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

  • *Corresponding Author

    *Corresponding Author 
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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  • We show that the operator

    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]. $

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

    Citation:

    \begin{equation} \\ \end{equation}
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    [2] M. W. Biccari Umberto and E. Zuazua, Local elliptic regularity for the dirichlet fractional laplacian, Adv. Nonlinear Stud., 17 (2017), 387-409.  doi: 10.1515/ans-2017-0014.
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