# Operator-valued multipliers in vector-valued weighted Besov spaces and applications

## TL;DR

This paper develops operator-valued multiplier theorems in weighted Besov spaces, leading to new embedding results, regularity estimates, and analysis of degenerate differential operators and their associated evolution equations.

## Contribution

It introduces new operator-valued multiplier theorems in weighted Besov spaces and applies them to establish embedding, regularity, and semigroup generation results for degenerate differential operators.

## Key findings

- Differential operators are positive and generate analytic semigroups.
- New embedding theorems for weighted Besov-Lions spaces are established.
- Regularity results for abstract elliptic and degenerate parabolic equations are proved.

## Abstract

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.