Czechoslovak Mathematical Journal, first online, pp. 1-31
On the characterization of harmonic functions with initial data in Morrey space
Bo Li, Jinxia Li, Bolin Ma, Tianjun Shen
Received September 8, 2023. Published online March 18, 2024.
Abstract: Let $(X,d,\mu)$ be a metric measure space satisfying the doubling condition and an $L^2$-Poincaré inequality. Consider the nonnegative operator $\mathcal{L}$ generalized by a Dirichlet form on $X$. We will show that a solution $u$ to $(-\partial^2_t+\mathcal{L})u=0$ on $X\times\mathbb{R}_+$ satisfies an $\alpha$-Carleson condition if and only if $u$ can be represented as the Poisson integral of the operator $\mathcal{L}$ with the trace in the generalized Morrey space $L^{2,\alpha}(X)$, where $\alpha$ is a nonnegative function defined on a class of balls in $X$. This result extends the analogous characterization founded by R. Jiang, J. Xiao, D. Yang (2016) from the classical Morrey space on Euclidean space to the generalized Morrey space on the metric measure space.
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Affiliations: Bo Li, College of Data Science, Jiaxing University, No. 899 Guangqiong Road, Jiaxing City 314001, Zhejiang Province, P. R. China, e-mail: bli@zjxu.edu.cn; Jinxia Li, School of Mathematics and Information Science, Henan Polytechnic University, No. 2001 Shiji Road, Jiaozuo City 454003, Henan Province, P. R. China, e-mail: jinxiali@hpu.edu.cn; Bolin Ma, College of Data Science, Jiaxing University, No. 899 Guangqiong Road, Jiaxing City 314001, Zhejiang Province, P. R. China, e-mail: blma@zjxu.edu.cn; Tianjun Shen (corresponding author), Center for Applied Mathematics, Tianjin University, No. 92 Weijin Road, Tianjin City 300072, P. R. China, e-mail: shentj@tju.edu.cn
