Embedding Theorems in B-Spaces and Applications

Abstract

This study focuses on the anisotropic Besov-Lions type spaces B _p,θ ^l(Ω;E ₀,E) associated with Banach spaces E ₀ and E. Under certain conditions, depending on l = (l ₁, l ₂,⋯, l _n) and α = (α₁, α₂, ⋯, α _n), the most regular class of interpolation space E _α between E ₀ and E are found so that the mixed differential operators D ^α are bounded and compact from B _p,θ ^l+s(Ω;E ₀,E) to B _p,θ ^s(Ω;E _α). These results are applied to concrete vector-valued function spaces and to anisotropic differential-operator equations with parameters to obtain conditions that guarantee the uniform B separability with respect to these parameters. By these results the maximal B-regularity for parabolic Cauchy problem is obtained. These results are also applied to infinite systems of the quasi-elliptic partial differential equations and parabolic Cauchy problems with parameters to obtain sufficient conditions that ensure the same properties.

Keywords

Embedding theorems / Banach-valued function spaces / Differential-operator equations / B-Separability / Operator-valued Fourier multipliers / Interpolation of Banach spaces

References

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

[2] Agranovich M. S., Vishik M. I.. Elliptic problems with a parameter and parabolic problems of general type. Uspekhi Mat. Nauk, 1964, 19(3): 53-159

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

[4] Amann, H., Linear and Quasi-linear Equations, 1, Birkhauser, Basel, 1995.

[5] Amann H.. Compact embedding of vector-valued Sobolev and Besov spaces. Glasnik Mathematicki, 2000, 35(55): 161-177

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

[7] Ashyralyev A.. On well-posedeness of the nonlocal boundary value problem for elliptic equations. Numer. Funct. Anal. Optim., 2003, 24(1–2): 1-15

[8] Besov, O. V., Ilin, V. P. and Nikolskii, S. M., Integral representations of functions and embedding theorems, Nauka, Moscow, 1975.

[9] Burkholder, D. L., A geometrical conditions that implies the existence certain singular integral of Banach space-valued Functions, Proc. Conf. Harmonic Analysis in Honor of Antonu Zigmund, Chicago, 1981, Wads Worth, Belmont, 1983, 270–286.

[10] Clement Ph., de Pagter B., Sukochev F. A., Witvlet H.. Schauder decomposition and multiplier theorems. Studia Math., 2000, 138: 135-163

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

[12] Denk R., Hieber M., Prüss J.. R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc., 2003, 166(788): viii+114

[13] Haller R., Heck H., Noll A.. Mikhlin’s theorem for operator-valued Fourier multipliers in n variables. Math. Nachr., 2002, 244: 110-130

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

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

[16] Karakas H. I., Shakhmurov V. B., Yakubov S.. Degenerate elliptic boundary value problems. Applicable Analysis, 1996, 60: 155-174

[17] Krein S. G.. Linear Differential Equations in Banach Space. Transi. Math. Monographs, 1971, Providence, RI: A. M. S.

[18] Kalderon A. P.. Intermediate spaces and interpolation, the complex method. Studia Math., 1964, 24: 113-190

[19] Komatsu H.. Fractional powers of operators. Pas. J. Math., 1966, 19: 285-346

[20] McConnell T. R.. On Fourier multiplier transformations of Banach-valued functions. Trans. Amer. Mat. Soc., 1984, 285(2): 739-757

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

[22] Lions J. L., Magenes E.. Problems and limites non homogenes. J. d’Analyse Math., 1963, 11: 165-188

[23] Lizorkin P. I.. (L _p, L _q)-Multiplicators of Fourier integrals. Dokl. Akad. Nauk SSSR, 1963, 152(4): 808-811

[24] Lizorkin P. I., Shakhmurov V. B.. Embedding theorems for classes of vector-valued functions. IZ. VUZ. USSR, Math., 1989, 12: 70-78

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

[26] Lindenstraus, J. and Tzafiri, L., Classical Banach Spaces II, Funcion Spaces, Springer-Verlag, Berlin, 1979.

[27] Sobolev, S. L., Certain Applications of Functional Analysis to Mathematical Physics, Novosibirski, 1962.

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

[29] Sobolevkii P. E.. Coerciveness inequalities for abstract parabolic equations. Dokl. Akad. Nauk. SSSR, 1964, 57(1): 27-40

[30] Shklyar A. Ya.. Complate Second Order Linear Differential Equations in Hilbert Spaces, 1997, Basel: Birkhauser Verlak

[31] Shakhmurov V. B.. Theorems about of compact embedding and applications. Dokl. Akad. Nauk. SSSR, 1978, 241(6): 1285-1288

[32] Shakhmurov V. B.. Embedding theorems in abstract function spaces and applications. Math. Sb., 1987, 134(176): 260-273

[33] Shakhmurov V. B.. Embedding theorems and their applications to degenerate equations. Diff. Equations, 1988, 24: 672-682

[34] Shakhmurov V. B.. Coercive boundary value problems for regular degenerate differential-operator equations. J. Math. Anal. Appl., 2004, 292(2): 605-620

[35] Shakhmurov V. B., Dzabrailov M. C.. About compactness of embedding in B-spaces and its applications. Dokl. Akad. Nauk. Azerb., TXLVI, 1990, 6(3): 7-9

[36] Shakhmurov V. B.. Embedding theorems in Banach-valued B-spaces and maximal B-regular differential operator equations. J. Inequal. and Appl., 2006, 2006: 1-22

[37] Schmeisser, H.-J., Vector-valued Sobolev and Besov spaces, Sem. Analysis, 1985/86, 4–44; Teubner Texte Math., 96, 1986.

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

[39] Yakubov S.. Completeness of Root Functions of Regular differential operators, 1994, New York: Longman, Scientific and Technical

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

[41] Yakubov S., Yakubov Ya.. Differential-Operator Equations, Ordinary and Partial Differential Equations, 2000, Boca Raton: Chapmen and Hall /CRC