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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 |
$ 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 $ |
$ K \in L^\infty( \mathbb{R}^n \times \mathbb{R}^n) $ |
$ s,s_1,s_2 \in (0,1) $ |
$ s_1+s_2 = 2s $ |
$ \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*} $ |
$ \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]. $ |
References:
[1] |
P. Auscher, S. Bortz, M. 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. Felsinger, M. 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. Kuusi, G. 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. |
show all references
References:
[1] |
P. Auscher, S. Bortz, M. 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. Felsinger, M. 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. Kuusi, G. 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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