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 L p(ℝn; X) by checking appropriate conical square function estimates and also a conical analogue ofBourgain’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.
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Hytönen, T., van Neerven, J. & Portal, P. Conical square function estimates in UMD Banach spaces and applications to H ∞-functional calculi. J Anal Math 106, 317–351 (2008). https://doi.org/10.1007/s11854-008-0051-3
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DOI: https://doi.org/10.1007/s11854-008-0051-3