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Spherical maximal operators on Heisenberg groups: Restricted dilation sets

Joris Roos, Andreas Seeger, Rajula Srivastava

Abstract

Consider spherical means on the Heisenberg group with a codimension two incidence relation, and associated spherical local maximal functions $M_Ef$ where the dilations are restricted to a set $E$. We prove $L^p\to L^q$ estimates for these maximal operators; the results depend on various notions of dimension of $E$.

Spherical maximal operators on Heisenberg groups: Restricted dilation sets

Abstract

Consider spherical means on the Heisenberg group with a codimension two incidence relation, and associated spherical local maximal functions where the dilations are restricted to a set . We prove estimates for these maximal operators; the results depend on various notions of dimension of .
Paper Structure (13 sections, 10 theorems, 165 equations, 1 figure)

This paper contains 13 sections, 10 theorems, 165 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The operators versus their derivatives
  4. Basic estimates
  5. The point Q4
  6. Proof of Proposition \ref{['prop:kernelest']}
  7. Curvature
  8. Proof of Proposition \ref{['prop:kernelest']}, continued
  9. Proof of Lemma \ref{['lem:oio']}
  10. Necessary Conditions
  11. The line connecting Q2 and Q3
  12. The line connecting Q1 and Q4
  13. The line connecting Q3 and Q4

Key Result

Theorem 1.1

Let $n\ge 2$, $E\subset [1,2]$ with $\mathrm{dim}_\mathrm{M}\,E=\beta$ and $\mathrm{dim}_\mathrm{qA}\,E=\gamma$. Then the following hold. (i) $M_E:L^p({\mathbb {H}}^n)\to L^q({\mathbb {H}}^n)$ is bounded for $(\frac{1}{p}, \frac{1}{q})$ in the interior of $\mathcal{R}(\beta,\gamma)$, and on the line

Figures (1)

  • Figure 1: The quadrilateral $\mathcal{R}(\beta,\gamma)$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 6 more