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Non-tangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators
Published online by Cambridge University Press: 20 November 2018
Abstract
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
$H_{w}^{p}\left( {{\mathbb{R}}^{n}} \right)$.
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- Copyright © Canadian Mathematical Society 2015
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