Square functions and spectral multipliers for Bessel operators in UMD 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 ${\Sigma }_{\vartheta }$.

Cited by in corpus (1)

• Conical square functions associated with Bessel, Laguerre and Schrödinger operators in UMD Banach spaces