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A Class of Integral Operators on the Unit Ball of \mathbb{C}^{n}

Authors:

Osman Kures Kehe Zhu

Institution:

(1) Department of Mathematics, State University of New York, Albany, NY 12222, USA

Abstract:

For real parameters a, b, c, and t, where c is not a nonpositive integer, we determine exactly when the integral operator

$Tf(z) = {\left( {1 - |z|^{2} } \right)}^{a} {\int_{\mathbb{B}_{n} } {\frac{{{\left( {1 - |w|^{2} } \right)}^{b} }} {{{\left( {1 - \langle z,w\rangle } \right)}^{c} }}f(w)dv(w)} }$

is bounded on $L^{p} {\left( {\mathbb{B}_{n} ,dv_{t} } \right)},$ where $\mathbb{B}_{n}$ is the open unit ball in $\mathbb{C}^{n} ,1 \leq p < \infty ,$ and dv_t (z) = (1 − |z| ²) ^t dv (z) with dv being volume measure on $\mathbb{B}_{n} .$ The characterization remains the same if we replace (1 − 〈z, w 〉) ^c in the integral kernel above by its modulus |1 − 〈z, w〉| ^c.

Keywords:

Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 47G10

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