Square functions and spectral multipliers for Bessel operators in UMD spaces

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 $\Sigma_\vartheta$.