Embedding Theorems in B-Spaces and Applications

This study focuses on the anisotropic Besov-Lions type spaces B _p,θ ^l (Ω;E ₀,E) associated with Banach spaces E ₀ and E. Under certain conditions, depending on l = (l ₁, l ₂,⋯, l _n ) and α = (α₁, α₂, ⋯, α _n ), the most regular class of interpolation space E _α between E ₀ and E are found so that the mixed differential operators D ^α are bounded and compact from B _p,θ ^l+s (Ω;E ₀,E) to B _p,θ ^s (Ω;E _α). 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.