Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The $\Pi$-operator on Some Conformally Flat Manifolds and the Upper Half
  Space   The $\Pi$-operator, also known as Ahlfors-Beurling transform, plays an
important role in solving the existence of locally quasiconformal solutions of
Beltrami equations. In this paper, we first construct the $\Pi$-operator on a
general Clifford-Hilbert module. This $\Pi$-operator is also an $L^2$ isometry.
Further, it can also be used for solving certain Beltrami equations when the
Hilbert space is the $L^2$ space of a measure space. Then, we show that this
technique can be applied to construct the classical $\Pi$-operator in the
complex plane and some other examples on some conformally flat manifolds, which
are constructed by $U/\Gamma$, where $U$ is a simply connected subdomain of
either $\mathbb{R}^{n}$ or $\mathbb{S}^{n}$, and $\Gamma$ is a Kleinian group
acting discontinuously on $U$. The $\Pi$-operators on those manifolds also
preserve the isometry property in certain $L^2$ spaces, and their $L^p$ norms
are bounded by the $L^p$ norms of the $\Pi$-operators on $\mathbb{R}^{n}$ or
$\mathbb{S}^{n}$, depending on where $U$ lies. The applications of the
$\Pi$-operator to solutions of the Beltrami equations on those conformally flat
manifolds are also discussed. At the end, we investigate the $\Pi$-operator
theory in the upper-half space with the hyperbolic metric.

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The categories of this paper are: hep-th
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The categories of this paper are: q-bio
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: cond-mat
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The categories of this paper are: gr-qc
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-ph
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The categories of this paper are: nlin
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The categories of this paper are: patt-sol
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The categories of this paper are: cmp-lg
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The categories of this paper are: astro-ph
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The categories of this paper are: econ
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The categories of this paper are: nucl-th
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The categories of this paper are: q-alg
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The categories of this paper are: plasm-ph