Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global estimates for the fundamental solution of homogeneous H\"ormander
  operators   Let $\mathcal{L}=\sum_{j=1}^{m}X_{j}^{2}$ be a H\"{o}rmander sum of squares
of vector fields in $\mathbb{R}^{n}$, where any $X_{j}$ is homogeneous of
degree $1$ with respect to a family of non-isotropic dilations in
$\mathbb{R}^{n}$. Then $\mathcal{L}$ is known to admit a global fundamental
solution $\Gamma (x;y)$, that can be represented as the integral of a
fundamental solution of a sublaplacian operator on a lifting space
$\mathbb{R}^{n}\times \mathbb{R}^{p}$, equipped with a Carnot group structure.
The aim of this paper is to prove global pointwise (upper and lower) estimates
of $\Gamma $, in terms of the Carnot-Carath\'{e}odory distance induced by
$X=\{X_{1},\ldots ,X_{m}\}$ on $\mathbb{R}^{n}$, as well as global pointwise
(upper) estimates for the $X$-derivatives of any order of $\Gamma $, together
with suitable integral representations of these derivatives. The least
dimensional case $n=2$ presents several peculiarities which are also
investigated. Applications to the potential theory for $\mathcal{L}$ and to
singular-integral estimates for the kernel $X_{i}X_{j}\Gamma $ are also
provided. Finally, most of the results about $\Gamma$ are extended to the case
of H\"{o}rmander operators with drift $\sum_{j=1}^{m}X_{j}^{2}+X_{0}$, where
$X_{0}$ is $2$-homogeneous and $X_{1},...,X_{m}$ are $1$-homogeneous.

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The categories of this paper are: plasm-ph
6
The categories of this paper are: astro-ph
5
The categories of this paper are: gr-qc
26
The categories of this paper are: bayes-an
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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: quant-ph
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The categories of this paper are: q-fin
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The categories of this paper are: nlin
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The categories of this paper are: nucl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-th
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The categories of this paper are: q-alg
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-ph