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

Abstract

We show that the operator \[ T_{K,s_1,s_2}f(z) := \int_{\mathbb{R}^n} A_{K,s_1,s_2}(z_1,z_2) f(z_2)\, dz_2 \] 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 \[ A_{K,s_1,s_2}(z_1,z_2) = \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{K(x,y) \left (|x-z_1|^{s_1-n} -|y-z_1|^{s_1-n} \right )\, \left (|x-z_2|^{s_2-n} -|y-z_2|^{s_2-n}\right )}{|x-y|^{n+2s}}\, dx\, dy. \] This operator is motivated by the recent work by Mengesha-Schikorra-Yeepo where it appeared as analogue of the Riesz transforms for the equation \[ \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{K(x,y) (u(x)-u(y))\, (φ(x)-φ(y))}{|x-y|^{n+2s}}\, dx\, dy = f[φ]. \]

References in corpus (2)

• Calderon-Zygmund type estimates for nonlocal PDE with Hölder continuous kernel
• Higher Hölder regularity for nonlocal equations with irregular kernel

Cited by in corpus (1)

• Calderon-Zygmund type estimates for nonlocal PDE with Hölder continuous kernel