Extended Cesàro operators on mixed norm spaces. (English) Zbl 1054.47023
Given a function \(g\) holomorphic on the unit ball \(B\) of \(\mathbb C^n\), let \({\mathcal R} g=\sum_{j=1}^n z_j{{\partial g}\over {\partial z_j}}\) be the radial derivative of \(g\). The extended Cesàro operator \(T_g\) with symbol \(g\) is then defined by
\[
T_gf(z)=\int_0^1 f(tz) {\mathcal R}g(tz)\, dt,\qquad z\in B,
\]
for functions \(f\) holomorphic on \(B\). In this paper, the author considers these operators acting on certain weighted mixed norm spaces \(H_{p,q}(\varphi)\). Roughly speaking, the space \(H_{p,q}(\varphi)\) is the space of all holomorphic functions on \(B\) whose \(q\)-integral means belong to \(L^p\) with respect to certain weights involving \(\varphi\). These mixed norm spaces contain the well-known weighted Bergman spaces as special cases.
The author shows that \(T_g\) is bounded (resp. compact) on \(H_{p,q}(\varphi)\) for some (resp. all) \(p,q\in(0,\infty]\) if and only if \(g\) is a (resp. little) Bloch function. This result extends earlier works byJ. H. Shi and G. P. Ren [Proc. Am. Math. Soc. 126, 3553–3560 (1998; Zbl 0905.47019)] and Z. Xiao [Acta Math. Sin., New Ser. 14, 647–654 (1998; Zbl 0927.30030)].
The author shows that \(T_g\) is bounded (resp. compact) on \(H_{p,q}(\varphi)\) for some (resp. all) \(p,q\in(0,\infty]\) if and only if \(g\) is a (resp. little) Bloch function. This result extends earlier works byJ. H. Shi and G. P. Ren [Proc. Am. Math. Soc. 126, 3553–3560 (1998; Zbl 0905.47019)] and Z. Xiao [Acta Math. Sin., New Ser. 14, 647–654 (1998; Zbl 0927.30030)].
Reviewer: Boo Rim Choe (Seoul)
MSC:
| 47B38 | Linear operators on function spaces (general) |
| 32A36 | Bergman spaces of functions in several complex variables |
References:
| [1] | Alexandru Aleman and Aristomenis G. Siskakis, An integral operator on \?^{\?}, Complex Variables Theory Appl. 28 (1995), no. 2, 149 – 158. · Zbl 0837.30024 |
| [2] | Alexandru Aleman and Aristomenis G. Siskakis, 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] | S. Bochner, Classes of holomorphic functions of several variables in circular domains, Proc. Nat. Acad. Sci. U.S.A. 46 (1960), 721 – 723. · Zbl 0092.29901 |
| [4] | Peter L. Duren, Theory of \?^{\?} spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. |
| [5] | T. M. Flett, The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl. 38 (1972), 746 – 765. · Zbl 0246.30031 · doi:10.1016/0022-247X(72)90081-9 |
| [6] | G. H. Hardy, Notes on some points in the integral calculus LXVI, Messenger of Math. 58(1929), 50-52. |
| [7] | Jie Miao, The Cesàro operator is bounded on \?^{\?} for 0<\?<1, Proc. Amer. Math. Soc. 116 (1992), no. 4, 1077 – 1079. · Zbl 0787.47029 |
| [8] | Ch. Pommerenke, Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation, Comment. Math. Helv. 52 (1977), no. 4, 591 – 602 (German). · Zbl 0369.30012 · doi:10.1007/BF02567392 |
| [9] | Walter Rudin, Function theory in the unit ball of \?\(^{n}\), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 241, Springer-Verlag, New York-Berlin, 1980. · Zbl 0495.32001 |
| [10] | Ji Huai Shi, On the rate of growth of the means \?_{\?} of holomorphic and pluriharmonic functions on bounded symmetric domains of \?\(^{n}\), J. Math. Anal. Appl. 126 (1987), no. 1, 161 – 175. · Zbl 0625.32003 · doi:10.1016/0022-247X(87)90083-7 |
| [11] | Ji-huai Shi and Guang-bin Ren, Boundedness of the Cesàro operator on mixed norm spaces, Proc. Amer. Math. Soc. 126 (1998), no. 12, 3553 – 3560. · Zbl 0905.47019 |
| [12] | A. L. Shields and D. L. Williams, Bonded projections, duality, and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc. 162 (1971), 287 – 302. · Zbl 0227.46034 |
| [13] | Aristomenis G. Siskakis, Composition semigroups and the Cesàro operator on \?^{\?}, J. London Math. Soc. (2) 36 (1987), no. 1, 153 – 164. · Zbl 0634.47038 · doi:10.1112/jlms/s2-36.1.153 |
| [14] | Richard M. Timoney, Bloch functions in several complex variables. II, J. Reine Angew. Math. 319 (1980), 1 – 22. · Zbl 0425.32008 · doi:10.1515/crll.1980.319.1 |
| [15] | Zhijing Xiao, Bergman type spaces and Cesàro operator, Acta Math. Sinica (N.S.) 14 (1998), no. suppl., 647 – 654. · Zbl 0927.30030 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
