UMD Banach spaces and square functions associated with heat semigroups for Schrödinger, Hermite and Laguerre operators

Abstract

In this paper we define square functions (also called Littlewood‐Paley‐Stein functions) associated with heat semigroups for Schrödinger and Laguerre operators acting on functions which take values in UMD Banach spaces. We extend classical (scalar) ‐boundedness properties for the square functions to our Banach valued setting by using γ‐radonifying operators. We also prove that these ‐boundedness properties of the square functions actually characterize the Banach spaces having the UMD property.