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Composition operators on small spaces. (English) Zbl 1114.47028

Let \(A^p_{\alpha,s}\) denote the space of holomorphic functions \(f\) on the unit disk such that \(\int_{\mathbb D}| {\mathcal R}^s f(z)| ^p(1-| z| ^2)^\alpha\,dA(z)<\infty\), that is to say that its \(s\)-fractional derivative \({\mathcal R}^sf(z)=\sum_{n=0}^\infty (n+1)^\alpha a_nz^n\) belongs to \(A_\alpha^p \). The authors analyze the boundedness and compactness of \(C_\phi\) on the spaces \(A^p_{\alpha,s}\) in some particular situations. Motivated by the notion of “suitably small” Banach space, they denote by \(\mathcal R\) the set of parameters \((s,p,\alpha)\) such that either \(sp>\alpha+2\) or \(sp=\alpha+2,\;0<p\leq 1\). Their main results establish that if \((s-1,p,\alpha)\in{\mathcal R}\) (where \(s\) is assumed to be a positive integer for \(\alpha=-1\)) and \(\phi\in A^p_{\alpha,s}\), then the composition operator \(C_\phi\) is bounded on \(A^p_{\alpha,s-k}\) for all \(k\in {\mathbb Z}^+,\;k\leq s\). In particular, they obtain in such a case that \(C_\phi\) is bounded on \(A^p_{\alpha,s}\) if and only if \(\phi\in A^p_{\alpha,s}\). As a byproduct, they get that \(A^p_{\alpha,s}\) is an algebra under multiplication for \((s,p,\alpha)\in {\mathcal R}\). Among other interesting results in the paper, the authors also give a complete description of symbols for the boundedness of \(C_\phi\) on the Zygmund class \(\Lambda_1\) of functions such that \(| f''(z)| =O\left(\frac{1}{1-| z| }\right)\). Finally, some examples concerning the boundedness of \(C_\phi\) on \(A^p_{\alpha,s}\) if \(\alpha+1\leq sp\leq \alpha+2+p\) are provided.

MSC:

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