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Atomic Characterization of Musielak–Orlicz–Lorentz Hardy Spaces and Its Applications to Real Interpolation and Boundedness of Calderón–Zygmund Operators

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Abstract

Let \(\varphi :\ {{\mathbb {R}}^n}\times [0,\infty )\longrightarrow [0,\infty )\) be a Musielak–Orlicz function satisfying the uniformly Muckenhoupt condition and be of uniformly lower type \(p_{\varphi }^-\) and of uniformly upper type \(p_{\varphi }^+\) with \(0<p_{\varphi }^-\le p_{\varphi }^+<\infty \), and let \(q\in (0,\infty ]\). In this article, the authors introduce the Musielak–Orlicz–Lorentz Hardy space \(H^{\varphi ,q}({\mathbb {R}}^n)\) which, when \(q=\infty \), coincides with the known weak Musielak–Orlicz Hardy space \(WH^{\varphi }({\mathbb {R}}^n)\). Then the authors establish both atomic and molecular characterizations of \(H^{\varphi ,q}({\mathbb {R}}^n)\). Applying these characterizations, the authors obtain the real interpolation and the boundedness of Calderón–Zygmund operators on \(H^{\varphi ,q}({\mathbb {R}}^n)\) when \(q\in (0,{\infty })\) or from the Musielak–Orlicz Hardy space \(H^{\varphi }({\mathbb {R}}^n)\) to \(H^{\varphi ,\infty }({\mathbb {R}}^n)\) in the critical case. The ranges of all the exponents under consideration are the best possible admissible ones which particularly improve all the known corresponding results for \(WH^{\varphi }({{\mathbb {R}}^n})\) via weakening the original assumption \(0<p_{\varphi }^-\le p_{\varphi }^+\le 1\) to the full range \(0<p_{\varphi }^-\le p_{\varphi }^+<\infty \), and all the results when \(q\in (0,{\infty })\) are new. The main novelty of this article is that the authors skillfully use the corresponding results related to weighted Lebesgue spaces via their relations with Musielak–Orlicz spaces to overcome those essential difficulties caused by the deficiency of both the explicit quasi-norm expression of Musielak–Orlicz spaces and the boundedness of the Hardy–Littlewood maximal operator on associate spaces of Musielak–Orlicz spaces.

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

    Hongchao Jia, Dachun Yang, Wen Yuan & Yangyang Zhang

  2. Department of Numerical Analysis, Eötvös L. University, Pázmány P. sétány 1/C., Budapest, 1117, Hungary

    Ferenc Weisz

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Jia, H., Weisz, F., Yang, D. et al. Atomic Characterization of Musielak–Orlicz–Lorentz Hardy Spaces and Its Applications to Real Interpolation and Boundedness of Calderón–Zygmund Operators. J Geom Anal 33, 188 (2023). https://doi.org/10.1007/s12220-023-01242-9

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