Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Multilinear estimates for Calder\'on commutators   In this paper, we investigate the multilinear boundedness properties of the
higher ($n$-th) order Calder\'on commutator for dimensions larger than two. We
establish all multilinear endpoint estimates for the target space
$L^{\frac{d}{d+n},\infty}(\mathbb{R}^d)$, including that Calder\'on commutator
maps the product of Lorentz spaces $L^{d,1}(\mathbb{R}^d)\times\cdots\times
L^{d,1}(\mathbb{R}^d)\times L^1(\mathbb{R}^d)$ to
$L^{\frac{d}{d+n},\infty}(\mathbb{R}^d)$, which is the higher dimensional
nontrivial generalization of the endpoint estimate that the $n$-th order
Calder\'on commutator maps $L^{1}(\mathbb{R})\times\cdots\times
L^{1}(\mathbb{R})\times L^1(\mathbb{R})$ to
$L^{\frac{1}{1+n},\infty}(\mathbb{R})$. When considering the target space
$L^{r}(\mathbb{R}^d)$ with $r<\frac{d}{d+n}$, some counterexamples are given to
show that these multilinear estimates may not hold. The method in the present
paper seems to have a wide range of applications and it can be applied to
establish the similar results for Calder\'on commutator with a rough
homogeneous kernel.

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The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: dg-ga
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The categories of this paper are: alg-geom
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The categories of this paper are: adap-org
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The categories of this paper are: funct-an
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: atom-ph
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The categories of this paper are: eess
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The categories of this paper are: cond-mat
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The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn