γ-Radonifying operators and UMD-valued Littlewood–Paley–Stein functions in the Hermite setting on BMO and Hardy spaces

In this paper we study Littlewood–Paley–Stein functions associated with the Poisson semigroup for the Hermite operator on functions with values in a UMD Banach space B. If we denote by H the Hilbert space L2((0,∞),dt/t), γ(H,B) represents the space of γ-radonifying operators from H into B. We prove that the Hermite square function defines bounded operators from BMOL(Rn,B) (respectively, ) into BMOL(Rn,γ(H,B)) (respectively, ), where BMOL and denote BMO and Hardy spaces in the Hermite setting. Also, we obtain equivalent norms in BMOL(Rn,B) and by using Littlewood–Paley–Stein functions. As a consequence of our results, we establish new characterizations of the UMD Banach spaces.