Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Riesz Transform Characterizations of Musielak-Orlicz-Hardy Spaces   Let $\varphi$ be a Musielak-Orlicz function satisfying that, for any
$(x,\,t)\in\mathbb{R}^n\times(0,\,\infty)$, $\varphi(\cdot,\,t)$ belongs to the
Muckenhoupt weight class $A_\infty (\mathbb{R}^n)$ with the critical weight
exponent $q(\varphi)\in[1,\,\infty)$ and $\varphi(x,\,\cdot)$ is an Orlicz
function with $0<i(\varphi)\le I(\varphi)\le 1$ which are, respectively, its
critical lower type and upper type. In this article, the authors establish the
Riesz transform characterizations of the Musielak-Orlicz-Hardy spaces
$H_\varphi (\mathbb{R}^n)$ which are generalizations of weighted Hardy spaces
and Orlicz-Hardy spaces. Precisely, the authors characterize $H_\varphi
(\mathbb{R}^n)$ via all the first order Riesz transforms when
$\frac{i(\varphi)}{q(\varphi)}>\frac{n-1}{n}$, and via all the Riesz transforms
with the order not more than $m\in\mathbb{N}$ when
$\frac{i(\varphi)}{q(\varphi)}>\frac{n-1}{n+m-1}$. Moreover, the authors also
establish the Riesz transform characterizations of $H_\varphi(\mathbb{R}^n)$,
respectively, by means of the higher order Riesz transforms defined via the
homogenous harmonic polynomials or the odd order Riesz transforms. Even if when
$\varphi(x,t):=tw(x)$ for all $x\in{\mathbb R}^n$ and $t\in [0,\infty)$, these
results also widen the range of weights in the known Riesz characterization of
the classical weighted Hardy space $H^1_w({\mathbb R}^n)$ obtained by R. L.
Wheeden from $w\in A_1({\mathbb R}^n)$ into $w\in A_\infty({\mathbb R}^n)$ with
the sharp range $q(w)\in [1,\frac n{n-1})$, where $q(w)$ denotes the critical
index of the weight $w$.

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The categories of this paper are: chao-dyn
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The categories of this paper are: q-fin
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The categories of this paper are: econ
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The categories of this paper are: bayes-an
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The categories of this paper are: acc-phys
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The categories of this paper are: quant-ph
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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: dg-ga
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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: stat
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The categories of this paper are: math-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th