Riesz Transform Characterizations of Musielak-Orlicz-Hardy Spaces

Abstract

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

38 pages, Trans. Amer. Math. Soc. (to appear)