On the Cauchy Problem for Integro-Differential Equations in the Scale of Spaces of Generalized Smoothness

  • R. Mikulevičius
  • C. Phonsom


Parabolic integro-differential model Cauchy problem is considered in the scale of L p -spaces of functions whose regularity is defined by a scalable Levy measure. Existence and uniqueness of a solution is proved by deriving apriori estimates. Some rough probability density function estimates of the associated Levy process are used as well.


Non-local parabolic integro-differential equations Lévy processes 

Mathematics Subject Classification (2010)

35R09 60J75 35B65 


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We are very grateful to our reviewers for valuable comments and suggestions.


  1. 1.
    Bergh, J., Löfstrom, J.: Interpolation Spaces. An Introduction. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  2. 2.
    Dong, H., Kim, D: On L p- estimates of non-local elliptic equations. J. Funct. Anal. 262, 1166–1199 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Garcia-Cuerva, J., Rubio De Francia, J.L.: Weighted Norm Inequalities and Related Topics. North-Holland (1985)Google Scholar
  4. 4.
    Farkas, W., Jacob, N., Schilling, R. L.: Function spaces related to continuous negative definite functions: ψ -Bessel potential spaces. Diss. Math., 1–60 (2001)Google Scholar
  5. 5.
    Farkas, W., Leopold, H. -G.: Characterisation of function spaces of generalized smoothness. Annali di Matematica 185, 1–62 (2006)CrossRefzbMATHGoogle Scholar
  6. 6.
    Kalyabin, G. A.: Description of functions in classes of Besov-Lizorkin-Triebel type. Trudy Mat. Inst. Steklov 156, 160–173 (1980)zbMATHGoogle Scholar
  7. 7.
    Kalyabin, G. A., Lizorkin, P. I.: Spaces of functions of generalized smoothness. Math. Nachr. 133, 7–32 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kim, I., Kim, K. -H.: An L p-theory for a class of non-local elliptic equations related to nonsymmetric measurable kernels. J. Math. Anal. Appl 434, 1302–1335 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kim, P., Song, R., Vondraček, Z.: Global uniform boundary Harnack principle with explicit decay rate and its application. Stoch. Proc. Appl. 124, 235–267 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kim, I., Kim, K. -H.: An L p,-boundedness of stochastic singular integral operators and its application to SPDEs, arXiv:1608.08728 (2016)
  11. 11.
    Mikulevicius, R., Phonsom, C: On L p −theory for parabolic and elliptic integro-differential equations with scalable operators in the whole space, Stochastics and PDEs: Anal Comp, 2017,; arXiv:1605.07086 (2016)
  12. 12.
    Stein, E.: Harmonic Analysis. Princeton University Press (1993)Google Scholar
  13. 13.
    Triebel, H.: Interpolation Theory, Function Spaces. Differential Operators. North-Holland (1978)Google Scholar
  14. 14.
    Xicheng, Zhang: L p-maximal regularity of nonlocal parabolic equations and applications. Ann. I.H. Poincaré-AN 30, 573–614 (2013)CrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of Southern CaliforniaLos AngelesUSA

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