Local comparability of measures, averaging and maximal averaging operators

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.