Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: H\"ormander's multiplier theorem for the Dunkl transform   For a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function
$k\geq 0$ let $\mathbf N=N+\sum_{\alpha \in R} k(\alpha)$. Denote by
$dw(\mathbf x)=\prod_{\alpha\in R}|\langle \mathbf
x,\alpha\rangle|^{k(\alpha)}\, d\mathbf x $ the associated measure in $\mathbb
R^N$. Let $\mathcal F$ stands for the Dunkl transform. Given a bounded function
$m$ on $\mathbb R^N$, we prove that if there is $s>\mathbf N$ such that $m$
satisfies the classical H\"ormander condition with the smoothness $s$, then the
multiplier operator $\mathcal T_mf=\mathcal F^{-1}(m\mathcal Ff)$ is of weak
type $(1,1)$, strong type $(p,p)$ for $1<p<\infty$, and bounded on a relevant
Hardy space $H^1$. To this end we study the Dunkl translations and the Dunkl
convolution operators and prove that if $F$ is sufficiently regular, for
example its certain Schwartz class seminorm is finite, then the Dunkl
convolution operator with the function $F$ is bounded on $L^p(dw)$ for $1\leq
p\leq \infty$. We also consider boundedness of maximal operators associated
with the Dunkl convolutions with Schwartz class functions.

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The categories of this paper are: physics
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th
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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: chao-dyn
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The categories of this paper are: adap-org
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: cs
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The categories of this paper are: comp-gas
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The categories of this paper are: alg-geom
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The categories of this paper are: econ
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-th