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Article

Li, Bo ; Li, Jinxia ; Ma, Bolin ; Shen, Tianjun
On the characterization of harmonic functions with initial data in Morrey space. (English). Czechoslovak Mathematical Journal, vol. 74 (2024), issue 2, pp. 461-491
MSC: 35J25, 42B35, 43A85 | DOI: 10.21136/CMJ.2024.0368-23
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Keywords:
harmonic function; Dirichlet problem; Morrey space; Carleson measure; metric measure space
Summary:
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 \hbox {$\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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