## 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.

### MSC:

 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

### Keywords:

Riemann-Stieltjes operator; Hardy space; BMOA space
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