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Extended Cesàro operators on mixed norm spaces

Author: Zhangjian Hu
Journal: Proc. Amer. Math. Soc. 131 (2003), 2171-2179
MSC (2000): Primary 47B38, 32A36
Published electronically: December 30, 2002
MathSciNet review: 1963765
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Abstract: We define an extended Cesàro operator $T_g$ with holomorphic symbol $g$ in the unit ball $B$ of $C^n$ as

\begin{displaymath}T_g(f)(z)=\int_0^1f(tz)\Re g(tz)\frac{dt}{t}, \qquad f\in H(B),z\in B, \end{displaymath}

where $\Re g(z)= \sum_{j=1}^{n} z_j\frac{\partial f}{\partial z_j}$ is the radial derivative of $g$. In this paper we characterize those $g$ for which $T_g$ is bounded (or compact) on the mixed norm space $H_{p,q}(w)$.

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  • 1. A. Aleman and A. G. Siskakis, An integral operator on $H^p$, Complex Variables, 28(1995), 149-158. MR 2000d:47050
  • 2. A. Aleman and A. G. Siskakis, Integration operators on Bergman spaces, Indiana University Math. J. 46(1997), 337-356. MR 99b:47039
  • 3. S. Bochner, Classes of holomorphic functions of several variables in circular domains, Proc. Nat. Acad. Sci. U.S.A. 46(1960), 721-723. MR 22:11144
  • 4. P. Duren, Theory of $H^p$ Spaces, Acad. Press, New York, 1970. MR 42:3552
  • 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. MR 46:3799
  • 6. G. H. Hardy, Notes on some points in the integral calculus LXVI, Messenger of Math. 58(1929), 50-52.
  • 7. J. Miao, The Cesàro operator is bounded on $H^p$ for $0< p<1$, Proc. Amer. Math. Soc. 116(1992), 1077-1079. MR 93b:47064
  • 8. Ch. Pommerenke, Schlichte funktionen und analytische funktionen von beschrankter mittler oszilation, Comment. Math. Helv. 52(1977), 122-129. MR 56:12268
  • 9. W. Rudin, Function Theory in the Unit Ball of ${ C}^n$, Springer-Verlag, New York, 1980. MR 82i:32002
  • 10. J. H. Shi, On the rate of growth of the mean $M_p$ of holomorphic and pluriharmonic functions on bounded symmetric domains of $C^n$, J. Math. Anal. Appl. 126(1987), 161-175. MR 89d:32011
  • 11. J. H. Shi and G. P. Ren, Boundedness of the Cesàro operator on mixed norm spaces, Proc. Amer. Math. Soc. 126(1998), 3553-3560. MR 99b:47047
  • 12. A. L. Shields and D. L. Williams, Bounded projections, duality and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc., 162(1971), 287-302. MR 44:790
  • 13. A. G. Siskakis, Composition semigroups and the Cesàro operator on $H\sp p$. J. London Math. Soc. (2) 36 (1987), no. 1, 153-164. MR 89a:47048
  • 14. R. M. Timoney, Bloch functions in several complex variables. II, J. Reine Angew. Math. 319(1980), 1-22. MR 83b:32005
  • 15. Z. Xiao, Bergman type spaces and Cesàro operator, Acta Math. Sinica, New Series, 14(1998), 647-654. MR 2001c:46049

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Additional Information

Zhangjian Hu
Affiliation: Department of Mathematics, Huzhou Teachers College, Huzhou, Zhejiang, 313000, People’s Republic of China — and — Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599

Keywords: Ces\`{a}ro operator, mixed norm space, normal weight
Received by editor(s): May 8, 2001
Received by editor(s) in revised form: February 26, 2002
Published electronically: December 30, 2002
Additional Notes: This research was partially supported by the 151 Projection and the Natural Science Foundation of Zhejiang Province
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2002 American Mathematical Society

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