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

Dostanić, Milutin R.

Rev. Mat. Iberoam. 23, No. 2, 421-436 (2007).

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)].

Reviewer: John Schmeelk (Doha)

MSC:

47B38

Linear operators on function spaces (general)

Keywords:

weighted Berman’s space; radial weight function; Tauberian theorem of Ingham; boundedness; compactness

Citations:

Zbl 0951.47039

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
[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
[4]	Dzhrbashyan, M. M.: Integral Transforms and Representation of Function in Complex Domain . Izdat. “Nauka”, Moscow, 1966.
[5]	Fedoryuk, M. V.: Asymptotics Integrals and Series (Russian). Nauka, Moscow, 1987. · Zbl 0641.41001
[6]	Hedenmalm, H., Korenblum, B. and Zhu, K.: Theory of Bergman Spaces . Graduate Texts in Mathematics 199 . Springer, New York, 2000. · Zbl 0955.32003
[7]	Oleinik, V. L.: Imbedding theorems for weighted classes of harmonic and analytic functions. Investigations on linear operators and the theory of functions, V. Zap.Nauč. Sem. Lenningrad. Otdel. Math. Inst. Steklov. (LOMI) 47 (1974), 120-137 (in Russian).
[8]	Pommerenke, Ch.: Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation. Comment. Math. Helv. 52 (1977), no. 4, 591-602. · Zbl 0369.30012
[9]	Postnikov, A. G.: Introduction to the analytic number theory . Izdat. “Nauka”, Moscow, 1971. · Zbl 0231.10001
[10]	Ingham, A. E.: A Tauberian theorem for partitions. Ann. of Math. (2) 42 (1941), 1075-1090. JSTOR: · Zbl 0063.02973