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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References:

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