 and Q 3 .Let B δ be the ball of radius δ centered at the origin.Let f δ be the characteristic function of B 100δ .Then For 1 ≤ t ≤ 2 we consider the set (here we have used the skew symmetry of the J).We get for x ∈ R and t = |x| K .Passing to the maximal operator yields the inequality δ 2n δ (2n+1)/p , and consequently, the necessary condition 2n ≥ 2n + 1 p , that is, (1/p, 1/q) lies on or above the line connecting Q 2 and Q 3 .9.3.The point Q 2 .For p = p 2 := 2n+1 2n the L p → L p bound fails.Here one uses a modification of Stein's example [28] for the Euclidean spherical maximal function.One considers the function f α defined by f One finds that if t(x) = |x| K then for α < 1 the integrals f * σ t(x) (x) are ∞ on a set of positive measure.9.4.The line connecting Q 1 and Q 4 .For this line we just use the counterexample for the individual averaging operator, using it to bound the maximal function from below.Given t ∈ [1, 2], let g δ,t be the characteristic function of the set {y = (y, ȳ) : ||y| K − t| ≤ 100δ}.Thus g δ,t p ∼ δ 1/p .Let x = (x, x) be such that |x| ≤ δ and |x| ≤ δ.For any ω = (ω, ω) with |ω| K = 1, we have t|x Jω| 2δ.Furthermore, This yields the inequality δ (2n+1)/q ≤ δ 1/p which leads to the necessary condition that is, (1/p, 1/q) lies on or above the line connecting Q 1 and Q 4 . δ.

Sparse Bounds
As mentioned in the introduction, the principal goal of [10] was to derive for the lacunary maximal operator M lac := sup k≥0 |A 2 k (x)| an inequality of the form (10.1) where the supremum is taken over sparse families of nonisotropic Heisenberg cubes (see [10] for precise definitions and constructions) and the sparse form Λ S,p 1 ,p 2 is given by Λ S,p 1 ,p 2 (f, w) By using the sharp L p → L q bounds in Theorem 1.1 and by appealing to the arguments in [2,10], we can prove the following theorem.
The above result is sharp up to the boundary and improves upon the main result in [10].Similarly, by applying the reasoning in [2,10] and using the L p → L q bounds in Theorem 1.2, we can prove the following result which is new and also sharp up to the boundary.The proof of sparse bounds for the global maximal operator by using L p → L q estimates for localized maximal functions was pioneered by Lacey [14] in his work on the Euclidean spherical maximal function.The recent papers [4] and [7] give very general results about this correspondence for Euclidean spaces and spaces of homogeneous type, respectively.x Jy, t).We prove the following lemma which says that the rank of the curvature matrices associated to the oscillatory integral operator corresponding to the phase function Φ is invariant under group translation.
Proof.Differentiating (11.1) and using the chain rule, we have ∇ (x,t) Φ(x, t, y) = ∇Φ (Θ(x,y),t) Here D 2 2n denotes the Hessian of Φ with respect to the first 2n coordinates.In particular, for a fixed t, the mixed Hessian of Φ at (x, y) is given by Φ x,y (x, t, y) = − D 2 2n Φ + Since the second matrix in the product on the right hand side has determinant equal to one, the rank of Φ x,y (x, t, y) is the same as the rank of x,y (0, t, Θ(x, y)).
This proves the first part.For the second part, we consider the surface Σ (x,t) := ∇ (x,t) Φ(x, t, y).
The tangent space of Σ (x,t) at a point y is spanned by the column vectors of the matrix Φ (x,t),y (x, t, y).Observe that (11.2) implies that the column vectors of the matrix . also form a basis for the tangent space to Σ (x,t) at y.For 1 ≤ j ≤ 2n + 1, let Ξ j denote the j-th column vector of the matrix above, and let N ∈ R 2n+2 be a unit vector normal to Σ (x,t) , so that N, Ξ j = 0 for 1 ≤ j ≤ 2n + 1.
Recall that the cinematic curvature matrix C(x, t, y) with respect to N is a (2n + 1) × (2n + 1) dimensional matrix with the (i, j)-th entry given by N, ∂ ∂y i Ξ j (Θ(x, y), t) .We use subscript i with i ∈ {1, . . ., 2n + 1} to denote partial derivative in the direction of e i .By the chain rule, the i-th row of C(x, t, y) is given by  This clearly implies that the rank of the cinematic curvature matrix for Φ at (x, t, y) is equal to the cinematic curvature matrix rank at (0, t, Θ(x, y)).

Figure 1 .
Figure 1.The Korányi sphere centred at the origin in H 1

3 .
Combining it with Lemma 4.2 for a = 3 , b = 4 and c = yields the following.

u
(|w|).We now prove (6.2) by induction on the dimension d.It can be easily verified for d = 1, 2. Suppose now that the claim is true for d − 1 with d ≥ 3. Expressing w ∈ R d as w = (w , w d ) with w ∈ R d−1 , we set r = |w |.Using Schur's complement, we have det

7 . 4 . 7 . 1 .
Proof of Propositions 4.3 and 5.1: Estimates at the Poles We begin by proving Proposition 4.3 about the fixed time L 2 bound on operators T k, t for 400n ≤ ≤ k Proof of Proposition 4.3.Recall that for 0 ≤ ≤ k 4 , the operator T k, t is defined as follows
Arguing as in Remark 8.2, using the equations above and the fact that |N | = 1, it is not difficult to see that |ᾱ| 1.
and thus u 0 is a point of maximum on the interval.Evaluating, we get Thus |D + λ −1 B B| ≥ 3 − 2.81 = 0.19.We conclude that the rank of the cinematic curvature matrix is 2n, thus verifying condition(8.3)inthe intermediate setting and finishing the proof of Proposition 5.2.8.3.Proof of Proposition 3.4 at the Poles.