Characterization of UMD Banach spaces by imaginary powers of Hermite and Laguerre operators

TL;DR

This paper characterizes UMD Banach spaces using boundedness properties of imaginary powers of Hermite and Laguerre operators, involving new pointwise representations of related multipliers.

Contribution

It introduces a novel characterization of UMD spaces via imaginary powers of classical operators, with new pointwise multiplier representations.

Findings

UMD spaces characterized by imaginary powers of Hermite and Laguerre operators

Established pointwise representations for Laplace transform type multipliers

Connected boundedness properties to Banach space geometry

Abstract

In this paper we characterize the Banach spaces with the UMD property by means of Lp-boundedness properties for the imaginary powers of the Hermite and Laguerre operators. In order to do this we need to obtain pointwise representations for the Laplace transform type multipliers associated with Hermite and Laguerre operators.