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 $$u=0$$.

MSC:

 47B33 Linear composition operators 32A35 $$H^p$$-spaces, Nevanlinna spaces of functions in several complex variables

Citations:

Zbl 1026.47016; Zbl 1042.47017
Full Text:

References:

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