Bounded operators on the weighted spaces of holomorphic functions on the polydiscs

Abstract

Assuming that S is the space of functions of regular variation and ω = (ω₁, …, ω _n ), ω _j ∈ S, by A ^p (ω) we denote the class of all holomorphic functions defined on the polydisc U ⁿ such that

where dm _2n (z) is the 2n-dimensional Lebesgue measure on U ⁿ . In this article, we characterize the multipliers of A ^p (ω) (0 < p ≤ 1) and study the boundedness of Toeplitz operators on this space. As an application, a theorem about the division by good-inner functions in A ^p (ω) is proved. We also consider the generalized little Hankel and Berezin-type operators on A ^p (ω) (and on L ¹(ω)) and prove some theorems about the boundedness of these operators.

Keywords:

• weighted spaces
• polydisc
• Toeplitz operator
• little Hankel operator
• Berezin operator
• multiplier

AMS Subject Classifications:

• 32A36
• 45P05
• 47B35

Acknowledgement

A.V. Harutyunyan was supported by DFG 436 17/2/06.