• Previous Article
    The curved symmetric $ 2 $– and $ 3 $–center problem on constant negative surfaces
  • CPAA Home
  • This Issue
  • Next Article
    Local well-posedness in Sobolev spaces for first-order barotropic causal relativistic viscous hydrodynamics
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

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]. $
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. 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.

show all references

References:
[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.

[1]

Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054

[2]

Philipp Reiter. Regularity theory for the Möbius energy. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1463-1471. doi: 10.3934/cpaa.2010.9.1463

[3]

Xavier Cabré. Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 425-457. doi: 10.3934/dcds.2008.20.425

[4]

Huilian Jia, Lihe Wang, Fengping Yao, Shulin Zhou. Regularity theory in Orlicz spaces for the poisson and heat equations. Communications on Pure and Applied Analysis, 2008, 7 (2) : 407-416. doi: 10.3934/cpaa.2008.7.407

[5]

Sun-Sig Byun, Hongbin Chen, Mijoung Kim, Lihe Wang. Lp regularity theory for linear elliptic systems. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 121-134. doi: 10.3934/dcds.2007.18.121

[6]

Marco Squassina. Preface: Recent progresses in the theory of nonlinear nonlocal problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : i-i. doi: 10.3934/dcdss.201803i

[7]

Mouhamed Moustapha Fall. Regularity estimates for nonlocal Schrödinger equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1405-1456. doi: 10.3934/dcds.2019061

[8]

Mostafa Fazly. Regularity of extremal solutions of nonlocal elliptic systems. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 107-131. doi: 10.3934/dcds.2020005

[9]

King-Yeung Lam. Dirac-concentrations in an integro-pde model from evolutionary game theory. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 737-754. doi: 10.3934/dcdsb.2018205

[10]

Yong Fang, Patrick Foulon, Boris Hasselblatt. Zygmund strong foliations in higher dimension. Journal of Modern Dynamics, 2010, 4 (3) : 549-569. doi: 10.3934/jmd.2010.4.549

[11]

Peter E. Kloeden, Stefanie Sonner, Christina Surulescu. A nonlocal sample dependence SDE-PDE system modeling proton dynamics in a tumor. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2233-2254. doi: 10.3934/dcdsb.2016045

[12]

Fatimzehrae Ait Bella, Aissam Hadri, Abdelilah Hakim, Amine Laghrib. A nonlocal Weickert type PDE applied to multi-frame super-resolution. Evolution Equations and Control Theory, 2021, 10 (3) : 633-655. doi: 10.3934/eect.2020084

[13]

Jean-Daniel Djida, Gisèle Mophou, Mahamadi Warma. Optimal control of mixed local-nonlocal parabolic PDE with singular boundary-exterior data. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022015

[14]

Chun Liu. Dynamic theory for incompressible Smectic-A liquid crystals: Existence and regularity. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 591-608. doi: 10.3934/dcds.2000.6.591

[15]

Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310

[16]

Jerrold E. Marsden, Alexey Tret'yakov. Factor analysis of nonlinear mappings: p-regularity theory. Communications on Pure and Applied Analysis, 2003, 2 (4) : 425-445. doi: 10.3934/cpaa.2003.2.425

[17]

Carlos Lizama, Marina Murillo-Arcila. Discrete maximal regularity for volterra equations and nonlocal time-stepping schemes. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 509-528. doi: 10.3934/dcds.2020020

[18]

Filomena Feo, Pablo Raúl Stinga, Bruno Volzone. The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3269-3298. doi: 10.3934/dcds.2018142

[19]

Sunra J. N. Mosconi. Optimal elliptic regularity: A comparison between local and nonlocal equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 547-559. doi: 10.3934/dcdss.2018030

[20]

Dinh-Ke Tran, Nhu-Thang Nguyen. On regularity and stability for a class of nonlocal evolution equations with nonlinear perturbations. Communications on Pure and Applied Analysis, 2022, 21 (3) : 817-835. doi: 10.3934/cpaa.2021200

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (176)
  • HTML views (188)
  • Cited by (0)

[Back to Top]