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

Abstract: In this paper we consider conical square functions in the Bessel, Laguerre and Schr\"odinger settings where the functions take values in UMD Banach spaces. Following a paper of Hyt\"onen, van Neerven and Portal, in order to define our conical square functions, we use $\gamma$-radonifying operators. We obtain new equivalent norms in the Lebesgue-Bochner spaces $L^p((0,\infty ),\mathbb{B})$ and $L^p(\mathbb{R}^n,\mathbb{B})$, $1<p<\infty$, in terms of our square functions, provided that $\mathbb{B}$ is a UMD Banach space. Our results can be seen as Banach valued versions of known scalar results for square functions.