<p>In this paper, it is shown that the harmonic Bergman projection <i>P</i><Stack> <sub><i>ω</i></sub> <sup><i>h</i></sup> </Stack>, induced by a radial weight <i>ω</i>, is bounded and onto from <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_414_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^{\infty}(\mathbb{D})\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mi>L</mi> <mrow> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="double-struck">D</mi> </mrow> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> to the harmonic Bloch space <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_414_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\cal{B}_{h}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mi mathvariant="script">B</mi> </mrow> <mrow> <mi>h</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>- if and only if <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_414_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(\omega\in \cal{D}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>ω</mi> <mo>∈</mo> <mrow> <mi mathvariant="script">D</mi> </mrow> </math></EquationSource> </InlineEquation>, which is a class of radial weights satisfying the two-sided doubling conditions. As an application, the bounded and compact positive Toeplitz operators <i>T</i><Stack> <sub><i>μ,ω</i></sub> <sup><i>h</i></sup> </Stack> on the endpoint case weighted harmonic Bergman space <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_414_Article_IEq4.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^1_{h,\omega}(\mathbb{D})\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msubsup> <mi>L</mi> <mrow> <mi>h</mi> <mo>,</mo> <mi>ω</mi> </mrow> <mn>1</mn> </msubsup> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="double-struck">D</mi> </mrow> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> are characterized.</p>

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Bergman projection and Toeplitz operators on weighted harmonic Bergman spaces induced by doubling weights

  • Yongjiang Duan,
  • Sawlet Junis,
  • Na Zhan

摘要

In this paper, it is shown that the harmonic Bergman projection P ω h , induced by a radial weight ω, is bounded and onto from \(L^{\infty}(\mathbb{D})\) L ( D ) to the harmonic Bloch space \(\cal{B}_{h}\) B h - if and only if \(\omega\in \cal{D}\) ω D , which is a class of radial weights satisfying the two-sided doubling conditions. As an application, the bounded and compact positive Toeplitz operators T μ,ω h on the endpoint case weighted harmonic Bergman space \(L^1_{h,\omega}(\mathbb{D})\) L h , ω 1 ( D ) are characterized.