Problems on averages and lacunary maximal functions

TL;DR

This paper investigates convolution operators and lacunary maximal functions related to dilates of measures, establishing boundedness conditions, regularity results, and posing open problems in harmonic analysis.

Contribution

It provides new bounds for lacunary maximal operators, characterizes $L^2$ boundedness for averages over convex curves, and proves $L^p$ regularity for these averages.

Findings

Established $H^1$ to $L^{1, }infty$ bounds under dimensional and regularity assumptions.

Derived a necessary and sufficient condition for $L^2$ boundedness of lacunary maximal operators.

Proved an $L^p$ regularity result for averages over convex curves.

Abstract

We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First, we obtain an $H^1$ to $L^{1,\infty}$ bound for lacunary maximal operators under a dimensional assumption on the underlying measure and an assumption on an $L^p$ regularity bound for some $p>1$. Secondly, we obtain a necessary and sufficient condition for $L^2$ boundedness of lacunary maximal operator associated to averages over convex curves in the plane. Finally we prove an $L^p$ regularity result for such averages. We formulate various open problems.