Abstract
The aim of this work is to develop a localization technique and to establish a regularity result for non-local integro-differential operators \({\fancyscript{L}}\) of order \({\alpha\in (0,2)}\) . Thereby we extend the De Giorgi–Nash–Moser theory to non-local integro-differential operators. The operators \({\fancyscript{L}}\) under consideration generate strong Markov processes via the theory of Dirichlet forms. As is well known, regularity properties of the resolvents are important for many aspects of the corresponding stochastic process. Therefore, this work is related to probability theory and analysis, especially partial differential equations, at the same time.
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References
Abels H. and Kassmann M. (2007). An analytic approach to purely nonlocal Bellman equations arising in models of stochastic control. J. Diff. Equ. 236(1): 29–56
Adams D.R. and Lewis J.L. (1982). On Morrey–Besov inequalities. Stud. Math. 74(2): 169–182
Barlow, M.T., Bass, R.F., Chen, Z.-Q., Kassmann, M.: Non-local Dirichlet forms and symmetric jump processes. Trans. Am. Math. Soc. (in press)
Bass R.F. and Kassmann M. (2005). Hölder continuity of harmonic functions with respect to operators of variable orders. Comm. Partial Diff. Equ. 30: 1249–1259
Bass R.F. and Levin D.A. (2002). Harnack inequalities for jump processes. Potential Anal. 17(4): 375–388
Bass, R.F., Levin, D.A.: Transition probabilities for symmetric jump processes. Trans. Am. Math. Soc. 354(7), 2933–2953 (electronic) (2002)
Chen Z.-Q. and Kumagai T. (2003). Heat kernel estimates for stable-like processes on d-sets. Stoch. Process. Appl. 108(1): 27–62
Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for jump processes of mixed types on metric measure spaces. Prob. Theor. Relat. Fields, doi:10.1007/s00440-007-0070-5 (2006)
De Giorgi E. (1957). Sulla differenziabilità e l’analiticitàdelle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3: 25–43
DiBenedetto E., Gianazza U. and Vespri V. (2006). Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equations. Electron. Res. Ann. AMS 12: 95–99
Fukushima M. (1977/78). On an L p. Publ. Res. Inst. Math. Sci. 13(1): 277–284
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, vol. 224, Grundlehren der Mathematischen Wissenschaften, 2nd edn. Springer, Berlin (1983)
Husseini, R., Kassmann, M.: Markov chain approximations to symmetric jump processes. Potential Anal. 27(4), 353–380 (2007)
Husseini, R., Kassmann, M.,: Jump processes, \({\fancyscript{L}}\) -harmonic functions, continuity estimates and the Feller property, see http://www.iam.uni-bonn.de/~kassmann (preprint) (2006)
John F. and Nirenberg L. (1961). On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14: 415–426
Komatsu T. (1995). Uniform estimates for fundamental solutions associated with non-local Dirichlet forms. Osaka J. Math. 32(4): 833–860
Krylov N.V. and Safonov M.V. (1979). An estimate for the probability of a diffusion process hitting a set of positive measure. Dokl. Akad. Nauk SSSR 245(1): 18–20
Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. Academic Press, New York (1968)
Landis, E.M.: Uravneniya vtorogo poryadka ellipticheskogo i parabolicheskogo tipov. Izdat. “Nauka”, Moscow (1971)
Moser J. (1961). On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14: 577–591
Nash J. (1958). Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80: 931–954
Netrusov, Yu.V.: Some imbedding theorems for spaces of Besov–Morrey type. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 139:139–147, 1984. Numerical methods and questions in the organization of calculations, 7
Silvestre L. (2006). Hölder estimates for solutions of integro-differential equations like the fractional Laplace. Indiana Univ. Math. J. 55(3): 1155–1174
Stampacchia, G.: Èquations elliptiques du second ordre à coefficients discontinus. Séminaire de Mathématiques Supérieures, No. 16 (Été, 1965). Les Presses de l’Université de Montréal, Montreal, Que. (1966)
Tomisaki M. (1979). Some estimates for solutions of equations related to non-local Dirichlet forms. Rep. Fac. Sci. Eng. Saga Univ. Math. 6: 1–7
Triebel, H.: Interpolation theory, function spaces, differential operators. VEB Deutscher Verlag der Wissenschaften, Berlin
Trudinger N.S. (1967). On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math. 20: 721–747
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This work has been support by the German Science Foundation DFG via SFB 611.
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Kassmann, M. A priori estimates for integro-differential operators with measurable kernels. Calc. Var. 34, 1–21 (2009). https://doi.org/10.1007/s00526-008-0173-6
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DOI: https://doi.org/10.1007/s00526-008-0173-6