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Integration operators on Bergman spaces with exponential weight. (English) Zbl 1146.47020

The paper studies operators of the form \(T_gf(x)= \int^\pi_0 f(\xi)g'(\xi)\,d(\xi)\), where \(g\) is an analytic function on the unit disc in the weighted Bergman space \(L^p(\omega)\). The special case is then studied where the weight function is of the form
\[ \omega(r)= \exp\Biggl({-a\over (1-r)^\beta}\Biggr)\qquad (a> 0,\;0<\beta\leq 1). \]
Then it is proven that the operator \(T_g\) is bounded (respectively, compact) in the space \(L^p_a(\omega)\) if and only if the condition \((1-|z|)^{\beta+ 1}|g'(z)|= O(1)\) (respectively, \(=o(1)\) as \(|z|\to 1-\)), thus solving a previous problem formulated in the paper by A. Aleman and A. G. Siskakis [Indiana Univ. Math. J. 46, No. 2, 337–356 (1997; Zbl 0951.47039)].

MSC:

47B38 Linear operators on function spaces (general)

Citations:

Zbl 0951.47039
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References:

[1] Aleman, A. and Siskakis, A. G.: An integral operator on \(H^p\). Complex Variables Theory Appl. 28 (1995), no. 2, 149-158. · Zbl 0837.30024
[2] Aleman, A. and Siskakis, A. G.: Integration Operators on Bergman Spaces. Indiana Univ. Math. J. 46 (1997), no. 2, 337-356. · Zbl 0951.47039 · doi:10.1512/iumj.1997.46.1373
[3] Aleman, A. and Cima, J. A.: An integral operator on \(H^p\) and Hardy’s inequality. J. Anal. Math. 85 (2001), 157-176. · Zbl 1061.30025 · doi:10.1007/BF02788078
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