Extended Cesàro operators on Bergman spaces. (English) Zbl 1072.47029

Denote by \(H(B)\) the class of all holomorphic functions on the open unit ball \(B\) of \(\mathbb{C}^n\) and, for \(g\in H(B)\), let \(\operatorname{Re} g(z)=\sum_{j=1}^n z_j (\partial g/\partial z_j)\) denote the radial derivative. For \(g\in H(B)\), the extended Cesàro operator \(T_g\) with symbol \(g\) is the operator on \(H(B)\) defined by \[ T_g(f)(z)=\int_0^1 f(tz)\operatorname{Re} g(tz) \frac{dt}{t},\quad f\in H(B),\; z\in B. \] For a positive Lebesgue measurable function \(w\) defined on \(B\) and for \(0<p<\infty\), the weighted Bergman space \(L_{a,w}^p(B)\) is the space of all functions \(f\in H(B)\) such that \[ \int_B | f(z)| ^pw(z)dm(z)<\infty. \] The main result of the paper under review characterizes (for some class of weights \(w\)) all symbols \(g\) for which \(T_g\) is a bounded operator from the Bergman space \(L_{a,w}^p(B)\) to \(L_{a,w}^q(B)\) for \(0<p, q<\infty\). A similar characterization is given for compact operators.


47B38 Linear operators on function spaces (general)
46E15 Banach spaces of continuous, differentiable or analytic functions
32A36 Bergman spaces of functions in several complex variables
Full Text: DOI


[1] Aleman, A; Cima, J, An integral operator on Hp and Hardy’s inequality, J. anal. math., 85, 157-176, (2001) · Zbl 1061.30025
[2] Aleman, A; Siskakis, A.G, An integral operator on Hp, Complex variables theory appl., 28, 149-158, (1995) · Zbl 0837.30024
[3] Aleman, A; Siskakis, A.G, Integration operators on Bergman spaces, Indiana univ. math. J., 46, 337-356, (1997) · Zbl 0951.47039
[4] Benke, G; Chang, D.C, A note on weighted Bergman spaces and the Cesàro operator, Nagoya math. J., 159, 25-43, (2000) · Zbl 0981.32001
[5] Duren, P, Theorem of Hp spaces, (1970), Academic Press New York
[6] Flett, T.M, The dual of an inequality of Hardy and Littlewood and some related inequalities, J. math. anal. appl., 38, 746-765, (1972) · Zbl 0246.30031
[7] Hardy, G.H, Notes on some points in the integral calculus LXVI, Messenger of math., 58, 50-52, (1929)
[8] Luecking, D.H, Multipliers of Bergman spaces into Lebesgue spaces, Proc. Edinburgh math. soc., 29, 125-131, (1986) · Zbl 0587.30048
[9] Miao, J, The Cesàro operator is bounded on Hp for 0<p<1, Proc. amer. math. soc., 116, 1077-1079, (1992) · Zbl 0787.47029
[10] Pommerenke, Ch, Schlichte funktionen und analytische funktionen von beschrankter mittlerer oszilation, Comment. math. helv., 52, 591-602, (1977) · Zbl 0369.30012
[11] Rochberg, R, Interpolation by functions in Bergman spaces, Michigan math. J., 29, 229-236, (1982) · Zbl 0496.32010
[12] Rudin, W, Function theory in the unit ball of Cn, (1980), Springer-Verlag New York
[13] Shi, J, On the rate of growth of the Mean Mp of holomorphic and pluriharmonic functions on bounded symmetric domains of cn, J. math. anal. appl., 126, 161-175, (1987)
[14] Shi, J.H; Ren, G.P, Boundedness of the Cesàro operator on mixed norm spaces, Proc. amer. math. soc., 126, 3553-3560, (1998) · Zbl 0905.47019
[15] Shields, A.L; Williams, D.L, Bounded projections, duality and multipliers in spaces of analytic functions, Trans. amer. math. soc., 167, 253-268, (1971)
[16] Siskakis, A.G, Composition semigroups and the Cesàro operator on Hp, J. London math. soc. (2), 36, 153-164, (1987) · Zbl 0634.47038
[17] Timoney, R.M, Bloch functions in several complex variables. II, J. reine angew. math., 319, 1-32, (1980) · Zbl 0425.32008
[18] Xiao, Z, Bergman type spaces and Cesàro operator, Acta math. sinica (N.S.), 14, 647-654, (1998) · Zbl 0927.30030
[19] Zhu, K.H, The Bergman spaces, the Bloch space, and Gleason’s problem, Trans. amer. math. soc., 309, 253-268, (1988) · Zbl 0657.32002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.