Circular average relative to fractal measures

Seheon Ham; Hyerim Ko; Sanghyuk Lee

doi:10.3934/cpaa.2022100

Article Contents

2022, Volume 21, Issue 10: 3283-3307. Doi: 10.3934/cpaa.2022100

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Circular average relative to fractal measures

Department of Mathematical Sciences and RIM, Seoul National University, Seoul 08826, Republic of Korea

^*Corresponding author
^*Corresponding author

Received: September 30, 2021

Revised: April 30, 2022

Early access: June 2022

Published: October 2022

This work was supported by the NRF (Republic of Korea) grants No. 2017R1C1B2002959 (Ham), No. 2022R1I1A1A01055527 (Ko), and No. 2022R1A4A1018904 (Lee)

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Abstract

We prove new $ L^p $–$ L^q $ estimates for averages over dilates of the circle with respect to fractal measures, which unify different types of maximal estimates for the circular average. Our results are consequences of $ L^p $–$ L^q $ smoothing estimates for the wave operator relative to fractal measures. We also discuss similar results concerning the spherical averages.

Keywords:

circular average,
general measures.

Mathematics Subject Classification: 42B25.

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