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Weighted Hardy spaces associated with elliptic operators. Part II: Characterizations of $H^1_L(w)$

José María Martell and Cruz Prisuelos-Arribas

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Given a Muckenhoupt weight $w$ and a second order divergence form elliptic operator $L$, we consider different versions of the weighted Hardy space $H^1_L(w)$ defined by conical square functions and non-tangential maximal functions associated with the heat and Poisson semigroups generated by $L$. We show that all of them are isomorphic and also that $H^1_L(w)$ admits a molecular characterization. One of the advantages of our methods is that our assumptions extend naturally the unweighted theory developed by S. Hofmann and S. Mayboroda in [19] and we can immediately recover the unweighted case. Some of our tools consist in establishing weighted norm inequalities for the non-tangential maximal functions, as well as comparing them with some conical square functions in weighted Lebesgue spaces.

Article information

Publ. Mat., Volume 62, Number 2 (2018), 475-535.

Received: 9 January 2017
Revised: 2 February 2017
First available in Project Euclid: 16 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B30: $H^p$-spaces 35J15: Second-order elliptic equations 42B37: Harmonic analysis and PDE [See also 35-XX] 42B25: Maximal functions, Littlewood-Paley theory 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30] 47G10: Integral operators [See also 45P05]

Hardy spaces second order divergence form elliptic operators heat and Poisson semigroups conical square functions non-tangential maximal functionss molecular decomposition Muckenhoupt weights off-diagonal estimates


Martell, José María; Prisuelos-Arribas, Cruz. Weighted Hardy spaces associated with elliptic operators. Part II: Characterizations of $H^1_L(w)$. Publ. Mat. 62 (2018), no. 2, 475--535. doi:10.5565/PUBLMAT6221806.

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