# American Institute of Mathematical Sciences

June  2013, 33(6): 2319-2347. doi: 10.3934/dcds.2013.33.2319

## Schauder estimates for a class of non-local elliptic equations

 1 Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912 2 Department of Applied Mathematics, Kyung Hee University, 1732 Deogyeong-daero, Giheung-gu, Yongin-si, Gyeonggi-do 446-701, South Korea

Received  January 2012 Revised  May 2012 Published  December 2012

We prove Schauder estimates for a class of non-local elliptic operators with kernel $K(y)=a(y)/|y|^{d+\sigma}$ and either Dini or Hölder continuous data. Here $0 < \sigma < 2$ is a constant and $a$ is a bounded measurable function, which is not necessarily to be homogeneous, regular, or symmetric. As an application, we prove that the operators give isomorphisms between the Lipschitz--Zygmund spaces $\Lambda^{\alpha+\sigma}$ and $\Lambda^\alpha$ for any $\alpha>0$. Several local estimates and an extension to operators with kernels $K(x,y)$ are also discussed.
Citation: Hongjie Dong, Doyoon Kim. Schauder estimates for a class of non-local elliptic equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2319-2347. doi: 10.3934/dcds.2013.33.2319
##### References:
 [1] H. Abels and M. Kassmann, The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels,, Osaka J. Math., 46 (2009), 661. [2] R. F. Bass, Regularity results for stable-like operators,, J. Funct. Anal., 257 (2009), 2693. doi: 10.1016/j.jfa.2009.05.012. [3] R. F. Bass and M. Kassmann, Harnack inequalities for non-local operators of variable order,, Trans. Amer. Math. Soc., 357 (2005), 837. doi: 10.1090/S0002-9947-04-03549-4. [4] R. F. Bass and M. Kassmann, Hölder continuity of harmonic functions with respect to operators of variable order,, Comm. Partial Differential Equations, 30 (2005), 1249. doi: 10.1080/03605300500257677. [5] R. F. Bass and D. A. Levin, Harnack inequalities for jump processes,, Potential Anal., 17 (2002), 375. doi: 10.1023/A:1016378210944. [6] G. Barles, E. Chasseigne and C. Imbert, Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations,, J. Eur. Math. Soc., 13 (2011), 1. doi: 10.4171/JEMS/242. [7] L. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations,", American Mathematical Society, (1995). [8] L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations,, Comm. Pure Appl. Math., 62 (2009), 597. doi: 10.1002/cpa.20274. [9] L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation,, Arch. Ration. Mech. Anal., 200 (2011), 59. doi: 10.1007/s00205-010-0336-4. [10] H. Dong and D. Kim, On $L_p$-estimates for a class of non-local elliptic equations,, J. Funct. Anal., 262 (2012), 1166. doi: 10.1016/j.jfa.2011.11.002. [11] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 ed. Springer-Verlag, (1998). [12] N. Jacob, "Pseudo Differential Operators and Markov Processes. Vol. I. Fourier Analysis and Semigroups,", Imperial College Press, (2001). doi: 10.1142/9781860949746. [13] N. Jacob, "Pseudo Differential Operators and Markov Processes. Vol. II. Generators and Their Potential Theory,", Imperial College Press, (2001). doi: 10.1142/9781860949562. [14] M. Kassmann, A priori estimates for integro-differential operators with measurable kernels,, Calc. Var. Partial Differential Equations, 34 (2009), 1. doi: 10.1007/s00526-008-0173-6. [15] Y. Kim and K. Lee, Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels,, Manuscripta Math., 139 (2012), 291. [16] N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Hölder Spaces,", American Mathematical Society, (1996). [17] G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific Publishing Co., (1996). [18] R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces,, Liet. Mat. Rink., 32 (1992), 299. doi: 10.1007/BF02450422. [19] R. Mikulevicius and H. Pragarauskas, On Hölder solutions of the integro-differential Zakai equation,, Stochastic Process. Appl., 119 (2009), 3319. doi: 10.1016/j.spa.2009.05.008. [20] R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the corresponding martingale problem,, preprint, (). [21] L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional laplace,, Indiana Univ. Math. J., 55 (2006), 1155. doi: 10.1512/iumj.2006.55.2706. [22] E. Sperner, Schauder's existence theorem for $\alpha$-Dini continuous data,, Ark. Mat., 19 (1981), 193. doi: 10.1007/BF02384477. [23] E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, (1970). [24] E. M. Stein, "Harmonic Analysis: Real-Variable Methods Orthogonality, and Oscillatory Integrals,", Princeton Mathematical Series, 43 (1993). [25] X.-J. Wang, Schauder estimates for elliptic and parabolic equations,, Chinese Ann. Math. Ser. B, 27 (2006), 637. doi: 10.1007/s11401-006-0142-3.

