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Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications

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Abstract

Let \(\mathcal{X}\) be a metric space with doubling measure and L a nonnegative self-adjoint operator in \(L^{2}(\mathcal{X})\) satisfying the Davies–Gaffney estimates. Let \(\varphi:\mathcal{X}\times[0,\infty)\to[0,\infty)\) be a function such that φ(x,⋅) is an Orlicz function, \(\varphi(\cdot,t)\in\mathbb{A}_{\infty}(\mathcal{X})\) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index I(φ)∈(0,1], and it satisfies the uniformly reverse Hölder inequality of order 2/[2−I(φ)]. In this paper, the authors introduce a Musielak–Orlicz–Hardy space \(H_{\varphi,L}(\mathcal{X})\), by the Lusin area function associated with the heat semigroup generated by L, and a Musielak–Orlicz BMO-type space \(\mathrm{BMO}_{\varphi,L}(\mathcal{X})\), which is further proved to be the dual space of \(H_{\varphi,L}(\mathcal{X})\) and hence whose φ-Carleson measure characterization is deduced. Characterizations of \(H_{\varphi,L}(\mathcal{X})\), including the atom, the molecule, and the Lusin area function associated with the Poisson semigroup of L, are presented. Using the atomic characterization, the authors characterize \(H_{\varphi,L}(\mathcal{X})\) in terms of the Littlewood–Paley \(g^{\ast}_{\lambda}\)-function \(g^{\ast}_{\lambda,L}\) and establish a Hörmander-type spectral multiplier theorem for L on \(H_{\varphi,L}(\mathcal{X})\). Moreover, for the Musielak–Orlicz–Hardy space H φ,L (ℝn) associated with the Schrödinger operator L:=−Δ+V, where \(0\le V\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})\), the authors obtain its several equivalent characterizations in terms of the non-tangential maximal function, the radial maximal function, the atom, and the molecule; finally, the authors show that the Riesz transform ∇L −1/2 is bounded from H φ,L (ℝn) to the Musielak–Orlicz space L φ(ℝn) when i(φ)∈(0,1], and from H φ,L (ℝn) to the Musielak–Orlicz–Hardy space H φ (ℝn) when \(i(\varphi)\in(\frac{n}{n+1},1]\), where i(φ) denotes the uniformly critical lower type index of φ.

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Acknowledgements

The first author is supported by the National Natural Science Foundation (Grant No. 11171027) of China and Program for Changjiang Scholars and Innovative Research Team in University of China.

The authors would like to express their deep thanks to the referees for their careful reading and many valuable remarks which made this article more readable.

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  1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, People’s Republic of China

    Dachun Yang & Sibei Yang

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Yang, D., Yang, S. Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications. J Geom Anal 24, 495–570 (2014). https://doi.org/10.1007/s12220-012-9344-y

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