On the characterization of harmonic functions with initial data in Morrey space

Bo Li; Jinxia Li; Bolin Ma; Tianjun Shen

Czechoslovak Mathematical Journal (2024)

  • Volume: 74, Issue: 2, page 461-491
  • ISSN: 0011-4642

Abstract

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Let ( X , d , μ ) be a metric measure space satisfying the doubling condition and an L 2 -Poincaré inequality. Consider the nonnegative operator ℒ generalized by a Dirichlet form on X . We will show that a solution u to ( - ∂ t 2 + ℒ ) u = 0 on X × ℝ + satisfies an α -Carleson condition if and only if u can be represented as the Poisson integral of the operator ℒ with the trace in the generalized Morrey space L 2 , α ( X ) , where α 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.

How to cite

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Li, Bo, et al. "On the characterization of harmonic functions with initial data in Morrey space." Czechoslovak Mathematical Journal 74.2 (2024): 461-491. <http://eudml.org/doc/299369>.

@article{Li2024,
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.},
author = {Li, Bo, Li, Jinxia, Ma, Bolin, Shen, Tianjun},
journal = {Czechoslovak Mathematical Journal},
keywords = {harmonic function; Dirichlet problem; Morrey space; Carleson measure; metric measure space},
language = {eng},
number = {2},
pages = {461-491},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the characterization of harmonic functions with initial data in Morrey space},
url = {http://eudml.org/doc/299369},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Li, Bo
AU - Li, Jinxia
AU - Ma, Bolin
AU - Shen, Tianjun
TI - On the characterization of harmonic functions with initial data in Morrey space
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 2
SP - 461
EP - 491
AB - 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.
LA - eng
KW - harmonic function; Dirichlet problem; Morrey space; Carleson measure; metric measure space
UR - http://eudml.org/doc/299369
ER -

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