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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

Junqiang Zhang ,
Jun Cao ,
Renjin Jiang  and
Dachun Yang
Junqiang Zhang
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China. [email protected], [email protected], [email protected]
Jun Cao
Affiliation:
Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310032, P. R. China. [email protected]
Renjin Jiang
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China. [email protected], [email protected], [email protected]
Dachun Yang
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China. [email protected], [email protected], [email protected]
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Abstract

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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 $H_{w}^{p}\left( {{\mathbb{R}}^{n}} \right)$.

Keywords

42B2542B3042B3535J70degenerate elliptic operatorHardy spacesquare functionmaximal functionmoleculeRiesz transform
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 5 , 01 October 2015 , pp. 1161 - 1200
Copyright
Copyright © Canadian Mathematical Society 2015

References

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