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September  2021, 20(9): 2915-2939. doi: 10.3934/cpaa.2021071

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

Sasikarn Yeepo 1, , Wicharn Lewkeeratiyutkul 1,, , Sujin Khomrutai 1, and  Armin Schikorra 2,

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

*Corresponding Author

Received  July 2020 Revised  March 2021 Published  September 2021 Early access  June 2021

Fund Project: 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

We show that the operator
$ 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 $
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]. $
Keywords: Calderon-Zygmund, regularity theory, nonlocal PDE.
Mathematics Subject Classification: Primary: 35R11 Secondary: 35J99 46E35 47G30.
Citation: Sasikarn Yeepo, Wicharn Lewkeeratiyutkul, Sujin Khomrutai, Armin Schikorra. On the Calderon-Zygmund property of Riesz-transform type operators arising in nonlocal equations. Communications on Pure and Applied Analysis, 2021, 20 (9) : 2915-2939. doi: 10.3934/cpaa.2021071
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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