Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Limits at Infinity of Weighted Sobolev Functions   We study necessary and sufficient conditions for a Muckenhoupt weight $w \in
L^1_{\mathrm{loc}}(\mathbb R^d)$ that yield almost sure existence of radial,
and vertical, limits at infinity for Sobolev functions $u \in
W^{1,p}_{\mathrm{loc}}(\mathbb R^d,w)$ with a $p$-integrable gradient $|\nabla
u|\in L^p(\mathbb R^d,w)$. The question is shown to subtly depend on the sense
in which the limit is taken.
  First, we fully characterize the existence of radial limits. Second, we give
essentially sharp sufficient conditions for the existence of vertical limits.
In the specific setting of product and radial weights, we give if and only if
statements. These generalize and give new proofs for results of Fefferman and
Uspenski\u{\i}.

26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
29
The categories of this paper are: funct-an
17
The categories of this paper are: q-fin
18
The categories of this paper are: econ
37
The categories of this paper are: supr-con
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The categories of this paper are: dg-ga
3
The categories of this paper are: physics
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The categories of this paper are: comp-gas
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The categories of this paper are: eess
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The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
12
The categories of this paper are: nucl-th
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The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
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The categories of this paper are: ao-sci