Local comparability of measures, averaging and maximal averaging operators

TL;DR

This paper investigates how a local measure comparability condition affects the boundedness of averaging and maximal operators, comparing it to doubling measures, with specific focus on Gaussian measures and their dimensional behavior.

Contribution

It characterizes when local comparability is equivalent to doubling measures and analyzes the boundedness of operators under Gaussian measures across dimensions.

Findings

Local comparability implies doubling in geometrically doubling spaces.

Averaging operators are uniformly bounded in $L^1$ for Gaussian measures, despite failure of local comparability.

Bounds grow exponentially with dimension under Gaussian measures.

Abstract

We explore the consequences for the boundedness properties of averaging and maximal averaging operators, of the following local comparabiliity condition for measures: Intersecting balls of the same radius have comparable sizes. Since in geometrically doubling spaces this property yields the same results as doubling, we study under which circumstances it is equivalent to the latter condition, and when it is more general. We also study the concrete case of the standard gaussian measure, where this property fails, but nevertheles averaging operators are uniformly bounded, with respect to the radius, in $L^1$. However, such bounds grow exponentially fast with the dimension, very much unlike the case of Lebesgue measure.