<p>In this paper, we study a Forelli-Rudin type operator on a generalized Hartogs triangle defined by <Equation ID="Equ48"> <EquationSource Format="TEX">$\begin{aligned} H_k=\{(z, w)\in {\mathbb {C}}^n\times {\mathbb {C}}:|z_1|^2+\cdots +|z_n|^2<|w|^{2k}<1\} \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>H</mi> <mi>k</mi> </msub> <mrow> <mo>=</mo> <mo stretchy="false">{</mo> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> </mrow> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> <mrow> <mo>×</mo> <mi mathvariant="double-struck">C</mi> <mo>:</mo> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mi>n</mi> </msub> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <msup> <mrow> <mo><</mo> <mo stretchy="false">|</mo> <mi>w</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msup> <mrow> <mo><</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$z=(z_1, \cdots , z_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>z</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$k\in {\mathbb {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>. We give a sufficient and necessary condition for the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-boundedness of the Forelli-Rudin type operators on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$H_k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation>.</p>

$L^p$-Boundedness of Forelli-Rudin Type Operators on the Generalized Hartogs Triangles

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发表日期：

2025年12月11日

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Qingyang Zou

摘要

In this paper, we study a Forelli-Rudin type operator on a generalized Hartogs triangle defined by $\begin{aligned} H_k=\{(z, w)\in {\mathbb {C}}^n\times {\mathbb {C}}:|z_1|^2+\cdots +|z_n|^2<|w|^{2k}<1\} \end{aligned}$ $\begin{matrix} H_{k} = {(z, w) \in C^{n} \times C : | z_{1} |^{2} + \dots + | z_{n} |^{2} {< | w |}^{2 k} < 1} \end{matrix}$ where $z=(z_1, \cdots , z_n)$ $z = (z_{1}, \dots, z_{n})$ and $k\in {\mathbb {N}}$ $k \in N$ . We give a sufficient and necessary condition for the $L^p$ $L^{p}$ -boundedness of the Forelli-Rudin type operators on $H_k$ $H_{k}$ .

\(L^p\)-Boundedness of Forelli-Rudin Type Operators on the Generalized Hartogs Triangles

摘要