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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.


    \begin{equation} \\ \end{equation}
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  • [1] P. AuscherS. BortzM. Egert and O. Saari, Nonlocal self-improving properties: a functional analytic approach, Tunisian J. Math., 1 (2019), 151-183.  doi: 10.2140/tunis.2019.1.151.
    [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.
    [3] J. Chaker and M. Kassmann, Nonlocal operators with singular anisotropic kernels, Commun. Partial Differ. Equ., 45 (2020), 1-31.  doi: 10.1080/03605302.2019.1651335.
    [4] M. Cozzi, Interior regularity of solutions of non-local equations in sobolev and nikol'skii spaces, Annali di Matematica Pura ed Applicata, 196 (2017), 555-578.  doi: 10.1007/s10231-016-0586-3.
    [5] H. Dong and D. Kim, On Lp-estimates for a class of non-local elliptic equations, J. Funct. Anal., 262 (2012), 1166-1199.  doi: 10.1016/j.jfa.2011.11.002.
    [6] M. M. Fall, Regularity results for nonlocal equations and applications, Calc. Var. Partial Differ. Equ., 59 (2020), 53pp. doi: 10.1007/s00526-020-01821-6.
    [7] M. FelsingerM. Kassmann and P. Voigt, The Dirichlet problem for nonlocal operators, Math. Z., 279 (2015), 779-809.  doi: 10.1007/s00209-014-1394-3.
    [8] L. Grafakos, Modern Fourier Analysis, Springer, New York, 2014. doi: 10.1007/978-1-4939-1230-8.
    [9] T. Iwaniec and C. Sbordone, Riesz transforms and elliptic PDEs with VMO coefficients, J. Anal. Math., 74 (1998), 183-212.  doi: 10.1007/BF02819450.
    [10] M. Kassmann, A priori estimates for integro-differential operators with measurable kernels, Calc. Var. Partial Differ. Equ., 34 (2009), 1-21.  doi: 10.1007/s00526-008-0173-6.
    [11] T. KuusiG. Mingione and Y. Sire, Nonlocal self-improving properties, Anal. Partial Differ. Equ., 8 (2015), 57-114.  doi: 10.2140/apde.2015.8.57.
    [12] T. Mengesha, A. Schikorra and S. Yeepo, Calderon-Zygmund type estimates for nonlocal PDE with HC6lder continuous kernel, Adv. Math., 383 (2021), 107692. doi: 10.1016/j. aim. 2021.107692.
    [13] S. Nowak, $H^{s, p}$ regularity theory for a class of nonlocal elliptic equations, Nonlinear Anal., 195 (2020), 111730, 28. doi: 10.1016/j. na. 2019.111730.
    [14] S. Nowak, Higher Hölder regularity for nonlocal equations with irregular kernel, Calc. Var. Partial Differ. Equ., 60 (2021), 24. doi: 10.1007/s00526-020-01915-1.
    [15] S. Nowak, Regularity theory for nonlocal equations with VMO coefficients, arXiv: 2101.11690.
    [16] T. Runst and W. Sickel, De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin, 1996. doi: 10.1515/9783110812411.
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