Local Comparability of Measures, Averaging and Maximal Averaging Operators

We study the boundedness properties of averaging and maximal averaging operators, under the following local comparability condition for measures: Intersecting balls of the same radius have comparable sizes. In geometrically doubling spaces, this property yields the weak type (1,1) of the uncentered...

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Bibliographic Details
Published inPotential analysis Vol. 49; no. 2; pp. 309 - 330
Main Author Aldaz, J. M.
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.08.2018
Springer Nature B.V
Subjects
Functional Analysis
Geometry
Mathematics
Mathematics and Statistics
Operators
Potential Theory
Probability Theory and Stochastic Processes
Averaging operators
Maximal averaging operators
42B25
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Summary:We study the boundedness properties of averaging and maximal averaging operators, under the following local comparability condition for measures: Intersecting balls of the same radius have comparable sizes. In geometrically doubling spaces, this property yields the weak type (1,1) of the uncentered maximal operator. We explore when local comparability implies doubling, and when it is more general. We also study the concrete case of the standard gaussian measure, where this property fails, but nevertheless averaging operators are uniformly bounded, with respect to the radius, in L 1 . However, such bounds grow exponentially fast with the dimension.
ISSN:0926-2601
1572-929X
DOI:10.1007/s11118-017-9658-2