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Compact embedding in Besov spaces and B-separable elleptic operators

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Abstract

Necessary and sufficient conditions for compactness of sets in Banach space valued Besov class B s p,q (Ω;E) is derived. The embedding theorems in Besov-Lions type spaces B l,sp,q (Ω;E 0, E) are studied, where E 0, E are two Banach spaces and E 0 ⊂ E. The most regular class of interpolation space E α , between E 0 and E are found such that the mixed differential operator D α is bounded and compact from B l,s p,q (Ω;E 0,E) to B s p,q (Ω;E α ) and Ehrling-Nirenberg-Gagliardo type sharp estimates established. By using these results the separability of differential operators with variable coefficients and the maximal B-regularity of parabolic Cauchy problem are obtained. In applications, the infinite systems of the elliptic partial differential equations and parabolic Cauchy problems are studied.

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References

  1. Agmon S, Nirenberg L. Properties of solutions of ordinary differential equations in Banach spaces. Comm Pure Appl Math, 1963, 16: 121–239

    Article  MATH  MathSciNet  Google Scholar 

  2. Agranovich M S, Vishik M I. Elliptic problems with a parameter and parabolic problems of general type. Uspekhi Mat Nauk, 1964, 19: 53–159

    MATH  Google Scholar 

  3. Amann H. Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications. Math Nachr, 1997, 186: 5–56

    Article  MATH  MathSciNet  Google Scholar 

  4. Amann H. Linear and Quasi-Linear Parabolic Equations, 1. Basel-Boston-Berlin: Birkhäuser, 1995

    Google Scholar 

  5. Amann H. Compact embedding of vector-valued Sobolev and Besov spaces. Glasnik Math, 2000, 35: 161–177

    MATH  MathSciNet  Google Scholar 

  6. Aubin J P. Abstract boundary-value operators and their adjoint. Rend Sem Padova, 1970, 43: 1–33

    MATH  MathSciNet  Google Scholar 

  7. Ashyralyev A. On well-posedeness of the nonlocal boundary value problem for elliptic equations. Numer Funct Anal Optim, 2003, 24: 1–15

    Article  MATH  MathSciNet  Google Scholar 

  8. Besov O V, Ilin V P, Nikolskii S M. Integral Representations of Functions and Embedding Theorems. New York: John Wiley, 1979

    Google Scholar 

  9. Burkholder D L. A geometrical conditions that implies the existence certain singular integral of Banach space-valued functions. In: Proc Conf Harmonic Analysis in Honor of Antonu Zigmund. Chicago: Wads Worth, 1981, 270–286

    Google Scholar 

  10. Clement P, Pagter B, Sukochev F A, et al. Schauder decomposition and multiplier theorems. Studia Math, 2000, 138: 135–163

    MATH  MathSciNet  Google Scholar 

  11. Denk R, Hieber M, Prüss J. R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. In: Mem Amer Math Soc, vol. 166. Providence, RI: Amer Math Soc, 2003

    Google Scholar 

  12. Dore C, Yakubov S. Semigroup estimates and non-coercive boundary value problems. Semigroup Form, 2000, 60: 93–121

    Article  MATH  MathSciNet  Google Scholar 

  13. Girardy M, Weis L. Operator-valued multiplier theorems on Besov spaces. Math Nachr, 2003, 251: 34–51

    Article  MathSciNet  Google Scholar 

  14. Gorbachuk V I, Gorbachuk M L. Boundary Value Problems for Differential-Operator Equations. Kiev: Naukova Dumka, 1984

    Google Scholar 

  15. Grisvard P. Elliptic Problems in Non-Smooth Domains. Boston: Pitman, 1985

    Google Scholar 

  16. Kalderon A P. Intermediate spaces and interpolation, the complex method. Studia Math, 1964, 24: 113–190

    MathSciNet  Google Scholar 

  17. Karakas H I, Shakhmurov V B, Yakubov S. Degenerate elliptic boundary value problems. Appl Anal, 1996, 60: 155–174

    Article  MATH  MathSciNet  Google Scholar 

  18. Komatsu H. Fractional powers of operators. Pacific J Math, 1966, 19: 285–346

    MATH  MathSciNet  Google Scholar 

  19. Krein S G. Linear Differential Equations in Banach Space. Providence, RI: Amer Math Soc, 1971

    Google Scholar 

  20. Lions J L, Magenes E. Problems and limits non homogenes. J d’Anal Math, 1963, 11: 165–188

    Article  MATH  MathSciNet  Google Scholar 

  21. Lions J L, Peetre J. Sur une classe d’espaces d’interpolation. Inst Hautes Etudes Sci Publ Math, 1964, 19: 5–68

    Article  MATH  MathSciNet  Google Scholar 

  22. Lizorkin P I. (L p, L q)-Multiplicators of Fourier integrals. Dokl Akad Nauk SSSR, 1963, 152: 808–811

    MathSciNet  Google Scholar 

  23. Lizorkin P I, Shakhmurov V B. Embedding theorems for classes of vector-valued functions. Izvestiya Vysshikh Uchebnukh Zavedennii Matematika, 1989, 1–2: 47–54, 70–78

