Weighted composition operators on weighted-type high-order growth spaces on the unit ball

Abstract

In this paper, we propose and investigate a class of weighted-type high-order growth spaces $H_{\omega }^{\left(n\right)}:=\{f\in H(B):{\sup }_{z\in B}\omega (z\left)\right|R^{\left(n\right)}f\left(z\right)|<\infty \}$ of holomorphic functions on the unit ball $B$ of $C^N$, where ω is a normal weight on $B$ and $R^{\left(n\right)}f$ is the n-order radial derivative of f. This class covers the class of classical iterated weighted spaces $V_n:=\{z\in D:{\sup }_{z\in D}(1-\left|z{|}^2\right)\left|f^{\left(n\right)}\right(z\left)\right|<\infty \}$ and contains the growth spaces $H_{\omega }^{\infty }\left(B\right)$, the Bloch-type spaces $B_{\omega }\left(B\right)$ and the Zygmund-type spaces $Z_{\omega }\left(B\right)$. We also characterize the boundedness, the compactness and estimate the essential norms of weighted composition operators $W_{\psi ,\phi }:H_{\nu }^{\left(k\right)}\to H_{\mu }^{\left(n\right)}$, $f\mapsto \psi \cdot (f\circ \phi )$, $k,n\ge 0$, in terms of the function-theoretic properties of the holomorphic function ψ on $B$ and the point evaluation functions ${\delta }_{\phi }^{H_{\nu }^{\left(k\right)}}$ on $H_{\nu }^{\left(k\right)}$, where φ is a holomorphic self-map of $B$.