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26 April 2024 |
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Article overview
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On the Calderon-Zygmund property of Riesz-transform type operators arising in nonlocal equations | Sasikarn Yeepo
; Wicharn Lewkeeratiyutkul
; Sujin Khomrutai
; Armin Schikorra
; | Date: |
8 Jul 2020 | 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 $K in L^infty(mathbb{R}^n imes
mathbb{R}^n)$, 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}
ight ), left (|x-z_2|^{s_2-n}
-|y-z_2|^{s_2-n}
ight )}{|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)),
(varphi(x)-varphi(y))}{|x-y|^{n+2s}}, dx, dy = f[varphi]. ] | Source: | arXiv, 2007.4173 | Services: | Forum | Review | PDF | Favorites |
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