Composition operators on generalized Hardy spaces (Q889965)

A domain \(\Omega \subset \overline{\mathbb{C}}\) is said to be Dini-smooth if its boundary \(\partial{\Omega}\) is a finite union of Jordan curves with non-singular Dini-parametrization, i.e., has a Dini-continous parametrization \(f\), that is, \(\int_0^{\varepsilon}\frac{\omega_f(t)}{t}\,dt < \infty\) for some \(\varepsilon > 0\), where \(\omega_f\) is the modulus of continuity of \(f\). For \(1 <p<\infty\), \(\nu \in {\mathcal{W}}_{\mathbb{R}}^{1,\infty}(\mathbb{D})\), the Sobolev space of all complex valued functions \(f \in {\mathcal{L}}^{\infty}(\mathbb{D})\) with distributional derivative in \({\mathcal{L}}^{\infty}(\mathbb{D})\), such that \(||\nu||_{{\mathcal{L}}}^{\infty}(\mathbb{D}) \leq k\) with \(k \in (0,1)\), the generalized Hardy space of the unit disk \(H_{\nu}^p(\mathbb{D})\) is defined as the set of all measurable functions \(f : \mathbb{D} \rightarrow \mathbb{C}\) such that \(\overline{\partial}f=\nu\overline{\partial}f\) in the sense of distribution in \(\mathbb{D}\) and \[ ||f||_{H_{\nu}^p(\mathbb{D})}=\left( \operatorname{ess}\sup\limits_{0<r<1}\frac{1}{2\pi}\int_0^{2\pi}|f(re^{it})|^pdt \right)^{\frac{1}{p}} < \infty. \] For a Dini-smooth domain \(\Omega\) with \[ \nu \in {\mathcal{W}}_{\mathbb{R}}^{1,\infty}(\Omega), \;\;||\nu||_{{\mathcal{L}}}^{\infty}(\mathbb{D}) \leq k \text{ with } k \in (0,1), \tag{\(*\)} \] the definition of \(H_{\nu}^p(\Omega)\) was further extended in [\textit{L. Baratchart} et al., Complex Var. Elliptic Equ. 59, No. 4, 504--538 (2014; Zbl 1293.30086)] with \[ ||f||_{H_{\nu}^p(\Omega)}=\sup\limits_{n \in \mathbb{N}}||f||_{ {\mathcal{L}}^p(\partial\Delta_n)}, \] where \((\Delta_n)\) is a fixed sequence of domains such that \(\overline{\Delta}_n \subset \Omega\) and \(\partial{\Delta_n}\) is a finite union of rectifiable Jordan curves of uniformly bounded length such that each subset of \(\Omega\) is eventually contained in \(\Delta_n\) for \(n\) large enough. Likewise, the generalized Hardy space \(G_{\alpha}^p(\Omega)\), for \(\alpha \in {\mathcal{L}}^{\infty}(\Omega)\), is defined as the set of all measurable functions \(\omega : \Omega \to \mathbb{C}\) such that \(\overline{\partial}\omega=\alpha\overline{\omega}\) in \(D'(\Omega)\), the dual space of the space \(D(\Omega)\) of smooth functions with compact support in \(\Omega\) (i.e, the space of distributions in \(\Omega\)), and \[ ||\omega||_{G_{\alpha}^p(\Omega)}=||tr (\omega)||_{ {\mathcal{L}}^p(\partial\Delta_n)} < \infty, \] where \(\operatorname{tr}(\omega)\) represents the non-tangential limit (a.e) of \(\omega\) on \(\partial{\Omega}\). The generalized Hardy spaces \(H_{\nu}^p(\Omega)\) and \(G_{\alpha}^p(\Omega)\) are real Banach spaces (they are complex Banach spaces for \(\nu=0\) and \(\alpha=0\), respectively). The paper under review investigates composition operators on these generalized Hardy spaces \(G_{\alpha}^p(\Omega)\) and \(H_{\nu}^p(\Omega)\) over Dini-smooth domains. For two bounded Dini-smooth domains \(\Omega_1\) and \(\Omega_2\) in \(\mathbb{C}\), \(\nu\) defined on \(\Omega_2\) satisfying (\(*\)) and \( \phi : \Omega_1 \rightarrow \Omega_2\) analytic with \(\phi \in {\mathcal{W}}_{\mathbb{R}}^{1,\infty}(\Omega_1)\), the authors prove the boundedness of \(C_{\phi} : H_{\nu}^p(\Omega_2) \rightarrow H_{\nu \circ \phi}^p(\Omega_1)\) and give estimates for \(||C_{\phi}||\) in the case of the unit disk and a doubly connected domain, respectively. They also prove that composition operators on \(H_{\nu}^p\) and \(G_{\alpha}^p\) are \(\mathbb{R}\)-isomorphic. In Section~5, the authors extend the characterization of invertibility of composition operators on the \(H_{\nu}^p\) and \(G_{\alpha}^p\) on \(\mathbb{D}\) as in [\textit{H. J. Schwartz}, Composition operators on \(H'(P)\). Toledo: University of Toledo (Ph.D. Thesis) (1969)] to the case of Dini-smooth domains. In Section 6, \(\phi \in {\mathcal{W}}_{\mathbb{D}}^{1,\infty}(\mathbb{D})\) and \(\overline{\alpha} =(\alpha \circ \phi)\overline{\partial{\phi}}\), the authors characterize isometries of composition operators \(C_{\phi} : G_{\alpha}^p(\mathbb{D}) \rightarrow G_{\overline{\alpha}}^p(\mathbb{D})\) (which also characterizes \(C_{\phi} : H^p(\mathbb{D}) \rightarrow H^p(\mathbb{D})\)). For the doubly connected domain \(\mathbb{A} = \mathbb{D}\cap(\mathbb{C}\setminus r_0\overline{\mathbb{D}})\), \(\phi : \mathbb{A} \rightarrow \mathbb{A}\) satisfying the condition \(\phi \in {\mathcal{W}}_{\mathbb{A}}^{1,\infty}(\mathbb{A})\), \(\alpha \in {\mathcal{L}}^{\infty}(\mathbb{A})\) and \(\nu\) satisfying \((*)\), and \(\overline{\nu} =(\nu \circ \phi)\overline{\partial{\phi}}\), the authors prove the equivalence of the following:{\parindent=8mm \begin{itemize}\item[(i)] \(C_{\phi}\) is an isometry from \(H^p(\mathbb{A})\) into \(H^p(\mathbb{A})\), \item[(ii)] \(C_{\phi}\) is an isometry from \(G_{\alpha}^p(\mathbb{A})\) into \(G_{\overline{\alpha}}^p(\mathbb{A})\), \item[(iii)] \(C_{\phi}\) is an isometry from \(G_{\nu}^p(\mathbb{A})\) into \(G_{\overline{\nu}}^p(\mathbb{A})\), and \item[(iv)] either there exists \(\lambda \in \mathbb{C}\) of unit modulus such that \(\phi(z)=\lambda z\) for all \(z \in \mathbb{A}\), or there exists \(\mu \in \mathbb{C}\) of unit modulus such that \(\phi(z)=\lambda\frac{r_0}{2}\) for all \(z \in \mathbb{A}\). \end{itemize}} It is interesting to note that (i) and (ii) are new even for the case of composition operators on \(H^p(\mathbb{A})\).