Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Neccessary conditions for two weight inequalities for singular integral
  operators   We prove necessary conditions on pairs of measures $(\mu,\nu)$ for a singular
integral operator $T$ to satisfy weak $(p,p)$ inequalities, $1\leq p<\infty$,
provided the kernel of $T$ satisfies a weak non-degeneracy condition first
introduced by Stein, and the measure $\mu$ satisfies a weak doubling condition
related to the non-degeneracy of the kernel. We also show similar results for
pairs of measures $(\mu,\sigma)$ for the operator $T_\sigma f = T(f\,d\sigma)$,
which has come to play an important role in the study of weighted norm
inequalities. Our major tool is a careful analysis of the strong type
inequalities for averaging operators; these results are of interest in their
own right. Finally, as an application of our techniques, we show that in
general a singular operator does not satisfy the endpoint strong type
inequality $T : L^1(\nu) \rightarrow L^1(\mu)$. Our results unify and extend a
number of known results.

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The categories of this paper are: bayes-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: math-ph
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The categories of this paper are: q-alg
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-th
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The categories of this paper are: adap-org
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The categories of this paper are: stat
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The categories of this paper are: chem-ph
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The categories of this paper are: funct-an
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The categories of this paper are: econ
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-lat