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Harmonic functions with BMO traces and their limiting behaviors on metric measure spaces

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Abstract

Let \((X,d,\mu )\) be a metric measure space satisfying a doubling condition and the \(L^2\)-Poincaré inequality. This paper is concerned with the boundary behavior of harmonic functions on the (open) upper half-space \(X\times {\mathbb {R}}_+\). We show that the traces of harmonic functions are in the bounded mean oscillation (BMO) space if and only if they satisfy the Carleson condition. This characterization is new even for uniformly elliptic operator on Euclidean spaces.

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Acknowledgements

Bo Li and Tianjun Shen would like to thank their advisor Prof. Renjin Jiang for proposing this joint work and for the useful discussions and advice on the topic of the paper. Yutong Jin was supported by the National Innovation and Entrepreneurship Training Program for Colleague Students (202310354015). Bo Li was supported by NNSF of China (12201250), NSF of Zhejiang Province (LQ23A010007) and NSF of Jiaxing (2023AY40003). Tianjun Shen was supported by NNSF of China (11922114 & 11671039) and NSF of Tianjin (20JCYBJC01410).

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Authors and Affiliations

  1. College of Data Science, Jiaxing University, Jiaxing, 314001, China

    Yutong Jin & Bo Li

  2. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China

    Bo Li & Tianjun Shen

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  1. Yutong Jin
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Correspondence to Tianjun Shen.

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Jin, Y., Li, B. & Shen, T. Harmonic functions with BMO traces and their limiting behaviors on metric measure spaces. Bull. Malays. Math. Sci. Soc. 47, 12 (2024). https://doi.org/10.1007/s40840-023-01603-1

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