Conical square function estimates in UMD Banach spaces and applications to H-infinity functional calculi

TL;DR

This paper develops conical square function estimates for Banach-valued functions and introduces vector-valued tent spaces, enabling new proofs of bounded H-infinity functional calculus for bisectorial operators in UMD Banach spaces.

Contribution

It introduces a vector-valued analogue of tent spaces and applies them to establish bounded functional calculus for operators on UMD Banach spaces, extending classical harmonic analysis tools.

Findings

Constructed vector-valued tent spaces for Banach-valued functions.

Proved bounded H-infinity calculus for bisectorial operators using conical square functions.

Extended Littlewood-Paley theory to UMD Banach spaces with refined p-dependent results.

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^{\infty})-functional calculus on these spaces. This provides a new way of proving functional calculus of (A) on the Bochner spaces (L^p(\R^n;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.