<p>In this paper, we provide geometric characterizations for the positive Borel measures <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> such that the weighted Bergman–Orlicz space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A^{\Phi _{1}}_\omega $</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>A</mi> <mi>ω</mi> <msub> <mi mathvariant="normal">Φ</mi> <mn>1</mn> </msub> </msubsup> </math></EquationSource> </InlineEquation> is continuously embedded into <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^{\Phi _{2}}_\mu $</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>L</mi> <mi>μ</mi> <msub> <mi mathvariant="normal">Φ</mi> <mn>2</mn> </msub> </msubsup> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Phi _{1},\Phi _{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Φ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">Φ</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> are Young functions satisfying certain growth conditions, and the radial weight <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> satisfies the doubling condition. This work is mainly inspired by Peláez and Rättyä’s complete characterization of Carleson measures for weighted Bergman spaces <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$A_{\omega }^{p}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>A</mi> <mrow> <mi>ω</mi> </mrow> <mi>p</mi> </msubsup> </math></EquationSource> </InlineEquation> induced by doubling weights. In addition to some classical techniques in harmonic analysis such as tent spaces and dyadic decompositions, the interpolation method given by Gustavsson and Peetre is highly involved. As a byproduct, the boundedness of the Bergman projection from weighted Lebesgue–Orlicz spaces to weighted Bergman–Orlicz spaces is obtained.</p>

Embedding theorems for weighted Bergman–Orlicz spaces induced by doubling weights

期刊论文
发表日期：

2025年9月27日

查看全文

Min Dong,
Yongjiang Duan,
Kunyu Guo,
Siyu Wang

摘要

In this paper, we provide geometric characterizations for the positive Borel measures $\mu $ $μ$ such that the weighted Bergman–Orlicz space $A^{\Phi _{1}}_\omega $ $A_{ω}^{Φ_{1}}$ is continuously embedded into $L^{\Phi _{2}}_\mu $ $L_{μ}^{Φ_{2}}$ , where $\Phi _{1},\Phi _{2}$ $Φ_{1}, Φ_{2}$ are Young functions satisfying certain growth conditions, and the radial weight $\omega $ $ω$ satisfies the doubling condition. This work is mainly inspired by Peláez and Rättyä’s complete characterization of Carleson measures for weighted Bergman spaces $A_{\omega }^{p}$ $A_{ω}^{p}$ induced by doubling weights. In addition to some classical techniques in harmonic analysis such as tent spaces and dyadic decompositions, the interpolation method given by Gustavsson and Peetre is highly involved. As a byproduct, the boundedness of the Bergman projection from weighted Lebesgue–Orlicz spaces to weighted Bergman–Orlicz spaces is obtained.