Abstract
Let \(({\mathbb {X}},d,\mu )\) be a space of homogeneous type in the sense of Coifman and Weiss, and \(X({\mathbb {X}})\) a ball quasi-Banach function space on \({\mathbb {X}}\). In this article, the authors introduce the weak Hardy space \(WH_X({\mathbb {X}})\) associated with \(X({\mathbb {X}})\) via the grand maximal function, and characterize \(WH_X({\mathbb {X}})\) by other maximal functions and atoms. The authors then apply these characterizations to obtain the real interpolation and the boundedness of Calderón–Zygmund operators in the critical case. The main novelties of this article exist in that the authors use the Aoki–Rolewicz theorem and both the dyadic system and the exponential decay of approximations of the identity on \({\mathbb {X}}\), which closely connect with the geometrical properties of \({\mathbb {X}}\), to overcome the difficulties caused by the deficiency of both the triangle inequality of \(\Vert \cdot \Vert _{X({\mathbb {X}})}\) and the reverse doubling assumption of the measure \(\mu \) under consideration, and also use the relation between the convexification of \(X({\mathbb {X}})\) and the weak ball quasi-Banach function space \(WX({\mathbb {X}})\) associated with \(X({\mathbb {X}})\) to prove that the infinite summation of atoms converges in the space of distributions on \({\mathbb {X}}\). Moreover, all these results have a wide range of generality and, particularly, even when they are applied to the weighted Lebesgue space, the Orlicz space, and the variable Lebesgue space, the obtained results are also new and, actually, some of them are new even on RD-spaces (namely, spaces of homogeneous type satisfying the additional reverse doubling condition).
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The authors would like to thank the referees for their carefully reading and several motivating and useful comments which indeed improve the presentation of this article.
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This project is partially supported by the National Key Research and Development Program of China (Grant No. 2020YFA0712900) and the National Natural Science Foundation of China (Grant Nos. 11971058, 12071197, 12122102 and 11871100)
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Sun, J., Yang, D. & Yuan, W. Weak Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Decompositions, Real Interpolation, and Calderón–Zygmund Operators. J Geom Anal 32, 191 (2022). https://doi.org/10.1007/s12220-022-00927-x
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DOI: https://doi.org/10.1007/s12220-022-00927-x
Keywords
- Space of homogeneous type
- Ball quasi-Banach function space
- Weak Hardy space
- Maximal function
- Atom
- Real interpolation
- Calderón–Zygmund operator