Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On $\Phi$-inequalities for martingale fractional integration and their
  Bellman functions   Inspired by a conjecture of Vladimir Maz'ya on $\Phi$-inequalities in the
spirit of Bourgain and Brezis, we establish some $\Phi$-inequalities for
fractional martingale transforms. These inequalities may be thought of as
martingale models of $\Phi$-inequalities for differential operators. The proofs
rest on new simple Bellman functions.

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The categories of this paper are: hep-ph
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The categories of this paper are: physics
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The categories of this paper are: funct-an
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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: cs
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The categories of this paper are: adap-org
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The categories of this paper are: quant-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: bayes-an
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The categories of this paper are: dg-ga
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The categories of this paper are: econ
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The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph