Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Improving $L^2$ estimates to Harnack inequalities   We consider operators of the form ${\mathcal L}=-L-V$, where $L$ is an
elliptic operator and $V$ is a singular potential, defined on a smooth bounded
domain $\Omega\subset \R^n$ with Dirichlet boundary conditions. We allow the
boundary of $\Omega$ to be made of various pieces of different codimension. We
assume that ${\mathcal L}$ has a generalized first eigenfunction of which we
know two sided estimates. Under these assumptions we prove optimal Sobolev
inequalities for the operator ${\mathcal L}$, we show that it generates an
intrinsic ultracontractive semigroup and finally we derive a parabolic Harnack
inequality up to the boundary as well as sharp heat kernel estimates.

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The categories of this paper are: stat
25
The categories of this paper are: patt-sol
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The categories of this paper are: q-fin
36
The categories of this paper are: cmp-lg
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The categories of this paper are: funct-an
13
The categories of this paper are: quant-ph
27
The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th
37
The categories of this paper are: supr-con
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-ex
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: nlin
20
The categories of this paper are: acc-phys