    MathSciNet  Google Scholar 

  24. Lindenstraus J, Tzafiri L. Classical Banach Spaces II: Function Spaces. Berlin: Springer-Verlag, 1979

    Google Scholar 

  25. McConnell, Terry R. On Fourier multiplier transformations of Banach-valued functions. Trans Amer Math Soc, 1984, 285: 739–757

    Article  MATH  MathSciNet  Google Scholar 

  26. Nazarov S A, Plammenevskii B A. Elliptic Problems in Domains with Piecewise Smooth Boundaries. New York: Walter de Gruyter, 1994

    MATH  Google Scholar 

  27. Schmeisser H J. Vector-valued Sobolev and Besov spaces. Teubner Texte Math, 1986, 96: 4–44

    MathSciNet  Google Scholar 

  28. Sobolev S L. Embedding theorems for abstract functions. Dok Akad Nauk USSR, 1957, 115: 55–59

    MathSciNet  Google Scholar 

  29. Sobolevkii P E. Coerciveness inequalities for abstract parabolic equations. Dokl Akad Nauk SSSR, 1964, 57: 27–40

    Google Scholar 

  30. Shakhmurov V B. Imbedding theorems for abstract functon-spaces and their applications. Math USSR-Sbornik, 1987, 134: 261–276

    MathSciNet  Google Scholar 

  31. Shakhmurov V B. Coercive boundary value problems for regular degenerate differential-operator equations. J Math Anal Appl, 2004, 292: 605–620

    Article  MATH  MathSciNet  Google Scholar 

  32. Shakhmurov V B. Embedding theorems in Banach-valued B-spaces and maximal B-regular differential operator equations. J Inequal Appl, 2006: 1–22;ID: 92134

    Google Scholar 

  33. Shakhmurov V B. Embedding and maximal regular differential operators in Banach-valued weighted spaces. Acta Math Sinica Engl Ser, 2006, 22: 1493–1508

    Article  MATH  MathSciNet  Google Scholar 

  34. Shakhmurov V B. Imbedding theorems and their applications to degenerate equations. Differential Equations, 1988, 24: 475–482

    MATH  MathSciNet  Google Scholar 

  35. Shklyar A Y. Complate Second-Order Linear Differential Equations in Hilbert Spaces. Basel: Birkhäuser Verlag, 1997

    Google Scholar 

  36. Sobolev S L. Some Applications of Functional Analysis to Mathematical Physics. Providence, RI: Amer Math Soc, 1962

    Google Scholar 

  37. Triebel H. Fractals and Spectra. Basel: Birkhäuser, 1997

    MATH  Google Scholar 

  38. Triebel H. Interpolation Theory, Function Spaces, Differential Operators. Amsterdam: North-Holland, 1978

    Google Scholar 

  39. Yakubov S. Completeness of Root Functions of Regular Differential Operators. New York: Longman Scientific and Technical, 1994

    MATH  Google Scholar 

  40. Yakubov S. A nonlocal boundary value problem for elliptic differential-operator equations and applications. Integr Equ Oper Theory, 1999, 35: 485–506

    Article  MATH  MathSciNet  Google Scholar 

  41. Yakubov S, Yakubov Y. Differential-operator equations. Ordinary and Partial Differential equations. Boca Raton: Chapman and Hall-CRC Press, 2000

    MATH  Google Scholar 

  42. Zimmermann F. On vector-valued Fourier multiplier theorems. Studia Math, 1989, 93: 201–222

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Electronics Engineering and Communication, Okan University, Akfirat, Tuzla, 34959, Istanbul, Turkey

    Veli B. Shakhmurov

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Shakhmurov, V.B. Compact embedding in Besov spaces and B-separable elleptic operators. Sci. China Math. 53, 1067–1084 (2010). https://doi.org/10.1007/s11425-009-0161-0

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