Cesàro-type operators on some spaces of analytic functions on the unit ball

Abstract

Let H (B) denote the space of all holomorphic functions on the unit ball B ⊂ Cn. In this paper we investigate the following integral operators:Tg (f) (z) = ∫01 f (tz) R g (tz) frac(dt, t) and Lg (f) (z) = ∫01 R f (tz) g (tz) frac(dt, t),where f ∈ H (B), z ∈ B, g ∈ H (B) and R h (z) = ∑j = 1n zj frac(∂ h, ∂ zj) (z) is the radial derivative of h. The operator Tg can be considered as an extension of the Cesàro operator on the unit disk. The boundedness and compactness of the operators Tg and Lg, on the Zygmund space and from the Zygmund space to the Bloch space are studied.