  The operator-valued multiplier theorems in weighted abstract Besov spaces are
studied. These results permit us to show embedding theorems in weighted
Besov-Lions type spaces. The most regular class of interpolation space is found
such that the mixed differential operator is bounded from Besov-Lions space to
this and Ehrling-Nirenberg-Gagliardo type sharp estimates are established. By
using these results the separability properties of degenerate differential
operators are studied. Especially, we prove that the associated differential
operators are positive and also are generators of analytic semigroups.
Moreover, regularity properties for abstract elliptic equation, Cauchy problem
for degenerate abstract parabolic equation and the infinite systems of
degenerate parabolic equations are studied.
