Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Heat flow on 1-forms under lower Ricci bounds. Functional inequalities,
  spectral theory, and heat kernel   We study the canonical heat flow $(\mathsf{H}_t)_{t\geq 0}$ on the cotangent
module $L^2(T^*M)$ over an $\mathrm{RCD}(K,\infty)$ space
$(M,\mathsf{d},\mathfrak{m})$, $K\in\mathbb{R}$. We show
Hess-Schrader-Uhlenbrock's inequality and, if $(M,\mathsf{d},\mathfrak{m})$ is
also an $\mathrm{RCD}^*(K,N)$ space, $N\in(1,\infty)$, Bakry-Ledoux's
inequality for $(\mathsf{H}_t)_{t\geq 0}$ w.r.t. the heat flow
$(\mathsf{P}_t)_{t\geq 0}$ on $L^2(M)$. A crucial tool is that the dimensional
vector $2$-Bochner inequality is self-improving, entailing a dimensional vector
$1$-Bochner inequality - a version of which is also available in the
dimension-free case - as a byproduct. Variable versions of these estimates are
discussed as well. In conjunction with a study of logarithmic Sobolev
inequalities for $1$-forms, the previous inequalities yield various
$L^p$-properties of $(\mathsf{H}_t)_{t\geq 0}$, $p\in [1,\infty]$.
  Then we establish explicit inclusions between the spectrum of its generator,
the Hodge Laplacian $\smash{\vec{\Delta}}$, of the negative functional
Laplacian $-\Delta$, and of the Schr\"odinger operator $-\Delta+K$. In the
$\mathrm{RCD}^*(K,N)$ case, we prove compactness of $\smash{\vec{\Delta}^{-1}}$
if $M$ is compact, and the independence of the $L^p$-spectrum of
$\smash{\vec{\Delta}}$ on $p \in [1,\infty]$ under a volume growth condition.
  We terminate by giving an appropriate interpretation of a heat kernel for
$(\mathsf{H}_t)_{t\geq 0}$. We show its existence in full generality without
any local compactness or doubling assumptions, and derive fundamental estimates
and properties of it.

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The categories of this paper are: dg-ga
30
The categories of this paper are: q-alg
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The categories of this paper are: comp-gas
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The categories of this paper are: chem-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: q-fin
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The categories of this paper are: quant-ph
1
The categories of this paper are: cs
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The categories of this paper are: supr-con
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The categories of this paper are: adap-org
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ex
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The categories of this paper are: nlin