## Nagoya Mathematical Journal

### A note on weighted Bergman spaces and the Cesáro operator

#### Abstract

Let $B$ denote the unit ball in $\mathbb{C}^n$, and $dV(z)$ normalized Lebesgue measure on $B$. For $\alpha > -1$, define $dV^\alpha (z)=(1-|z|^2)^\alpha dV(z)$. Let ${\cal H}(B)$ denote the space of holomorhic functions on $B$, and for $0 < p <\infty$, let ${\mathcal{A}}^p(dV_\alpha)$ denote $L^p(dV_\alpha)\cap {\cal H}(B)$. In this note we characterize ${\mathcal{A}}^p(dV_\alpha)$ as those functions in ${\cal H}(B)$ whose images under the action of a certain set of differential operators lie in $L^p(dV_\alpha)$. This is valid for $1 \le p <\infty$. We also show that the Ces\`aro operator is bounded on ${\mathcal{A}}^p(dV_\alpha)$ for $0<p<\infty$. Analogous results are given for the polydisc.

#### Article information

Source
Nagoya Math. J., Volume 159 (2000), 25-43.

Dates
First available in Project Euclid: 27 April 2005

https://projecteuclid.org/euclid.nmj/1114631450

Mathematical Reviews number (MathSciNet)
MR1783562

Zentralblatt MATH identifier
0981.32001

#### Citation

Benke, George; Chang, Der-Chen. A note on weighted Bergman spaces and the Cesáro operator. Nagoya Math. J. 159 (2000), 25--43. https://projecteuclid.org/euclid.nmj/1114631450

#### References

• S. Bell and E. Ligocka, A simplification and extension of Fefferman's theorem on biholomorphic mappings , Invent. Math. (1980, 57 ), 283–289.
• A. Bonami and S. Grellier, Weighted Bergman projections in domains of finite type in $\C^2$ , Contemporary Math. (1995, 189 ), 65–80.
• A. Bonami, D.C. Chang and S. Grellier, Commutation Properties and Lipschitz estimates for the Bergman and Szegö projections , Math. Zeit., 223 (1996), 275–302.
• D. Catlin, Subelliptic estimates for the $\bar\partial$-Neumann problem on pseudoconvex domains , Ann. of Math., 126 (1987), 131–191.
• D.C. Chang and B.Q. Li, Sobolev and Lipschitz Estimates for weighted Bergman projections , Nagoya Mathematical Journal, 147 (1997), 147–178.
• D.C. Chang, A. Nagel and E.M. Stein, Estimates for the $\bar\partial$-Neumann problem in pseudoconvex domains of finite type in ${\bf C}^2$ , Acta Mathemtica, 169 (1992), 153–228.
• P.L. Duren, Theory of $H^p$ Spaces, Academic Press, New York (1970).
• A.E. Djrbashian and F.A. Shamoian, Topics in the Theory of $A^p_\alpha$ Spaces, Teubner Verlagsgellschaft, Leibzig (1988).
• C.L. Fefferman, The Bergman kernel and biholomorphic mappings of pseudo-convex domains , Invent. Math. (1974, 26 ), 1–65.
• F. Forelli, Measures whose Poisson integrals are plurisubharmonic , Illinois J. Math. (1974, 18 ), 373–388.
• G.H. Hardy and J.E. Littlewoood, Some properties of fractional integrals II , Math. Zeit., 34 (1932), 403–439.
• S.G. Krantz, Function Theory of Several Complex Variables (2nd edition), Wadsworth & Brooks/Cole, Pacific Grove, California (1992).
• J. Miao, The Cesáro operator is bounded on $H^p$ for $0<p\le 1$ , Proc. Amer. Math. Soc., 116 (1992), 1077–1079.
• W. Rudin, Function Theory on the Unit Ball of $\C^n$, Springer-Verlag, Berlin$\cdot$New York$\cdot$Heidelberg (1980).
• A.G. Siskakis, The Cesáro operator is bounded on $H^1$ , Proc. Amer. Math. Soc., 110 (1990), 461–462.
• E.C. Titchmarsh, The Theory of Functions, Oxford University Press, London (1968).
• K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, Inc., New York$\cdot$ Basel (1990).