Shakhmurov, Veli B.2024-05-252024-05-2520101674-72831869-186210.1007/s11425-009-0161-02-s2.0-77952282685https://doi.org/10.1007/s11425-009-0161-0https://hdl.handle.net/20.500.14517/690Necessary and sufficient conditions for compactness of sets in Banach space valued Besov class B (p,q) (s) (Omega;E) is derived. The embedding theorems in Besov-Lions type spaces B (p,q) (l,s) (Omega;E (0), E) are studied, where E (0), E are two Banach spaces and E (0) aS, E. The most regular class of interpolation space E (alpha) , between E (0) and E are found such that the mixed differential operator D (alpha) is bounded and compact from B (p,q) (l,s) (Omega;E (0),E) to B (p,q) (s) (Omega;E (alpha) ) 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.eninfo:eu-repo/semantics/closedAccessembedding theoremsBanach-valued function spacesdifferential-operator equationsmaximal B-regularityoperator-valued Fourier multipliersinterpolation of Banach spacesabstract parabolic Cauchy problemArticleQ1Q153410671084WOS:0002769125000151