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Weighted estimates for lacunary maximal functions on homogeneous groups

Abhishek Ghosh, Rajesh K. Singh

Abstract

In this article, we study weighted estimates for a general class of lacunary maximal functions on homogeneous groups. As an application we derive improved weighted estimates for the lacunary maximal function associated to the Korányi spherical means as well as for the lacunary maximal function associated to codimension two spheres in the Heisenberg group.

Weighted estimates for lacunary maximal functions on homogeneous groups

Abstract

In this article, we study weighted estimates for a general class of lacunary maximal functions on homogeneous groups. As an application we derive improved weighted estimates for the lacunary maximal function associated to the Korányi spherical means as well as for the lacunary maximal function associated to codimension two spheres in the Heisenberg group.

Paper Structure

This paper contains 7 sections, 11 theorems, 94 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. Statement of main results
  3. Comparison with previously known results
  4. Proof of main results
  5. Proof of Theorem \ref{['main:koranyi']} and Theorem \ref{['main:hori']}
  6. Proof of Theorem \ref{['main:koranyi']}
  7. Proof of Theorem \ref{['main:hori']}

Key Result

Theorem 1.1

Let $G$ be a homogeneous group and let $\mu$ be a finite, compactly supported Borel measure on $G$ satisfying the Curavture assumption (CA). Then $\mathcal{S}_{lac}[\mu]$ maps $L^p(G)$ to $L^p(G)$ for $1<p\leq \infty.$

Theorems & Definitions (21)

  • Theorem 1.1: HickmanJFA
  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.1
  • Theorem 1.5
  • Theorem 1.6: Rajula
  • Lemma 1.7: Lemma 6, Sato2013
  • proof : Proof of Theorem \ref{['main:suff']}
  • ...and 11 more