Riesz transform characterizations of Musielak-Orlicz-Hardy spaces

Abstract

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 $w$.