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April 2022 Variational inequalities for hypersingular integrals with variable kernels
Yanping Chen, Zhenbing Gong
Rocky Mountain J. Math. 52(2): 445-470 (April 2022). DOI: 10.1216/rmj.2022.52.445

Abstract

We systematically study variational inequalities for hypersingular integral operators. More precisely, we show the variational inequalities for the families 𝒯α:={Tα,𝜀}𝜀>0 of truncated hypersingular integrals with variable kernels, which are defined by

Tα,𝜀f(x)=|xy|>𝜀Ω(x,xy)|xy|n+αf(y)dy,

where α0 and the kernel Ω belongs to L(n)×L1(𝕊n1). We first prove that the variation of the hypersingular integral with variable kernel is bounded from the Sobolev space L˙α2 to the Lebesgue space L2 when ΩL(n)×Lq(𝕊n1) for q>max{1,2(n1)(n+2α)} and satisfies some cancellation condition in its second variable. The result is sharp in the sense that the (L˙α2,L2) boundedness of 𝒯α fails if q2(n1)(n+2α). After strengthening the smoothness of Ω(x,z) in its second variable, we give the weighted boundedness of the variation of the hypersingular integrals with smooth variable kernels from L˙αp(w) to Lp(w) for 1<p< and wAp. Finally, we extend the result to the Sobolev–Morrey spaces.

Citation

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Yanping Chen. Zhenbing Gong. "Variational inequalities for hypersingular integrals with variable kernels." Rocky Mountain J. Math. 52 (2) 445 - 470, April 2022. https://doi.org/10.1216/rmj.2022.52.445

Information

Received: 17 March 2021; Accepted: 14 July 2021; Published: April 2022
First available in Project Euclid: 17 May 2022

Digital Object Identifier: 10.1216/rmj.2022.52.445

Subjects:
Primary: 42B20 , 42B25

Keywords: Ap weight , hypersingular integrals , Sobolev–Morrey space , variable kernel , variational inequalities

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium

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Vol.52 • No. 2 • April 2022
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