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Hörmander functional calculus on UMD lattice valued \(L^p\) spaces under generalized Gaussian estimates. (English) Zbl 1481.42012

Summary: We consider self-adjoint semigroups \(T_t = \exp(- tA )\) acting on \(L^2(\Omega )\) and satisfying (generalized) Gaussian estimates, where \(\Omega\) is a metric measure space of homogeneous type of dimension \(d\). The aim of the article is to show that \(A \bigotimes \mathrm{Id}_Y\) admits a Hörmander type \(\mathcal{H}_2^\beta\) functional calculus on \(L^p (\Omega; Y)\) where \(Y\) is a UMD lattice, thus extending the well-known Hörmander calculus of \(A\) on \(L^p (\Omega )\). We show that if \(T_t\) is lattice positive (or merely admits an \(H^\infty\) calculus on \(L^p (\Omega; Y)\)) then this is indeed the case. Here the derivation exponent has to satisfy \(\beta > \alpha \cdot d + \frac{1}{2} \), where \(\alpha \in (0, 1)\) 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 \(L^p(\Omega; Y)\). Moreover, our spectral multipliers satisfy square function estimates in \(L^p(\Omega; Y)\). In a variant, we show that if \(e^{itA}\) satisfies a dispersive \(L^1(\Omega) \rightarrow L^\infty (\Omega )\) estimate, then \(\beta > \frac{d + 1}{2}\) above is admissible independent of convexity and concavity of \(Y\). Finally, we illustrate these results in a variety of examples.

MSC:

42B15 Multipliers for harmonic analysis in several variables
42B25 Maximal functions, Littlewood-Paley theory
47B38 Linear operators on function spaces (general)
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