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**Riemann-Stieltjes operators on weighted Bloch and Bergman spaces of the unit ball.**
*(English)*
Zbl 1064.47034

Given a holomorphic function \(g\) on the unit ball \(B_n\) of \(\mathbb C^n\), let \({\mathcal R} g\) denotes the radial derivative of \(g\). More explicitly, \({\mathcal R} g(z)=\sum_{j=1}^n z_j\,{{\partial g(z)}\over{\partial z_j}}\) for \(z=(z_1, \dots, z_n)\in B_n\). Associated with \(g\) is the so-called Riemann-Stieltjes operator \(T_g\) defined by
\[
T_gf(z)=\int_0^1 f(tz)\,{{dg(tz)}\over{dt}}=\int_0^1 f(tz){\mathcal R} g(tz)\,{{dt}\over{t}}
\]
for holomorphic functions \(f\) on \(B_n\). The Riemann-Stieltjes operators may be viewed as generalizations of the well-known Cesàro operator on the unit disk.

Main results of this paper are characterizations for \(T_g\) to be bounded, compact and of Schatten-von Neumann’s ideal on well-known holomorphic function spaces such as weighted Bloch spaces \(B^\alpha\) and weighted Bergman spaces \(A^p_\beta\). More explicitly, the weighted Bloch spaces \(B^\alpha\) (\(0<\alpha<\infty\)) is the space all holomorphic functions \(f\) on \(B_n\) with \(\| f\| _{B^\alpha}=| f(0)| +\sup(1-| z| ^2)^\alpha| {\mathcal R}f(z)| <\infty\). So, the space \(B^1\) is the well-known Bloch space. Also, the weighted Bergman space \(A^p_\beta\) (\(1\leq p<\infty\), \(-1<\beta<\infty\)) is the space of all holomorphic functions \(f\) on \(B_n\) such that \(\| f\| _{A^p_\beta}^p=\int_{B_n}| f(z)| ^p(1-| z| ^2)^\beta dV(z)<\infty\) where \(dV\) is the volume measure on \(B_n\).

When dealing with the weighted Bloch spaces, the author proves that \(T_g\) is bounded on \(B^\alpha\) if and only if \(g\in B^\gamma\) where \(\gamma=\alpha\) for \(\alpha\in(0,1)\) and \(\gamma=1\) for \(\alpha\in(1,\infty)\). The case \(\alpha=1\) remains open. On the unit disk, the case \(\alpha=1\) was characterized earlier by Z. J. Hu [Acta Math. Sci. (English Ed.) Ser. B 23, 561–566 (2003; Zbl 1044.47023)]. Hu’s characterization involves some extra logarithmic growth factor. The author indicates some difficulties in extending his method to prove the ball version of Hu’s characterization.

When dealing with the weighted Bergman spaces, the author proves that \(T_g\) is bounded on \(A^p_\beta\) if and only if \(g\in B^1\). It is interesting to see that the boundedness on an arbitrary \(A^p_\beta\) and \(B^\alpha\), \(\alpha\in(1,\infty)\), are the same. The author also gives a characterization for \(T_g\) on \(A^2_\beta\) to be in the Schatten-von Neumann ideals in terms of a certain integrability condition. As is well expected, the author shows that all the characterizations for boundedness become those for compactness, once big oh conditions are replaced by corresponding little oh conditions.

Main results of this paper are characterizations for \(T_g\) to be bounded, compact and of Schatten-von Neumann’s ideal on well-known holomorphic function spaces such as weighted Bloch spaces \(B^\alpha\) and weighted Bergman spaces \(A^p_\beta\). More explicitly, the weighted Bloch spaces \(B^\alpha\) (\(0<\alpha<\infty\)) is the space all holomorphic functions \(f\) on \(B_n\) with \(\| f\| _{B^\alpha}=| f(0)| +\sup(1-| z| ^2)^\alpha| {\mathcal R}f(z)| <\infty\). So, the space \(B^1\) is the well-known Bloch space. Also, the weighted Bergman space \(A^p_\beta\) (\(1\leq p<\infty\), \(-1<\beta<\infty\)) is the space of all holomorphic functions \(f\) on \(B_n\) such that \(\| f\| _{A^p_\beta}^p=\int_{B_n}| f(z)| ^p(1-| z| ^2)^\beta dV(z)<\infty\) where \(dV\) is the volume measure on \(B_n\).

When dealing with the weighted Bloch spaces, the author proves that \(T_g\) is bounded on \(B^\alpha\) if and only if \(g\in B^\gamma\) where \(\gamma=\alpha\) for \(\alpha\in(0,1)\) and \(\gamma=1\) for \(\alpha\in(1,\infty)\). The case \(\alpha=1\) remains open. On the unit disk, the case \(\alpha=1\) was characterized earlier by Z. J. Hu [Acta Math. Sci. (English Ed.) Ser. B 23, 561–566 (2003; Zbl 1044.47023)]. Hu’s characterization involves some extra logarithmic growth factor. The author indicates some difficulties in extending his method to prove the ball version of Hu’s characterization.

When dealing with the weighted Bergman spaces, the author proves that \(T_g\) is bounded on \(A^p_\beta\) if and only if \(g\in B^1\). It is interesting to see that the boundedness on an arbitrary \(A^p_\beta\) and \(B^\alpha\), \(\alpha\in(1,\infty)\), are the same. The author also gives a characterization for \(T_g\) on \(A^2_\beta\) to be in the Schatten-von Neumann ideals in terms of a certain integrability condition. As is well expected, the author shows that all the characterizations for boundedness become those for compactness, once big oh conditions are replaced by corresponding little oh conditions.

Reviewer: Boo Rim Choe (Seoul)

### MSC:

47B38 | Linear operators on function spaces (general) |

32A18 | Bloch functions, normal functions of several complex variables |

32A36 | Bergman spaces of functions in several complex variables |

30D55 | \(H^p\)-classes (MSC2000) |

46E15 | Banach spaces of continuous, differentiable or analytic functions |

47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |