Composition operators from Bloch type spaces to Hardy and Besov spaces
The space of all analytic functions on the unit disk \(\mathcal D\) is denoted by \(H(\mathcal D)\). Every analytic self-map \(\varphi : \mathcal D\to \mathcal D\) of the unit disk induces through composition a linear composition operator \(C_{\varphi}\) from \(H(\mathcal D)\) to itself and thus \(C_{\varphi}\) is defined by \(C_{\varphi}(f)=f\circ\varphi\) for \(f\in H(\mathcal D)\). An analytic function \(f\) on \(\mathcal D\) is said to be in the \(\alpha\)-Bloch space \(\mathcal B^{\alpha}, 0<\alpha<\infty\), if \(\|f \|_{\mathcal B^{\alpha}}=\sup_{z\in \mathcal D}|f'(z)|(1-|z|^2)^{\alpha}<\infty\). Correspondingly, \(f\) is in the little \(\alpha\)-Bloch space \(\mathcal B_0^{\alpha}\), if \(\lim_{|z|\to 1}|f'(z)|(1-|z|^2)^{\alpha}=0\). For \(0<p<\infty\) and \(-1<q<\infty\) an analytic function \(f\) is in the Besov type space \(B_{p,q}\) if \(\|f\|_{p,q}=(\int_\mathcal D|f'(z)|^p(1-|z|^2)^q dm(z))^{1/p}<\infty\), where \( dm(z)\) denotes the Lebesgue area measure on \(\mathcal D\). The author's main result is the following theorem: Let \(\varphi : \mathcal D\to \mathcal D\) be an analytic self-map, let \(0<p<\infty, -1<q<\infty\), and \(1\leq \beta<\infty\). Then the following statements are equivalent: (i) \(C_{\varphi} : \mathcal B^{1/\beta} \to B_{p\beta,q}\) is bounded (ii) \(C_{\varphi} : \mathcal B^{1/\beta} \to B_{p\beta, q}\) is compact (iii) \(C_{\varphi} : \mathcal B_0^{1/\beta} \to B_{p\beta,q}\) is bounded (iv) \(C_{\varphi} : \mathcal B_0^{1/\beta} \to B_{p\beta,q}\) is compact (v) \(\int_{\mathcal D}|\varphi'(z)|^{p\beta}(1-|z|^2)^q/ (1-|\varphi(z)|^2)^p dm(z)<\infty\). The hyperbolic Besov class \(B_p^h\) is defined as the set of analytic self-maps \(\varphi : \mathcal D\to \mathcal D\) such that \(\int_{\mathcal D}(\varphi^h(z))^p(1-|z|^2)^{p-2} dm(z)<\infty\), where \(\varphi^h(z)=|\varphi'(z)|/(1-|\varphi(z)|^2)\). Noting that \(\mathcal B^1=\mathcal B\) (the Bloch space), \(\mathcal B_0^1=\mathcal B_0\) (the little Bloch space) and \(B_{p,p-2}=B_p\) (the classical Besov space) the above theorem gives as a special case for \(\beta =1\) and \(q=p-2\): Let \(\varphi : \mathcal D\to \mathcal D\) be an analytic self-map and let \(1<p<\infty\). Then the following statements are equivalent: (i) \(C_{\varphi} : \mathcal B \to B_p\) is bounded (ii) \(C_{\varphi} : \mathcal B \to B_p\) is compact (iii) \(C_{\varphi} : \mathcal B_0 \to B_p\) is bounded (iv) \(C_{\varphi} : \mathcal B_0 \to B_p\) is compact (v) \(\varphi\in B_p^h\). The equivalence of (i), (ii) and (v) of this special case was independently proved by \textit{S. Makhmutov} [Proc. Japan Acad., Ser. A 72, No. 10, 232-234 (1996, Zbl 0873.30017)] and \textit{M. Tjani} [``Compact composition operators on some Möbius invariant Banach spaces'', Ph.D. Dissertation, Michigan State Univ., East Lansing, MI, 1996]. A corresponding consideration is made for composition operators \(C_{\varphi}\) from \(\mathcal B\)(or \(\mathcal B_0)\) to \(B_1\), where \(B_1\) is the minimal Besov space such that an analytic function \(f\in B_1\) if and only if \(\int_{\mathcal D}|f''(z)|dm(z)<\infty\). Then the equivalence of boundedness and compactness for composition operators from Bloch type spaces to Besov type spaces is explained from the general Banach space theory. Finally the boundedness and compactness of composition operators from Bloch type spaces to Hardy spaces are studied and, by the criterion for univalent self-map \(\varphi\) of \(\mathcal D\) to belong to \(B_p^h\), the following example is constructed: Let \(1<p_1<p_2<\infty\). Then there exists an analytic and the univalent self-map \(\varphi\) of \(\mathcal D\) such that (i) \(\|\varphi\|_{H^\infty}=1\) (ii) \(C_{\varphi} : \mathcal B \to B_{p_2}\) (or \(\mathcal B_0\to B_{p_2})\) is compact (iii) \(C_{\varphi} : \mathcal B\to B_{p_1}\) (or \(\mathcal B_0 \to B_{p_1}\)) is not compact.
scientific article; zbMATH DE number 1295277
scientific article