Square functions and spectral multipliers for Bessel operators in UMD spaces

TL;DR

This paper investigates square functions related to Bessel operators in UMD Banach spaces, characterizing the UMD property via boundedness of spectral multipliers and imaginary powers, with implications for harmonic analysis.

Contribution

It introduces new characterizations of the UMD property using Bessel-Poisson semigroup square functions and spectral multipliers, extending harmonic analysis tools to Bessel operators in Banach spaces.

Findings

UMD property characterized by boundedness of g-functions

Imaginary powers of Bessel operators characterize UMD spaces

Spectral multipliers are bounded in UMD Banach spaces

Abstract

In this paper we consider square functions (also called Littlewood-Paley g-functions) associated to Hankel convolutions acting on functions in the Bochner-Lebesgue space $L^p((0,\infty),B)$, where $B$ is a UMD Banach space. As special cases we study square functions defined by fractional derivatives of the Poisson semigroup for the Bessel operator $\Delta_\lambda=-x^{-\lambda}\frac{d}{dx}x^{2\lambda}\frac{d}{dx}x^{-\lambda}$, $\lambda >0$. We characterize the UMD property for a Banach space $B$ by using $L^p((0,\infty),B)$-boundedness properties of g-functions defined by Bessel-Poisson semigroups. As a by product we prove that the fact that the imaginary power $\Delta_\lambda ^{iw}$, $w\in \mathbb{R}\setminus\{0\}$, of the Bessel operator $\Delta_\lambda$ is bounded in $L^p ((0,\infty),B)$, $1<p<\infty$, characterizes the UMD property for the Banach space $B$. As applications of our results for square functions we establish the boundedness in $L^p((0,\infty),B)$ of spectral multipliers $m(\Delta_\lambda)$ of Bessel operators defined by functions $m$ which are holomorphic in sectors $\Sigma_\vartheta$.