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

  Published:2015-02-11
 Printed: Oct 2015
  • Junqiang Zhang,
    School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China
  • Jun Cao,
    Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310032, P. R. China
  • Renjin Jiang,
    School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China
  • Dachun Yang,
    School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China
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Abstract

Let $w$ be either in the Muckenhoupt class of $A_2(\mathbb{R}^n)$ weights or in the class of $QC(\mathbb{R}^n)$ weights, and $L_w:=-w^{-1}\mathop{\mathrm{div}}(A\nabla)$ 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 $H_{L_w}^p(\mathbb{R}^n)$ associated with $L_w$ for $p\in (0,1]$ and, when $p\in (\frac{n}{n+1},1]$ and $w\in A_{q_0}(\mathbb{R}^n)$ 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 $H_{L_w}^p(\mathbb{R}^n)$ to the weighted classical Hardy space $H_w^p(\mathbb{R}^n)$.
Keywords: degenerate elliptic operator, Hardy space, square function, maximal function, molecule, Riesz transform degenerate elliptic operator, Hardy space, square function, maximal function, molecule, Riesz transform
MSC Classifications: 42B30, 42B35, 35J70 show english descriptions $H^p$-spaces
Function spaces arising in harmonic analysis
Degenerate elliptic equations
42B30 - $H^p$-spaces
42B35 - Function spaces arising in harmonic analysis
35J70 - Degenerate elliptic equations
 

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