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On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces. (English) Zbl 0795.45007
Lith. Math. J. 32, No. 2, 238-264 (1992) and Liet. Mat. Rink. 32, No. 2, 299-331 (1992).
We consider the Cauchy problem in Sobolev and Hölder spaces for the operators \(L^{(\alpha)} = A^{(\alpha)} + B^{(\alpha)}\), \(0<\alpha \leq 2\), where the principal part of the operator \(L^{(\alpha)}\) is the pseudo-differential operator \(A^{(\alpha)}\) of the order \(\alpha\), and \(B^{(\alpha)}\) is the integro-differential perturbing operator. We prove several existence and uniqueness theorems as well as error estimates for the solution of the Cauchy problem \(({\partial \over \partial t} + L^{(\alpha)} - \lambda) u=f\).

MSC:
45K05 Integro-partial differential equations
35S05 Pseudodifferential operators as generalizations of partial differential operators
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[1] A. Bensoussan and J.L. Lions,Contrôle Impulsionnel et Inequations Quasi-Variationnelles, Dunod, Paris (1981). · Zbl 0491.93002
[2] J. Bergh and J. Löfström,Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-Heidelberg-New York (1976).
[3] J.M. Bony, Problème de Dirichlet et semi-groupe fortement fellérien associes à un opérateur intégro-différentiel,C.R. Acad. Sci.,263 361–364 (1967). · Zbl 0168.12601
[4] R.R. Coifman and Y. Meyer, Au dela des opérateurs pseudo-différentiels, Société mathématique de France,Astérisque (1978).
[5] A. N. Kochubei, Parabolic pseudo-differential equations, hypersingular integrals and Markov processes,Izv. Akad. Nauk SSSR, Ser. Math.,52, 909–934 (1988).
[6] N. Dunford and J.T. Schwartz,Linear Operators. Part II, Interscience Publishers, New York-London (1963).
[7] G.I. Eskin,Boundary Value Problems for Elliptic Pseudo-differential Equations, [in Russian], Nauka, Moscow (1973). · Zbl 0292.35068
[8] E.B. Fabes and N.M. Riviere, Singular integrals with mixed homogeneity,Stud. Math.,27, 19–38 (1966). · Zbl 0161.32403
[9] A.D. Gadjiev, On differentiability properties of symbols of multidimensional singular integral operator,Math. USSR Sb.,114, 483–510 (1981).
[10] F. Gimbert and P.L. Lions, Existence and regularity results for solutions of second order, elliptic, integro-differential operators,Ric. Mat.,33, 315–358 (1984). · Zbl 0579.45010
[11] J.M. Drin’, Actions of certain parabolic pseudo-differential operators in the spaces of Hölder functions,Dokl. Akad. Nauk Ukr. SSR A1, 19–21 (1974). · Zbl 0269.35075
[12] T. Komatsu, On the martingale problem for generators of stable processes with perturbations,Osaka J. Math.,21, 113–132 (1984). · Zbl 0535.60063
[13] T. Komatsu, Pseudo-differential operators and Markov processes,J. Math. Soc. Jpn.,36, 384–418 (1984). · Zbl 0539.60081 · doi:10.2969/jmsj/03630387
[14] E.M. Stein,Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton (1970). · Zbl 0207.13501
[15] M.I. Vishik and G.I. Eskin, Parabolic convolution equations,Math. USSR Sb.,71, 162–190 (1966).
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