UMD Banach spaces and square functions associated with heat semigroups   for Schr\"odinger and Laguerre operators
  In this paper we define square functions (also called Littlewood-Paley-Stein functions) associated with heat semigroups for Schr\"odinger and Laguerre operators acting on functions which take values in UMD Banach spaces. We extend classical (scalar) L^p-boundedness properties for the square functions to our Banach valued setting by using \gamma-radonifying operators. We also prove that these L^p-boundedness properties of the square functions actually characterize the Banach spaces having the UMD property. 
J.J. Betancor, A.J. Castro, J.C.Fari\~na and L. Rodr\'iguez-Mesa
math.CA