On Lp-estimates for a class of non-local elliptic equations

Journal of Functional Analysis

Volumn 262, Issue 3, 2012, Pages 1166-1199

  Dong, Hongjie ^a   Kim, Doyoon ^b  

^a BROWN UNIVERSITY   (United States)

^b Kyung Hee University   (South Korea)

Author keywords

Bessel potential spaces; L vy processes; Non local elliptic equations; The martingale problem

Indexed keywords

EID: 84155167485     PISSN: 00221236     EISSN: 10960783     Source Type: Journal    
DOI: 10.1016/j.jfa.2011.11.002     Document Type: Article

References (28)

1. Abels H., Kassmann M. The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels. Osaka J. Math. 2009, 46(3):661-683.
2. Bañuelos R., Bogdan K. Lévy processes and Fourier multipliers. J. Funct. Anal. 2007, 250(1):197-213.
3. Barles G., Chasseigne E., Ciomaga A., Imbert C. Lipschitz regularity of solutions for mixed integro-differential equations arxiv:1107.3228.
4. Barles G., Chasseigne E., Imbert C. Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations. J. Eur. Math. Soc. (JEMS) 2011, 13(1):1-26.
5. Bass R.F., Kassmann M. Harnack inequalities for non-local operators of variable order. Trans. Amer. Math. Soc. 2005, 357(2):837-850.
6. Bass R.F., Kassmann M. Hölder continuity of harmonic functions with respect to operators of variable order. Comm. Partial Differential Equations 2005, 30(7-9):1249-1259.
7. Bass R.F., Levin D.A. Harnack inequalities for jump processes. Potential Anal. 2002, 17(4):375-388.
8. Caffarelli L., Silvestre L. Regularity theory for fully nonlinear integro-differential equations. Comm. Pure Appl. Math. 2009, 62(5):597-638.
9. Cont R., Tankov P. Financial Modelling with Jump Processes. Chapman & Hall/CRC Financ. Math. Ser. 2004, Chapman & Hall/CRC, Boca Raton, FL.
10. DiBenedetto E., Manfredi J. On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems. Amer. J. Math. 1993, 115(5):1107-1134.
11. p-solvability of higher order parabolic and elliptic systems with BMO coefficients. Arch. Ration. Mech. Anal. 2011, 199(3):889-941.
12. N. Guillen, R. Schwab, Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations, preprint, 2011, arxiv:1101.0279.
13. W. Hoh, Pseudo differential operators generating Markov processes, PhD thesis, Univ. Erlangen, Nürnberg, 1992.
14. p-estimates for degenerated elliptic operators. Studia Math. 1983, 75(3):293-312.
15. Jacob N. Pseudo Differential Operators and Markov Processes, vols. I, II. Fourier Analysis and Semigroups 2001, Imperial College Press, London.
16. Kassmann M. A priori estimates for integro-differential operators with measurable kernels. Calc. Var. Partial Differential Equations 2009, 34:1-21.
17. Y. Kim, K. Lee, Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels, preprint, 2010, arxiv:1011.3565v2.
18. Komatsu T. On the martingale problem for generators of stable processes with perturbations. Osaka J. Math. 1984, 21(1):113-132.
19. Komatsu T. Pseudodifferential operators and Markov processes. J. Math. Soc. Japan 1984, 36:387-418.
20. Krylov N.V. Parabolic and elliptic equations with VMO coefficients. Comm. Partial Differential Equations 2007, 32(1-3):453-475.
21. Krylov N.V. Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms. J. Funct. Anal. 2007, 250(2):521-558.
22. Krylov N.V. Lectures on Elliptic and Parabolic Equations in Sobolev Spaces 2008, Amer. Math. Soc.
23. Mikulevicius R., Pragarauskas H. On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces. Liet. Mat. Rink.. Lith. Math. J. 1992/1993, 32(2):238-264. translation in.
24. Mikulevicius R., Pragarauskas H. On the uniqueness of solutions of the martingale problem that is associated with degenerate Lévy operators. Liet. Mat. Rink.. Lith. Math. J. 1993/1994, 33(4):352-367. translation in.
25. R. Mikulevicius, H. Pragarauskas, On Lp theory for Zakai equation with discontinuous observation process, preprint, 2010, arxiv:1012.5816v1.
26. Negoro A., Tsuchiya M. Stochastic processes and semigroups associated with degenerate Lévy generating operators. Stochastics Stochastics Rep. 1989, 26:29-61.
27. Silvestre L. Hölder estimates for solutions of integro-differential equations like the fractional Laplace. Indiana Univ. Math. J. 2006, 55(3):1155-1174.
28. Stein E.M. Singular Integrals and Differentiability Properties of Functions. Princeton Math. Ser. 1970, vol. 30. Princeton University Press, Princeton, NJ, xiv+290 pp.