On families between the Hardy–Littlewood and spherical maximal functions

We study a family of maximal operators that provides a continuous link connecting the Hardy–Littlewood maximal function to the spherical maximal function. Our theorems are proved in the multilinear setting but may contain new results even in the linear case. For this family of operators we obtain bounds between Lebesgue spaces in the optimal range of exponents.

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The authors acknowledge the support of the Simons Foundation (Grant 624733). The second author acknowledges the Simons Fellowship 819503.

In the published form of the paper, statement (14) is missing a condition while the proofs of (13) and (14) in Theorem 1 contain errors. The corrected statement and proofs can be accessed via the arXiv link: (arxiv.org/abs/2005.02437v2).

Received 5 May 2020

Received revised 16 November 2020

Accepted 27 November 2020

Published 11 November 2021