Weighted Estimates for Operator-Valued Fourier Multipliers

TL;DR

This paper develops weighted Littlewood-Paley theory and extends the Mikhlin multiplier theorem to two-weight settings for operator-valued multipliers in UMD spaces, including H"ormander-type conditions.

Contribution

It introduces weighted decompositions and proves new two-weight multiplier theorems for operator-valued functions in UMD spaces, advancing harmonic analysis techniques.

Findings

Established Littlewood-Paley decompositions for Muckenhoupt weights in UMD spaces.

Derived two-weight Mikhlin multiplier theorems for operator-valued multipliers.

Proved two-weight estimates for multipliers satisfying H"ormander conditions.

Abstract

We establish Littlewood-Paley decompositions for Muckenhoupt weights in the setting of UMD spaces. As a consequence we obtain two-weight variants of the Mikhlin multiplier theorem for operator-valued multipliers. We also show two-weight estimates for multipliers satisfying H\"ormander type conditions.