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Improved endpoint bounds for the lacunary spherical maximal operator

Laura Cladek, Ben Krause

Abstract

We prove new endpoint bounds for the lacunary spherical maximal operator and as a consequence obtain almost everywhere pointwise convergence of lacunary spherical means for functions locally in $L\log\log\log L(\log\log\log\log L)^{1+ε}$ for any $ε>0$.

Improved endpoint bounds for the lacunary spherical maximal operator

Abstract

We prove new endpoint bounds for the lacunary spherical maximal operator and as a consequence obtain almost everywhere pointwise convergence of lacunary spherical means for functions locally in for any .

Paper Structure

This paper contains 4 sections, 14 theorems, 128 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary Reductions
  3. Proof of Proposition \ref{['mainprop2']}
  4. Proof of Proposition \ref{['mainpropo2']}

Key Result

Proposition 1.1

There exists a constant $C$ such that for all measurable $f=\chi_E$ and all $\alpha>0$ we have

Figures (3)

  • Figure 1: The scales $c_j$. Here $c_0=\gamma$ and the circle has radius $2^k$ for some $k$ with $2^k\ge \max(l(q), (\gamma\cdot\alpha^{-1})^{1/(d-1)})$.
  • Figure 2: Domination of the kernel $\sigma_k\ast\sigma_k$ in the sphere of radius $\frac{1}{100}\max(l(q), (\gamma\alpha^{-1})^{1/(d-1))}$ centered at the origin.
  • Figure 3: The two blue rectangles represent the set $x-y+(R)_1$.

Theorems & Definitions (29)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3: Structural decomposition lemma
  • Proposition 2.4
  • Remark 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 19 more