##
**Compact weighted composition operators and multiplication operators between Hardy spaces.**
*(English)*
Zbl 1167.47020

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. M.D.Contreras and 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 \(u=0\).

\[ \sup_{0<r<1}\int_{\partial B_N}|f(r\zeta)|^p\,d\sigma(\zeta)= \int_{\partial B_N}|f^*(\zeta)|^p\,d\sigma(\zeta)=\|f\|^p <\infty, \]

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. M.D.Contreras and 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 \(u=0\).

Reviewer: Héctor N. Salas (Mayagüez)

### MSC:

47B33 | Linear composition operators |

32A35 | \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables |

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