Embedding and Maximal Regular Differential Operators in Sobolev-Lions Spaces

This study focuses on vector-valued anisotropic Sobolev-Lions spaces associated with Banach spaces E0, 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 E0 and E. In particular, the most regular class of interpolation spaces Eα between E0, E depending on α and the order of space are found and the boundedness of differential operators D^α from this space to Eα-valued Lp,γ spaces is proved. These results are applied to partial differential-operator equations with parameters to obtain conditions that guarantee the maximal Lp,γ regularity and R-positivity uniformly with respect to these parameters.

Embedding and Maximal Regular Differential Operators in Sobolev–Lions Spaces

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 _α between E ₀, E depending on α and the order of space are found and the boundedness of differential operators D ^α from this space to Eα–valued L _p,γ spaces is proved. These results are applied to partial differential–operator equations with parameters to obtain conditions that guarantee the maximal L _p,γ regularity and R–positivity uniformly with respect to these parameters. This work is supported by the grant of Istanbul University (Project UDP-227/18022004)