Compact weighted composition operators and multiplication operators between Hardy spaces
For \(N\in \mathbb N,\) let \(B_N\) be the unit ball of \(\mathbb C^N\). For each \(p\), \(1\leq p < \infty,\) the Hardy space \(H^p(B_N)\) consists of the holomorphic functions \(f\) in the ball such that \[ \sup_{0<r<1}\int_{\partial B_N}|f(r\zeta)|^p\,d\sigma(\zeta)= \int_{\partial B_N}|f^*(\zeta)|^p\,d\sigma(\zeta)=\|f\|^p <\infty, \] where \(d\sigma\) is the normalized Lebesgue measure on the boundary of \(B_N\) and \(f^*\) is the radial limit of \(f\), which exists for almost every \(\zeta \in \partial B_N\). Given a holomorphic self-map \(\varphi\) of \(B_N\) and a holomorphic map \(u\) in \(B_N,\) the weighted composition operator \(M_uC_{\varphi}\) is defined by \(M_uC_{\varphi}f=u(f\circ\varphi),\) where \(f\) is holomorphic. Let \(X\) and \(Y\) be Banach spaces and \(T\) be a bounded linear operator from \(X\) into \(Y\). The essential norm \(\|T\|_{e,X \to Y}\) is the distance from \(T\) to the set of compact operators from \(X\) into \(Y\). The pull-back measure \(\mu_{\varphi,u}\) induced by the self-map \(\varphi\) and \(u\in H^q(B_N)\) is the finite positive Borel measure on \(\overline B_N\) defined by \[ \mu_{\varphi,u}(E)=\int_{\varphi^{*-1}(E)}|u^*|^q\,d\sigma, \] for all Borel sets \(E\). Notice that \(\varphi^*\) is a map of \(\partial B_N\) into \(\overline{B_N}\). For each \(\zeta \in \partial B_N\) and \(t>0,\) let the Carleson \(S(\zeta,t)\) be \(\{z \in \overline{B_N}:|1-\langle z,\zeta \rangle|<1\}\). Many authors have studied weighted composition operators on different holomorphic function spaces. \textit{M.\,D.\thinspace Contreras} and \textit{A.\,Hernandez--Díaz} [J.~Math.\ Anal.\ Appl.\ 263, No.\,1, 224--233 (2001; Zbl 1026.47016); Integral Equations Oper.\ Theory 46, No.\,2, 165--188 (2003; Zbl 1042.47017)] characterized the compactness of \(M_uC_{\varphi}\) from \(H^p(B_1)\) into \(H^q(B_1)\) with \(1<p\leq q<\infty\) in terms of the pull-back measure, but they didn't estimate \(\|M_uC_{\varphi}\|_{e, H^p\to H^q}\). The authors' main result is as follows. Let \(1<p\leq q<\infty\). If \(M_uC_{\varphi}\) is a bounded weighted composition operator from \(H^p(B_N)\) into \(H^q(B_N),\) then \[ \begin{aligned} \| M_uC_{\varphi}\|_{e, H^p\to H^q}&\thicksim \limsup _{|w|\to 1^-}\int_{\partial B_N} |u^*(\zeta)|^q\left(\frac{1-|w|^2}{|1-\langle\varphi^*(\zeta),w\rangle|} \right)^{qN/p}d\sigma(\zeta)\\ &\thicksim \limsup_{t \to 0}\sup_{\zeta \in \partial B_N}\frac{\mu_{\varphi,u} (S(\zeta,t))}{t^{qN/p}}.\end{aligned} \] The notation \(\thicksim\) means that the ratio of two terms are bounded below and above by constants dependent on the dimension \(N\) and other parameters. They also show that a multiplication operator \(M_u\) from \(H^p(B_N)\) into \(H^q(B_N)\) with \(1<p\leq q<\infty\) is compact if and only if \(u=0\).
scientific article; zbMATH DE number 5313174