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

Abstract

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-M c Intosh-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 (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 UMDvalued context. Even when X = C, our approach gives refined p-dependent versions of known results.

The Banach Space-valued BMO, Carleson’s Condition, and Paraproducts

“…The A functionals have already been studied in the vector-valued context in [10], and we quote two inequalities from there. Both results involve the UMD property of the underlying Banach space X, and our use of UMD in the present paper will be mainly via the application of these estimates.…”

“…For such spaces, descriptions of the L p (R n ; X) norm in terms of discrete versions of the randomized vertical square functions date back to the works of Bourgain [2] and McConnell [15] in the 80's. More recently, also continuous-parameter quadratic estimates for f ∈ L p (R n ; X) have been developed, both in the vertical [8,13,14] and the conical [10] forms.…”

“…The "Carleson-type" characterization of BMO(R n ; X) from Theorem 1.1, together with a conical square function description of L p (R n ; X) from [10], allow a new and clean proof of the following basic mapping property of the paraproduct. Theorem 1.2 Let X be a UMD space, and p ∈ (1, ∞).…”

“…Hytönen et al[10, Theorem 4.8]) Let X be a UMD space, andp ∈ (1, ∞). Let ψ ∈ S (R n ) have a vanishing integral and F (x, t) := f * ψ t (x) for f : R n → X.…”

Racines Carrées d'OPÉRATEURS Elliptiques Et Espaces De Hardy

“…Cette remarque signifie aussi que notre construction d'espaces de Hardy permet, pour p > 2, de prouver la continuité L p d'une généralisation appropriée de l'intégrale d'aire de Lusin. Un tel point de vue constitue une des motivations de [204], voir la section 5.5.3 ci-dessous.…”