Conical Square Function Estimates in Umd Banach Spaces and Applications to H∞-functional Calculi

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–MIntosh–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 L(R;X) by checking appropriate conical square function estimates, and also a conical analogue of Bourgain’s extension of the Littlewood-Paley theory to the UMDvalued context. Even when X = C, our approach gives refined p-dependent versions of known results.