On the Calderon-Zygmund property of Riesz-transform type operators arising in nonlocal equations

Sasikarn Yeepo; Wicharn Lewkeeratiyutkul; Sujin Khomrutai; Armin Schikorra

doi:10.3934/cpaa.2021071

Article Contents

2021, Volume 20, Issue 9: 2915-2939. Doi: 10.3934/cpaa.2021071

This issue Previous Article Local well-posedness in Sobolev spaces for first-order barotropic causal relativistic viscous hydrodynamics Next Article The curved symmetric

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On the Calderon-Zygmund property of Riesz-transform type operators arising in nonlocal equations

1.
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
2.
Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA

^*Corresponding Author
^*Corresponding Author

Received: July 2020

Revised: March 2021

Early access: June 2021

Published: September 2021

A. Schikorra is supported by Simons foundation, grant no 579261.S. Yeepo is supported by Science Achievement Scholarship of Thailand (SAST) and CU Graduate School Thesis Grant

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Abstract

We show that the operator

$ T_{K,s_1,s_2}f({z_1}) : = \int_{ \mathbb{R}^n} A_{K,s_1,s_2}(z_1,z_2) f(z_2)\, dz_2 $

is a Calderon-Zygmund operator. Here for $ K \in L^\infty( \mathbb{R}^n \times \mathbb{R}^n) $, and $ s,s_1,s_2 \in (0,1) $ with $ s_1+s_2 = 2s $ we have

$ \begin{align*} & A_{K,s_1,s_2}(z_1,z_2) \\ = &\!\int_{ \mathbb{R}^n}\!\! \int_{ \mathbb{R}^n}\!\! \frac{K(x,y)\! \left ({|x\!-\!z_1|^{s_1-n} \!-\!|y\!-\!z_1|^{s_1-n}} \right ) \!\left ({|x\!-\!z_2|^{s_2-n} \!-\!|y\!-\!z_2|^{s_2-n}} \right )}{|x\!-\!y|^{n+2s}} dx dy. \end{align*} $

This operator is motivated by the recent work [12] where it appeared as analogue of the Riesz transforms for the equation

$ \int_{ \mathbb{R}^n} \int_{ \mathbb{R}^n} \frac{K(x,y) (u(x)-u(y))\, (\varphi(x)-\varphi(y))}{|x-y|^{n+2s}}\, dx\, dy = f[\varphi]. $

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Mathematics Subject Classification: Primary: 35R11 Secondary: 35J99 46E35 47G30.

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