The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 451–509 | Cite as

Weighted Hardy Spaces Associated with Elliptic Operators. Part III: Characterisations of \(H_{L}^{p}({w})\) and the Weighted Hardy Space Associated with the Riesz Transform

  • Cruz Prisuelos-ArribasEmail author


We consider Muckenhoupt weights w, and define weighted Hardy spaces \(H^p_{\mathcal {T}}(w)\), where \(\mathcal {T}\) denotes a conical square function or a non-tangential maximal function defined via the heat or the Poisson semigroup generated by a second-order divergence form elliptic operator L. In the range \(0<p< 1\), we give a molecular characterisation of these spaces. Additionally, in the range \(p\in \mathcal {W}_w(p_-(L),p_+(L))\), we see that these spaces are isomorphic to the \(L^p(w)\) spaces. We also consider the Riesz transform \(\nabla L^{-\frac{1}{2}}\), associated with L, and show that the Hardy spaces \(H^p_{\nabla L^{-1/2},q}(w)\) and \(H^p_{\mathcal {S}_{ \mathrm {H} },q}(w)\) are isomorphic, in some range of \(p'\)s, and \(q\in \mathcal {W}_w(q_-(L),q_+(L))\).


Conical square functions Riesz transform Muckenhoupt weights Elliptic operators Heat and Poisson semigroups Off-diagonal estimates Complex interpolation Tent spaces Hardy spaces 

Mathematics Subject Classification

47B06 47B38 30H10 47G10 44A15 46M35 



I want to thank my advisor José María Martell for his useful comments and corrections, Li Chen for some conversations and help with references, and Pascal Auscher for some valuable comments.

Funding The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC Agreement No. 615112 HAPDEGMT. The author acknowledges receiving financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554).


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© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCMConsejo Superior de Investigaciones CientíficasMadridSpain

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