Embedding theorems in <i>B</i>-spaces and applications

Abstract

This study focuses on the anisotropic Besov-Lions type spaces B-p,theta(l)(Omega; E-0, E) associated with Banach spaces E-0 and E. Under certain conditions, depending on l=(l(1),l(2),..., l(n)) and alpha=(alpha(1), alpha(2),..., alpha(n)), the most regular class of interpolation space E-alpha between E-0 and E are found so that the mixed differential operators D-alpha are bounded and compact from B-p,theta(l+s) (Omega; E-0, E) to B-p,theta(s) (Omega; E-alpha). 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.