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 $L^p\left(\right(0,\infty ),\mathbb{B})$, where $\mathbb{B}$ 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 $\mathbb{B}$ by using $L^p\left(\right(0,\infty ),\mathbb{B})$-boundedness properties of $g$-functions defined by Bessel–Poisson semigroups. As a by-product, we prove that the fact that the imaginary power ${\Delta }_{\lambda }^{i\omega }$, $\omega \in \mathbb{R}\setminus \left\{0\right\}$, of the Bessel operator ${\Delta }_{\lambda }$ is bounded in $L^p\left(\right(0,\infty ),\mathbb{B})$, $1<p<\infty$, characterizes the UMD property for the Banach space $\mathbb{B}$. As applications of our results for square functions, we establish the boundedness in $L^p\left(\right(0,\infty ),\mathbb{B})$ of spectral multipliers $m\left({\Delta }_{\lambda }\right)$ of Bessel operators defined by functions $m$ which are holomorphic in sectors ${\Sigma }_{\vartheta }$.