Square functions in the Hermite setting for functions with values in UMD spaces

Abstract

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 $$H=L^2((0,\infty ),\frac{\mathrm{d}t}{t})$$ . 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.