Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Probabilistic Approach to Fractional Integrals and the
  Hardy-Littlewood-Sobolev Inequality   We give a short summary of Varopoulos' generalised Hardy-Littlewood-Sobolev
inequality for self-adjoint $C_{0}$ semigroups and give a new probabilistic
representation of the classical fractional integral operators on $\R^n$ as
projections of martingale transforms. Using this formula we derive a new proof
of the classical Hardy-Littlewood-Sobolev inequality based on Burkholder-Gundy
and Doob's inequalities for martingales.

13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
29
The categories of this paper are: funct-an
20
The categories of this paper are: acc-phys
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
4
The categories of this paper are: cond-mat
23
The categories of this paper are: solv-int
11
The categories of this paper are: hep-lat
36
The categories of this paper are: cmp-lg
9
The categories of this paper are: nlin