Non-tangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators

Zhang, Junqiang ; Cao, Jun ; Jiang, Renjin ; Yang, Dachun

Canadian journal of mathematics, Tome 67 (2015) no. 5, pp. 1161-1200

Résumé

Let $w$ be either in the Muckenhoupt class of ${{A}_{2}}\left( {{\mathbb{R}}^{n}} \right)$ weights or in the class of $QC\left( {{\mathbb{R}}^{n}} \right)$ weights, and let ${{L}_{w}}\,:=\,-{{w}^{-1}}\,\text{div}\left( A\nabla\right)$ be the degenerate elliptic operator on the Euclidean space ${{\mathbb{R}}^{n}}$ , $n\,\ge \,2$ . In this article, the authors establish the non-tangential maximal function characterization of the Hardy space $H_{{{L}_{w}}}^{p}\,\left( {{\mathbb{R}}^{n}} \right)$ associated with ${{L}_{w}}$ for $p\,\in \,(0,\,1]$ , and when $p\,\in \,(\frac{n}{n+1},\,1]$ and $w\,\in \,{{A}_{{{q}_{0}}}}\left( {{\mathbb{R}}^{n}} \right)$ with ${{q}_{0}}\,\in \,[1,\,\frac{p(n+1)}{n})$ , the authors prove that the associated Riesz transform $\nabla L_{w}^{-1/2}$ is bounded from $H_{{{L}_{w}}}^{p}\,\left( {{\mathbb{R}}^{n}} \right)$ to the weighted classical Hardy space $H_{w}^{p}\left( {{\mathbb{R}}^{n}} \right)$ .

Détail

DOI : 10.4153/CJM-2014-038-1

Mots-clés : 42B25, 42B30, 42B35, 35J70, degenerate elliptic operator, Hardy space, square function, maximal function, molecule, Riesz transform