<p>We study a Forelli-Rudin type operator on a generalized Hartogs triangle defined by <Equation ID="Equ1"> <EquationSource Format="TEX">\(H_n^k=\{z \in \mathbb C^n: \vert z_1 \vert^2+\cdots+\vert z_k \vert^2&lt;\vert z_{k+1}\vert^2&lt;\vert z_{k+2}\vert^2&lt;\cdots&lt;\vert z_{n}\vert^2&lt;1\},\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msubsup> <mi>H</mi> <mi>n</mi> <mi>k</mi> </msubsup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>z</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> <mo>:</mo> <mo fence="false" stretchy="false">∣</mo> <msub> <mi>z</mi> <mn>1</mn> </msub> <msup> <mo fence="false" stretchy="false">∣</mo> <mn>2</mn> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mo fence="false" stretchy="false">∣</mo> <msub> <mi>z</mi> <mi>k</mi> </msub> <msup> <mo fence="false" stretchy="false">∣</mo> <mn>2</mn> </msup> <mo>&lt;</mo> <mo fence="false" stretchy="false">∣</mo> <msub> <mi>z</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msup> <mo fence="false" stretchy="false">∣</mo> <mn>2</mn> </msup> <mo>&lt;</mo> <mo fence="false" stretchy="false">∣</mo> <msub> <mi>z</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <msup> <mo fence="false" stretchy="false">∣</mo> <mn>2</mn> </msup> <mo>&lt;</mo> <mo>⋯</mo> <mo>&lt;</mo> <mo fence="false" stretchy="false">∣</mo> <msub> <mi>z</mi> <mrow> <mi>n</mi> </mrow> </msub> <msup> <mo fence="false" stretchy="false">∣</mo> <mn>2</mn> </msup> <mo>&lt;</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </math></EquationSource> </Equation> where <i>z</i> = (<i>z</i><sub>1</sub>, …, <i>z</i><sub><i>n</i></sub>), <i>n</i> ⩾ <i>k</i> + 2 and <i>k, n</i> ∈ ℕ. We give a sufficient and necessary condition for the <i>L</i><sup><i>p</i></sup>-boundedness of the Forelli-Rudin type operators on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H_n^k\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msubsup> <mi>H</mi> <mi>n</mi> <mi>k</mi> </msubsup> </math></EquationSource> </InlineEquation>.</p>

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Lp-boundedness of the Forelli-Rudin type operators on generalized Hartogs triangles

  • Qingyang Zou

摘要

We study a Forelli-Rudin type operator on a generalized Hartogs triangle defined by \(H_n^k=\{z \in \mathbb C^n: \vert z_1 \vert^2+\cdots+\vert z_k \vert^2<\vert z_{k+1}\vert^2<\vert z_{k+2}\vert^2<\cdots<\vert z_{n}\vert^2<1\},\) H n k = { z C n : z 1 2 + + z k 2 < z k + 1 2 < z k + 2 2 < < z n 2 < 1 } , where z = (z1, …, zn), nk + 2 and k, n ∈ ℕ. We give a sufficient and necessary condition for the Lp-boundedness of the Forelli-Rudin type operators on \(H_n^k\) H n k .