CONICAL SQUARE FUNCTION ESTIMATES IN UMD BANACH SPACES AND APPLICATIONS TO H∞-FUNCTIONAL CALCULI By

Abstract. We study conical square function estimates for Banach-valued functions and introduce a vector-valued analogue of the Coifman–Meyer–Stein tent spaces. Following recent work of Auscher–McIntosh–Russ, the tent spaces in turn are used to construct a scale of vector-valued Hardy spaces associated with a given bisectorial operator A with certain off-diagonal bounds such that A always has a bounded H∞-functional calculus on these spaces. This provides a new way of proving functional calculus of A on the Bochner spaces Lp(Rn;X) by checking appropriate conical square function estimates and also a conical analogue of Bourgain’s extension of the Littlewood-Paley theory to the UMD-valued context. Even when X = C, our approach gives refined p-dependent versions of known results.