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Composition operators acting on holomorphic Sobolev spaces. (English) Zbl 1031.47017

Let \(\mathbf{D}\) be the unit disc in the complex plane and \(A_\alpha^p\) the weighted Bergman space, \(0<p<\infty\), \(\alpha>-1\) of functions holomorphic in \(\mathbf{D}\) with \[ \|f\|_{A_\alpha^p}=\int_{\mathbf{D}}|f(z)|^p(1-|z|^2)^\alpha dA(z)<\infty, \] and let \(A_{-1}^p=H^p\) denote the Hardy space. The holomorphic Sobolev space \(A_{\alpha,s}^p\) is treated as the space of functions \(f\) holomorphic in \(\mathbf{D}\) for which \(\mathcal{R}^sf\in A_\alpha^p\), where the fractional differentiation operator \(\mathcal{R}^s\) for \(f(z)=\sum\limits_{n=0}^\infty a_nz^n\) is defined by \[ \mathcal{R}^sf(z)=\sum\limits_{n=0}^\infty (n+1)^s a_nz^n. \] For the composition operator \(C_\varphi f(z)=f[\varphi(z)]\), where \(\varphi\) is a holomorphic mapping of \(\mathbf{D}\) onto itself, the authors prove some statements on boundedness and compactness. One of the typical results is the following theorem.
Theorem 1.1. Let \(p>0, s\geq 0\) and \(\alpha\geq -1\).
(a) If \(s<\frac{\alpha+1}{p}\), then every composition operator is bounded on \(A_{\alpha,s}^p\).
(b) If \(s=\frac{\alpha+1}{p}\) and
(i) \(p\geq 2\) or \(\alpha=-1\), then every composition operator is bounded on \(A_{\alpha,s}^p\);
(ii) \(p < 2\) and \(\alpha>-1\), then there exist composition operators not bounded on \(A_{\alpha,s}^p\).
(c) If \(s<\frac{\alpha+1}{p}\), then there exist composition operators not bounded on \(A_{\alpha,s}^p\).
The case when \(\varphi\) is of bounded valence is treated separately. Other results are also proved, in particular the boundedness of the operator \(C_\varphi\) in the space VMOA when \(ps=\alpha+2.\)

MSC:

47B33 Linear composition operators
30D55 \(H^p\)-classes (MSC2000)
46E15 Banach spaces of continuous, differentiable or analytic functions
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