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

By Tuomas Hytönen, J. V. Neerven, Pierre Portal. 2007

71 citations recorded by the source corpus.

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 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5cdd90cfe4eeb3066ca658c36fe4aa182a35808f

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Citations in the corpus

Change of angle in tent spaces (2011, background): “inating from [6], does not give optimal dependence. The inequality kgkTp,2 α ≤ C(1 +logα)αn/τkgk Tp,2 1 , for 1 < p < ∞ and α ≥ 1, where τ = min(p,2) and C depends only on n and p, is proved in [8] as a special case of a Banach space valued result, and, moreover, the polynomial growth αn/τ is ”
Second order elliptic operators with complex bounded measurable coefficients in L p , Sobolev and Hardy spaces (2010, background): “We would like to mention that in [41] the authors developed an alternative approach to (4.”
Tent space boundedness via extrapolation (2016, methodology): “s appear. First, T(F)(x,t) := Tt(F(·,t))(x), where Tt acts on functions on Rn. Second, T(F)(x,t) := Z∞ 0 Tt,s(F(·,s))(x) ds s where Tt,s acts on functions on Rn. For the second type, we refer to [7], [31], [8]. Positive results on Tq 2 all rely on the use of L 2 oﬀ-diagonal estimates (or improvedLmi”