Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: H\"ormander functional calculus on UMD lattice valued $L^p$ spaces under
  generalised Gaussian estimates   We consider self-adjoint semigroups $T_t = \exp(-tA)$ acting on $L^2(\Omega)$
and satisfying (generalised) Gaussian estimates, where $\Omega$ is a metric
measure space of homogeneous type of dimension $d$. The aim of the article is
to show that $A \otimes \mathrm{Id}_Y$ admits a H\"ormander type
$\mathcal{H}^\beta_2$ functional calculus on $L^p(\Omega;Y)$ where $Y$ is a UMD
lattice, thus extending the well-known H\"ormander 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 $\beta > \alpha \cdot d + \frac12$, 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 $L^1(\Omega) \to L^\infty(\Omega)$
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.

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The categories of this paper are: stat
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The categories of this paper are: atom-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-th
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The categories of this paper are: q-fin
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The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-bio
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The categories of this paper are: eess
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The categories of this paper are: hep-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: solv-int
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The categories of this paper are: supr-con
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The categories of this paper are: alg-geom
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The categories of this paper are: astro-ph