We study the boundedness of the linear operator S on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^{p}_{a}(dA_{\alpha })~(0<p<\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>a</mi> <mi>p</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <msub> <mi>A</mi> <mi>α</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mspace width="3.33333pt" /> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In particular, we obtain a sufficient and necessary condition for the compactness of the linear operator S on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^{p}_{a}(dA_{\alpha })~(1<p<\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>a</mi> <mi>p</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <msub> <mi>A</mi> <mi>α</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mspace width="3.33333pt" /> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Our results weaken the assumptions of earlier results of J. Miao and D. Zheng in a certain sense.

A class of linear operators on Bergman spaces

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2026年4月20日

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Zengjian Lou,
Antti Rasila,
Senhua Zhu

摘要

We study the boundedness of the linear operator S on $L^{p}_{a}(dA_{\alpha })~(0<p<\infty )$ $L_{a}^{p} (d A_{α}) (0 < p < \infty)$ . In particular, we obtain a sufficient and necessary condition for the compactness of the linear operator S on $L^{p}_{a}(dA_{\alpha })~(1<p<\infty )$ $L_{a}^{p} (d A_{α}) (1 < p < \infty)$ . Our results weaken the assumptions of earlier results of J. Miao and D. Zheng in a certain sense.