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##### References:
 [1] H. Abels and M. Kassmann, The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels,, Osaka J. Math., 46 (2009), 661. [2] R. F. Bass, Regularity results for stable-like operators,, J. Funct. Anal., 257 (2009), 2693. doi: 10.1016/j.jfa.2009.05.012. [3] R. F. Bass and M. Kassmann, Harnack inequalities for non-local operators of variable order,, Trans. Amer. Math. Soc., 357 (2005), 837. doi: 10.1090/S0002-9947-04-03549-4. [4] R. F. Bass and M. Kassmann, Hölder continuity of harmonic functions with respect to operators of variable order,, Comm. Partial Differential Equations, 30 (2005), 1249. doi: 10.1080/03605300500257677. [5] R. F. Bass and D. A. Levin, Harnack inequalities for jump processes,, Potential Anal., 17 (2002), 375. doi: 10.1023/A:1016378210944. [6] G. Barles, E. Chasseigne and C. Imbert, Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations,, J. Eur. Math. Soc., 13 (2011), 1. doi: 10.4171/JEMS/242. [7] L. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations,", American Mathematical Society, (1995). [8] L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations,, Comm. Pure Appl. Math., 62 (2009), 597. doi: 10.1002/cpa.20274. [9] L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation,, Arch. Ration. Mech. Anal., 200 (2011), 59. doi: 10.1007/s00205-010-0336-4. [10] H. Dong and D. Kim, On $L_p$-estimates for a class of non-local elliptic equations,, J. Funct. Anal., 262 (2012), 1166. doi: 10.1016/j.jfa.2011.11.002. [11] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 ed. Springer-Verlag, (1998). [12] N. Jacob, "Pseudo Differential Operators and Markov Processes. Vol. I. Fourier Analysis and Semigroups,", Imperial College Press, (2001). doi: 10.1142/9781860949746. [13] N. Jacob, "Pseudo Differential Operators and Markov Processes. Vol. II. Generators and Their Potential Theory,", Imperial College Press, (2001). doi: 10.1142/9781860949562. [14] M. Kassmann, A priori estimates for integro-differential operators with measurable kernels,, Calc. Var. Partial Differential Equations, 34 (2009), 1. doi: 10.1007/s00526-008-0173-6. [15] Y. Kim and K. Lee, Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels,, Manuscripta Math., 139 (2012), 291. [16] N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Hölder Spaces,", American Mathematical Society, (1996). [17] G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific Publishing Co., (1996). [18] R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces,, Liet. Mat. Rink., 32 (1992), 299. doi: 10.1007/BF02450422. [19] R. Mikulevicius and H. Pragarauskas, On Hölder solutions of the integro-differential Zakai equation,, Stochastic Process. Appl., 119 (2009), 3319. doi: 10.1016/j.spa.2009.05.008. [20] R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the corresponding martingale problem,, preprint, (). [21] L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional laplace,, Indiana Univ. Math. J., 55 (2006), 1155. doi: 10.1512/iumj.2006.55.2706. [22] E. Sperner, Schauder's existence theorem for $\alpha$-Dini continuous data,, Ark. Mat., 19 (1981), 193. doi: 10.1007/BF02384477. [23] E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, (1970). [24] E. M. Stein, "Harmonic Analysis: Real-Variable Methods Orthogonality, and Oscillatory Integrals,", Princeton Mathematical Series, 43 (1993). [25] X.-J. Wang, Schauder estimates for elliptic and parabolic equations,, Chinese Ann. Math. Ser. B, 27 (2006), 637. doi: 10.1007/s11401-006-0142-3.
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