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

This study focuses on anisotropic Sobolev type spaces associated with Banach spaces , . 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 and . In particular, the most regular class of interpolation spaces between , , depending of and order of spaces are found that mixed derivatives belong with values; the boundedness and compactness of differential operators from this space to -valued spaces are proved. These results are applied to partial differential-operator equations with parameters to obtain conditions that guarantee the maximal regularity uniformly with respect to these parameters.