# Multipliers and embedding operators with application to abstract differential equat{\i}ons

## TL;DR

This paper develops operator-valued multiplier theorems in weighted Lebesgue-Bochner spaces and applies them to establish embedding theorems and maximal regularity for abstract elliptic and parabolic equations.

## Contribution

It introduces new multiplier theorems of Mikhlin and Marcinkiewicz--Lizorkin type for operator-valued functions in weighted spaces, leading to advances in regularity theory.

## Key findings

- Established Mikhlin and Marcinkiewicz--Lizorkin type multiplier theorems in weighted Lebesgue-Bochner spaces.
- Derived embedding theorems in Sobolev-Lions type spaces using these multiplier results.
- Proved maximal regularity properties for abstract elliptic and parabolic equations.

## Abstract

In this paper, Mikhlin and Marcinkiewicz--Lizorkin type operator-valued multiplier theorems in weighted Lebesgue-Bochner spaces are studied. By using this results embedding theorems in Sobolev-Lions type spaces is obtained. Moreover, maximal regularity properties of abstract elliptic and parabolic equations are derived