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 Bp,qs(Ω;E) is derived. The embedding theorems in Besov-Lions type spaces Bp,ql,s(Ω;E0, E) are studied, where E0, E are two Banach spaces and E0 ⊂ E. The most regular class of interpolation space Eα, between E0 and E are found such that the mixed differential operator Dα is bounded and compact from Bp,ql,s(Ω;E0,E) to Bp,qs(Ω;Eα) 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.