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Register a new userOn the lacunary spherical maximal function on the Heisenberg group
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In this paper we investigate the $L^p$ boundedness of the lacunary maximal function $ M_{Ha}^{lac} $ associated to the spherical means $ A_r f$ taken over Koranyi spheres on the Heisenberg group. Closely following an approach used by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions leading to new unweighted and weighted estimates. The key ingredients in the proof are the $L^p$ improving property of the operator $A_rf$ and a continuity property of the difference $A_rf-tau_y A_rf$, where $tau_yf(x)=f(xy^{-1})$ is the right translation operator.
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We prove new $ell ^{p} (mathbb Z ^{d})$ bounds for discrete spherical averages in dimensions $ d geq 5$. We focus on the case of lacunary radii, first for general lacunary radii, and then for certain kinds of highly composite choices of radii. In particular, if $ A _{lambda } f $ is the spherical average of $ f$ over the discrete sphere of radius $ lambda $, we have begin{equation*} bigllVert sup _{k} lvert A _{lambda _k} f rvert bigrrVert _{ell ^{p} (mathbb Z ^{d})} lesssim lVert frVert _{ell ^{p} (mathbb Z ^{d})}, qquad tfrac{d-2} {d-3} < p leq tfrac{d} {d-2}, dgeq 5, end{equation*} for any lacunary sets of integers $ {lambda _k ^2 }$. We follow a style of argument from our prior paper, addressing the full supremum. The relevant maximal operator is decomposed into several parts; each part requires only one endpoint estimate.
We prove sparse bounds for the spherical maximal operator of Magyar, Stein and Wainger. The bounds are conjecturally sharp, and contain an endpoint estimate. The new method of proof is inspired by ones by Bourgain and Ionescu, is very efficient, and has not been used in the proof of sparse bounds before. The Hardy-Littlewood Circle method is used to decompose the multiplier into major and minor arc components. The efficiency arises as one only needs a single estimate on each element of the decomposition.
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this paper, we use the expansion method to prove a quantitative version of this characterization.
We prove an analogue of Chernoffs theorem for the Laplacian $ Delta_{mathbb{H}} $ on the Heisenberg group $ mathbb{H}^n.$ As an application, we prove Ingham type theorems for the group Fourier transform on $ mathbb{H}^n $ and also for the spectral projections associated to the sublaplacian.
We study maximal operators related to bases on the infinite-dimensional torus $mathbb{T}^omega$. {For the normalized Haar measure $dx$ on $mathbb{T}^omega$ it is known that $M^{mathcal{R}_0}$, the maximal operator associated with the dyadic basis $mathcal{R}_0$, is of weak type $(1,1)$, but $M^{mathcal{R}}$, the operator associated with the natural general basis $mathcal{R}$, is not. We extend the latter result to all $q in [1,infty)$. Then we find a wide class of intermediate bases $mathcal{R}_0 subset mathcal{R} subset mathcal{R}$, for which maximal functions have controlled, but sometimes very peculiar behavior.} Precisely, for given $q_0 in [1, infty)$ we construct $mathcal{R}$ such that $M^{mathcal{R}}$ is of restricted weak type $(q,q)$ if and only if $q$ belongs to a predetermined range of the form $(q_0, infty]$ or $[q_0, infty]$. Finally, we study the weighted setting, considering the Muckenhoupt $A_p^mathcal{R}(mathbb{T}^omega)$ and reverse Holder $mathrm{RH}_r^mathcal{R}(mathbb{T}^omega)$ classes of weights associated with $mathcal{R}$. For each $p in (1, infty)$ and each $w in A_p^mathcal{R}(mathbb{T}^omega)$ we obtain that $M^{mathcal{R}}$ is not bounded on $L^q(w)$ in the whole range $q in [1,infty)$. Since we are able to show that [ bigcup_{p in (1, infty)}A_p^mathcal{R}(mathbb{T}^omega) = bigcup_{r in (1, infty)} mathrm{RH}_r^mathcal{R}(mathbb{T}^omega), ] the unboundedness result applies also to all reverse Holder weights.
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