Riemann–Stieltjes operators on Hardy spaces in the unit ball of \(\mathbb C^ n\). (English) Zbl 1136.47023

Let \(B\) be the open unit ball in \({\mathbb C}^{n}\) and \(f:B\rightarrow {\mathbb C}^{1}\) be a holomorphic function with the Taylor expansion \(f(z)=\sum_{| \beta| \geq0}a_{\beta}z^{\beta}\). Denote by \(({\mathfrak R}f)(z)=\sum_{| \beta| \geq0}| \beta| a_{\beta}z^{\beta}\) the radial derivative of \(f\). The authors study the integral operators
\[ (T_{g}f)(z)=\int_{0}^{1}f(tz)({\mathfrak R}g)(tz)\,\frac{dt}{t} \quad \text{and} \quad (L_{g}f)(z)=\int_{0}^{1}({\mathfrak R}f)(tz)g(tz)\,\frac{dt}{t}, \quad z\in B. \]
The boundedness and compactness of the operators \(T_{g}\) and \(L_{g}\) on the Hardy space \(H^{2}\) in the unit ball are discussed.


47B38 Linear operators on function spaces (general)
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
46E15 Banach spaces of continuous, differentiable or analytic functions
Full Text: Euclid