Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Duality of capacities and Sobolev extendability in the plane   We reveal relations between the duality of capacities and the duality between
Sobolev extendability of Jordan domains in the plane, and explain how to read
the curve conditions involvd in the Sobolev extendability of Jordan domains via
the duality of capacities. Finally as an application, we give an alternative
proof of the necessary condition for a Jordan planar domain to be $W^{1,
q}$-extension domain when $2 < q < \infty$.

3
The categories of this paper are: physics
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg
29
The categories of this paper are: funct-an
18
The categories of this paper are: econ
5
The categories of this paper are: gr-qc
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
23
The categories of this paper are: solv-int
31
The categories of this paper are: ao-sci
14
The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
26
The categories of this paper are: bayes-an
25
The categories of this paper are: patt-sol
32
The categories of this paper are: atom-ph
19
The categories of this paper are: eess