Compact embedding in Besov spaces and B-separable elleptic operators

Abstract

Necessary 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.