Difference Norms for Vector-Valued Bessel Potential Spaces with an Application to Pointwise Multipliers

TL;DR

This paper establishes a new randomized difference norm characterization for vector-valued Bessel potential spaces in UMD Banach spaces, with applications to pointwise multipliers and Fourier multiplier operators.

Contribution

It introduces a novel difference norm characterization for these function spaces and applies it to analyze pointwise multiplier properties in weighted settings.

Findings

Characterization of Bessel potential spaces via randomized difference norms

Identification of pointwise multipliers for these spaces

Development of $ ext{R}$-boundedness results for Fourier multipliers

Abstract

In this paper we prove a randomized difference norm characterization for Bessel potential spaces with values in UMD Banach spaces. The main ingredients are $\mathcal{R}$-boundedness results for Fourier multiplier operators, which are of independent interest. As an application we characterize the pointwise multiplier property of the indicator function of the half-space on these spaces. All results are proved in the setting of weighted spaces.