Extended Cesàro operators between different Hardy spaces. (English) Zbl 1163.32004

Avetisyan, Karen; Stević, Stevo

Appl. Math. Comput. 207, No. 2, 346-350 (2009).

Summary: Let \(H^p\) denote the Hardy space of holomorphic functions on the unit ball \(\mathbb B\).
This note gives some sufficient and necessary conditions for the boundedness and compactness of the following extended Cesàro operators
\[ T_gf(z)=\int^1_0 f(tz) \operatorname{Re} g(tz) \frac{dt}{t}\quad\text{and}\quad L_gf(z)= \int^1_0 \operatorname{Re} f(tz) g(tz) \frac{dt}{t}\,, \]
where \(z\in\mathbb B\) and \(g\) is a fixed holomorphic map on \(\mathbb B\), acting from the space \(H^p\) into the space \(H^q\), for the case \(p<q\).
Our results extend and simplify some one-dimensional results.

MSC:

32A35

\(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables

Keywords:

extended Cesàro operators; unit ball; Hardy space; Bloch space; boundedness; compactness

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