Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Critical weak-$L^{p}$ differentiability of singular integrals   We establish that for every function $u \in L^1_\mathrm{loc}(\Omega)$ whose
distributional Laplacian $\Delta u$ is a signed Borel measure in an open set
$\Omega$ in $\mathbb{R}^{N}$, the distributional gradient $\nabla u$ is
differentiable almost everywhere in $\Omega$ with respect to the
weak-$L^{\frac{N}{N-1}}$ Marcinkiewicz norm. We show in addition that the
absolutely continuous part of $\Delta u$ with respect to the Lebesgue measure
equals zero almost everywhere on the level sets $\{u = \alpha\}$ and $\{\nabla
u = e\}$, for every $\alpha \in \mathbb{R}$ and $e \in \mathbb{R}^N$. Our
proofs rely on an adaptation of Calder\'on and Zygmund's singular-integral
estimates inspired by subsequent work by Hajlasz.

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The categories of this paper are: hep-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
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The categories of this paper are: funct-an
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The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: chem-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: gr-qc
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The categories of this paper are: stat
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The categories of this paper are: hep-lat
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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The categories of this paper are: q-bio