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Addendum to the paper “A note on weighted Bergman spaces and the Cesàro operator”. (English) Zbl 1090.32500

Summary: Let \(H({\mathbb D}_n)\) be the space of holomorphic functions on the unit polydisk \({\mathbb D}_n\), and let \({\mathcal L}_\alpha^{p,q}({\mathbb D}_n)\), where \(p,q>0\), \(\alpha=(\alpha_1,\dots,\alpha_n)\) with \(\alpha_j>-1\), \(j=1,\dots,n\), be the class of all measurable functions \(f\) defined on \({\mathbb D}_n\) such that \[ \int_{[0, 1)^n}M^q_p(f,r)\prod^n_{j=1}(1-r_j)^{\alpha_j}dr_j<\infty, \] where \(M_p(f,r)\) denote the \(p\)-integral means of the function \(f\). Denote the weighted Bergman space on \({\mathbb D}_n\) by \({\mathcal A}_\alpha^{p,q}({\mathbb D}_n)\cap H({\mathbb D}_n)\).
We provide a characterization for a function \(f\) being in \({\mathcal A}_\alpha^{p,q}({\mathbb D}_n)\). Using the characterization we prove the following result: Let \(p>1\), then the Cesàro operator is bounded on the space \({\mathcal A}_\alpha^{p,p}({\mathbb D})\).
[For the original paper see G. Benke and D.-C. Chang, Nagoya Math. J. 159, 25–43 (2000; Zbl 0981.32001)].

MSC:

32A17 Special families of functions of several complex variables
32A36 Bergman spaces of functions in several complex variables

Citations:

Zbl 0981.32001
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References:

[1] Nagoya Math. J. 159 pp 25– (2000) · Zbl 0981.32001 · doi:10.1017/S0027763000007406
[2] Houston J. Math. 30 pp 511– (2004)
[3] Acta Sci. Math. 69 pp 109– (2003)
[4] DOI: 10.1080/02781070310001650047 · Zbl 1053.47020 · doi:10.1080/02781070310001650047
[5] Studia Sci. Math. Hung. 39 pp 87– (2002)
[6] Acta Sci. Math. 66 pp 651– (2000)
[7] DOI: 10.1080/02781070290032216 · Zbl 1027.32005 · doi:10.1080/02781070290032216
[8] Math. Nachr. 248 pp 185– (2003)
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