Embedding Operators in Vector-Valued Weighted Besov Spaces and Applications

Abstract

The embedding theorems in weighted Besov-Lions type spaces 𝐵𝑙,𝑠𝑝,𝑞,𝛾 (Ω;𝐸0,𝐸) in which 𝐸0,𝐸 are two Banach spaces and 𝐸0⊂𝐸 are studied. The most regular class of interpolation space 𝐸𝛼 between 𝐸0 and E is found such that the mixed differential operator 𝐷𝛼 is bounded from 𝐵𝑙,𝑠𝑝,𝑞,𝛾 (Ω;𝐸0,𝐸) to 𝐵𝑠𝑝,𝑞,𝛾 (Ω;𝐸𝛼) and Ehrling-Nirenberg-Gagliardo type sharp estimates are established. By using these results, the uniform separability of degenerate abstract differential equations with parameters and the maximal B-regularity of Cauchy problem for abstract parabolic equations are obtained. The infinite systems of the degenerate partial differential equations and Cauchy problem for system of parabolic equations are further studied in applications.