Riesz Transform Characterizations of Musielak-Orlicz-Hardy Spaces

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