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Riesz Transform Characterizations of Musielak-Orlicz-Hardy Spaces

Let $φ$ be a Musielak-Orlicz function satisfying that, for any $(x,\,t)\in\mathbb{R}^n\times(0,\,\infty)$, $φ(\cdot,\,t)$ belongs to the Muckenhoupt weight class $A_\infty (\mathbb{R}^n)$ with the critical weight exponent $q(φ)\in[1,\,\infty)$ and $φ(x,\,\cdot)$ is an Orlicz function with $0<i(φ)\le I(φ)\le 1$ 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 $H_φ(\mathbb{R}^n)$ which are generalizations of weighted Hardy spaces and Orlicz-Hardy spaces. Precisely, the authors characterize $H_φ(\mathbb{R}^n)$ via all the first order Riesz transforms when $\frac{i(φ)}{q(φ)}>\frac{n-1}{n}$, and via all the Riesz transforms with the order not more than $m\in\mathbb{N}$ when $\frac{i(φ)}{q(φ)}>\frac{n-1}{n+m-1}$. Moreover, the authors also establish the Riesz transform characterizations of $H_φ(\mathbb{R}^n)$, respectively, by means of the higher order Riesz transforms defined via the homogenous harmonic polynomials or the odd order Riesz transforms. Even if when $φ(x,t):=tw(x)$ 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 $w$.