Conical square functions associated with Bessel, Laguerre and Schrödinger operators in UMD Banach spaces

Abstract

In this paper we consider conical square functions in the Bessel, Laguerre and Schrödinger settings where the functions take values in UMD Banach spaces. Following a recent paper of Hytönen, van Neerven and Portal [36], in order to define our conical square functions, we use γ-radonifying operators. We obtain new equivalent norms in the Lebesgue–Bochner spaces Lp((0,∞),B) and Lp(Rn,B), 1<p<∞, in terms of our square functions, provided that B is a UMD Banach space. Our results can be seen as Banach valued versions of known scalar results for square functions.