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Riemann–Stieltjes operators between different weighted Bergman spaces. (English) Zbl 1169.47026

Let \(B\) be the unit ball in \(\mathbb C^n\) and \(\mathcal R f(z)=\sum_{j=1}^nz_j \frac {\partial f}{\partial z_j}(z)\), \(z\in B\), denote the radial derivative of an holomorphic function \(f:B\to \mathbb C\). For a given holomorphic function \(g\), one can consider the Riemann–Stieltjes operator \(T_g(f)(z)= \int_0^1 f(tz)\mathcal R g(tz) \frac{dt}{t}\) and the operator \(L_g(f)(z)=\int_0^1 g(tz)\mathcal R f(tz) \frac{dt}{t}\).
J.Xiao [J. Lond.Math.Soc., II.Ser.70, No.1, 199–214 (2004; Zbl 1064.47034)] gave a characterization of the symbols \(g\) for the boundedness and compacteness of \(T_g\) on the weighted Bergman spaces \(A_\alpha^p\). This was later extended to more general weighted Bergman spaces by Z.Hu [J. Math.Anal.Appl.296, No.2, 435–454 (2004; Zbl 1072.47029)].
The authors present here an extension of Xiao’s results for the boundedness and compactness of \(T_g\) and also \(L_g\) from \(A_\alpha^p\) into \(A_\beta^q\). The proofs use the standard approaches. The arguments used in them differ according to the cases \(p\leq q\) or \(q<p\).

MSC:

47B38 Linear operators on function spaces (general)
30H05 Spaces of bounded analytic functions of one complex variable
46E15 Banach spaces of continuous, differentiable or analytic functions
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Full Text: Euclid