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Local Comparability of Measures, Averaging and Maximal Averaging Operators

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Abstract

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.

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Acknowledgments

The author was partially supported by Grant MTM2015-65792-P of the MINECO of Spain, and also by by ICMAT Severo Ochoa project SEV-2015-0554 (MINECO).

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Authors and Affiliations

  1. Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Madrid, Spain

    J. M. Aldaz

  2. Departamento de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, 28049, Madrid, Spain

    J. M. Aldaz

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  1. J. M. Aldaz
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Correspondence to J. M. Aldaz.

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Aldaz, J.M. Local Comparability of Measures, Averaging and Maximal Averaging Operators. Potential Anal 49, 309–330 (2018). https://doi.org/10.1007/s11118-017-9658-2

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  • DOI: https://doi.org/10.1007/s11118-017-9658-2

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