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Pré-Publication, Document De Travail Année : 2018

Hörmander functional calculus on UMD lattice valued Lp spaces under generalised Gaussian estimates

Résumé

We consider self-adjoint semigroups Tt = exp(−tA) acting on L 2 (Ω) and satisfying (generalised) Gaussian estimates, where Ω is a metric measure space of homogeneous type of dimension d. The aim of the article is to show that A ⊗ IdY admits a Hörmander type H β 2 functional calculus on L p (Ω; Y) where Y is a UMD lattice, thus extending the well-known Hörmander calculus of A on L p (Ω). We show that if Tt is lattice positive (or merely admits an H ∞ calculus on L p (Ω; Y)) then this is indeed the case. Here the derivation exponent has to satisfy β > α · d + 1 2 , where α ∈ (0, 1) depends on p, and on convexity and concavity exponents of Y. A part of the proof is the new result that the Hardy-Littlewood maximal operator is bounded on L p (Ω; Y). Moreover, our spectral multipliers satisfy square function estimates in L p (Ω; Y). In a variant, we show that if e itA satisfies a dispersive L 1 (Ω) → L ∞ (Ω) estimate, then β > d+1 2 above is admissible independent of convexity and concavity of Y. Finally, we illustrate these results in a variety of examples.
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Dates et versions

hal-01807058 , version 1 (04-06-2018)
hal-01807058 , version 2 (20-11-2019)

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Luc Deleaval, Mikko Kemppainen, Christoph Kriegler. Hörmander functional calculus on UMD lattice valued Lp spaces under generalised Gaussian estimates. 2018. ⟨hal-01807058v1⟩

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