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<p>We define the harmonic Bergman space on locally finite trees with respect to a suitable probabilistic Laplacian and a class of weighted flow measures. We characterise the corresponding Bergman projection and prove that it is bounded on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^p\)</EquationSource> </InlineEquation> for every <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(p&gt;1\)</EquationSource> </InlineEquation>, and of weak type (1,&#xa0;1). We also prove necessary and sufficient conditions for the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L^p\)</EquationSource> </InlineEquation>-boundedness of the extension of a class of Toeplitz-type operators.</p>

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Harmonic Bergman spaces on locally finite trees

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  • 出版模式：

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  • 发表日期：

    2026年1月16日

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  • Alessandro Ottazzi,
  • Federico Santagati

摘要

We define the harmonic Bergman space on locally finite trees with respect to a suitable probabilistic Laplacian and a class of weighted flow measures. We characterise the corresponding Bergman projection and prove that it is bounded on \(L^p\) for every \(p>1\) , and of weak type (1, 1). We also prove necessary and sufficient conditions for the \(L^p\) -boundedness of the extension of a class of Toeplitz-type operators.