Embedding operators and maximal regular differential-operator equations in Banach-valued function spaces

Abstract

This study focuses on anisotropic Sobolev type spaces associated with Banach spaces E-0, E. Several conditions are found that ensure the continuity and compactness of embedding operators that are optimal regular in these spaces in terms of interpolations of E-0 and E. In particular, the most regular class of interpolation spaces E-alpha between E-0, E, depending of a and order of spaces are found that mixed derivatives D-alpha belong with values; the boundedness and compactness of differential operators D-alpha from this space to E-alpha-valued L-p spaces are proved. These results are applied to partial differential-operator equations with parameters to obtain conditions that guarantee the maximal L-p regularity uniformly with respect to these parameters.