<p>We study the class of weighted Hadamard–Bergman operators <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {K}_{g,a}^{(\lambda )}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">K</mi> <mrow> <mi>g</mi> <mo>,</mo> <mi>a</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation> with general holomorphic kernels <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$g\in \mathcal {H}(\mathbb {D})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>∈</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">D</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and radial symbols <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$a\in L_{\lambda }^{1}(\mathbb {D})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <msubsup> <mi>L</mi> <mrow> <mi>λ</mi> </mrow> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> acting on general holomorphic function spaces <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {X}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">X</mi> </math></EquationSource> </InlineEquation>. By imposing a set of natural assumptions on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {X}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">X</mi> </math></EquationSource> </InlineEquation>, together with the boundedness of the maximal Bergman projection, we derive a sufficient condition for the boundedness of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {K}_{g,a}^{(\lambda ) }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">K</mi> <mrow> <mi>g</mi> <mo>,</mo> <mi>a</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation>. This condition relates the behavior of certain derivatives of the holomorphic kernel to specific integral averages of the symbol <i>a</i>. Furthermore, we establish the validity of our results for mixed-norm spaces and Orlicz spaces</p>

Weighted Hadamard-Bergman operators on general holomorphic function spaces: boundedness conditions and Berezin transform

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发表日期：

2026年7月14日

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Alexey Karapetyants,
Ali Raza

摘要

We study the class of weighted Hadamard–Bergman operators $\mathbb {K}_{g,a}^{(\lambda )}$ $K_{g, a}^{(λ)}$ with general holomorphic kernels $g\in \mathcal {H}(\mathbb {D})$ $g \in H (D)$ and radial symbols $a\in L_{\lambda }^{1}(\mathbb {D})$ $a \in L_{λ}^{1} (D)$ acting on general holomorphic function spaces $\mathcal {X}$ $X$ . By imposing a set of natural assumptions on $\mathcal {X}$ $X$ , together with the boundedness of the maximal Bergman projection, we derive a sufficient condition for the boundedness of $\mathbb {K}_{g,a}^{(\lambda ) }$ $K_{g, a}^{(λ)}$ . This condition relates the behavior of certain derivatives of the holomorphic kernel to specific integral averages of the symbol a. Furthermore, we establish the validity of our results for mixed-norm spaces and Orlicz spaces