Embedding and maximal regular differential operators in Sobolev-Lions spaces

Abstract

This study focuses on vector-valued anisotropic Sobolev-Lions 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 spaces E-0 and E. In particular, the most regular class of interpolation spaces E-alpha between E-0, E depending on alpha and the order of space are found and the boundedness of differential operators D-alpha from this space to E-alpha-valued L-p,L-gamma spaces is proved. These results are applied to partial differential-operator equations with paxameters to obtain conditions that guarantee the maximal L-p,L-gamma regularity and R-positivity uniformly with respect to these parameters.