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Sparse bounds for spherical maximal functions

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Abstract

We consider the averages of a function f on ℝn over spheres of radius 0 < r < ∞ given by \({A_r}f(x) = \int_{{\mathbb{S}^{n - 1}}} {f(x - ry)d\sigma (y)} \), where σ is the normalized rotation invariant measure on 𝕊n−1. We prove a sharp range of sparse bounds for two maximal functions, the first the lacunary spherical maximal function, and the second the full maximal function.

$${M_{1ac}}f = \mathop {\sup }\limits_{j \in \mathbb{Z}} {A_{{2^j}}}f,\;\;\;{M_{\text{full}}}f = \mathop {\sup }\limits_{r > 0} {A_r}f.$$

The sparse bounds are very precise variants of the known Lp bounds for these maximal functions. They are derived from known Lp-improving estimates for the localized versions of these maximal functions, and the indices in our sparse bound are sharp. We derive novel weighted inequalities for weights in the intersection of certain Muckenhoupt and reverse Hölder classes.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Michael T. Lacey

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  1. Michael T. Lacey
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Correspondence to Michael T. Lacey.

Additional information

Research supported in part by National Science Foundation grant DMS-1600693, and by Australian Research Council grant DP160100153. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140, while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 Semester.

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Lacey, M.T. Sparse bounds for spherical maximal functions. JAMA 139, 613–635 (2019). https://doi.org/10.1007/s11854-019-0070-2

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  • DOI: https://doi.org/10.1007/s11854-019-0070-2

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