Multipliers and imaginary powers of the schr\"odinger operators characterizing UMD Banach spaces

TL;DR

This paper investigates the boundedness of spectral multipliers and imaginary powers of Schr"odinger operators on UMD Banach spaces, providing new characterizations and integral representations.

Contribution

It establishes $L^p$-boundedness for Laplace transform spectral multipliers and characterizes UMD Banach spaces via the boundedness of imaginary powers of Schr"odinger operators.

Findings

Spectral multipliers are represented as principal value integrals.

Boundedness of imaginary powers characterizes UMD Banach spaces.

New $L^p$ bounds for Schr"odinger operator multipliers.

Abstract

In this paper we establish $L^p$-boundedness properties for Laplace type transform spectral multipliers associated with the Schr\"odinger operator $\mathcal{L}=-\Delta +V$. We obtain for this type of multipliers pointwise representation as principal value integral operators. We also characterize the UMD Banach spaces in terms of the $L^p$-boundedness of the imaginary powers $\mathcal{L}^{i\gamma}$, $\gamma \in \mathbb{R}$, of $\mathcal{L}$.