Square functions in the Hermite setting for functions with values in UMD   spaces
  In this paper we characterize the Lebesgue Bochner spaces $L^p(\mathbb{R}^n,B)$, $1<p<\infty$, by using Littlewood-Paley $g$-functions in the Hermite setting, provided that $B$ is a UMD Banach space. We use $\gamma$-radonifying operators $\gamma (H,B)$ where $H=L^2((0,\infty),\frac{dt}{t})$. We also characterize the UMD Banach spaces in terms of $L^p(\mathbb{R}^n,B)$-$L^p(\mathbb{R}^n,\gamma (H,B))$ boundedness of Hermite Littlewood-Paley $g$-functions. 
Jorge J. Betancor, Alejandro J. Castro, Jezabel Curbelo, Juan C.   Fari\~na and Lourdes Rodr\'iguez-Mesa
math.CA math.FA