Jump and variational inequalities for averaging operators with variable kernels

Zhenbing Gong; Yanping Chen; Wenyu Tao

doi:10.3934/cpaa.2021045

Article Contents

2021, Volume 20, Issue 5: 1851-1866. Doi: 10.3934/cpaa.2021045

This issue Previous Article Spatial asymptotics of mild solutions to the time-dependent Oseen system Next Article Strict solutions to stochastic semilinear evolution equations in M-type 2 Banach spaces

Jump and variational inequalities for averaging operators with variable kernels

Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

^* Corresponding author
^* Corresponding author

Received: August 31, 2020

Revised: December 31, 2020

Early access: March 2021

Published: May 2021

The second author is supported by NSF-China grant-11871096 and grant-11471033

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Abstract

In this paper, we prove that the jump function and variation of averaging operators with rough variable kernels are bounded on $ L^{2}(\mathbb{R}^{n}) $ if $ \Omega\in L^{\infty}(\mathbb{R}^{n})\times L^{q}(\mathbb{S}^{n-1}) $ for $ q>2(n-1)/n $ and $ n\geq2 $. Moreover, we obtain the boundedness on weighted $ L^{p}(\mathbb{R}^{n}) $ spaces of the jump function and $ \rho $-variations for averaging operators with smooth variable kernels. Finally, we extend the result to the Morrey spaces.

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Mathematics Subject Classification: Primary: 42B25; Secondary: 42B20.

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