Products of radial derivative and multiplication operators from \(F(p,q,s)\) to weighted-type spaces on the unit ball
Given \(0<p,s<\infty\), \(-n-1<q<\infty\), and a weight \(\mu\) on \([0,1)\), that is, a positive and continuous function, the authors deal with the weighted-type spaces \[ H^{\infty}_{\mu}(\mathbb{B}):=\{f\in H(\mathbb{B}) \text{ such that } \|f\|:=\sup_{z\in \mathbb{B}}\mu(|z|)|f(z)|<\infty\}, \] \[ H^{\infty}_{\mu,0}(\mathbb{B}):=\{f\in H(\mathbb{B}) \text{ such that } \lim_{|z|\rightarrow 1}\mu(|z|)|f(z)|=0\}, \] and the general function space \[ \begin{multlined} F(p,q,s)(\mathbb{B}):=\{f\in H(\mathbb{B}) \text{ such that } \|f\|^p:\\ =|f(0)|^p+\sup_{a\in \mathbb{B}}\int_{\mathbb{B}}|\mathcal{R}f(z)|^p(1-|z|^2)^qg^s(z,a)\, dV(z)<\infty\},\end{multlined} \] where \(H(\mathbb{B})\) denotes the class of all holomorphic functions on the open unit ball in \(\mathbb{C}^n,\) \(g(z,a)=\log|\varphi_a(z)|^{-1}\) is the Green function for \(\mathbb{B}\) with logarithmic singularity at \(a,\) \(dV\) is the normalized Lebesgue measure on \(\mathbb{C}^n,\) and \(\mathcal{R}\) is the radial derivative of \(f\) at \(z\) defined by \(\mathcal{R}f(z)=\sum_{j=1}^nz_j\frac{\delta f}{\delta z_j}(z)\) for every \(f\in H(\mathbb{B})\). For special parameters \(p, q, s\), we can get many function spaces, such as the BMOA space, the Bergman space, the Hardy space, or the Bloch space. For normal weights, that is, such that there is \(\delta\in [0,1)\) and \(a\) and \(b\), \(0<a<b\), such that \[ \frac{\mu(r)}{(1-r)^a} \text{ is decreasing on } [\delta, 1) \text{ and } \lim_{r\rightarrow 1}\frac{\mu(r)}{(1-r)^a}=0, \] \[ \frac{\mu(r)}{(1-r)^b} \text{ is increasing on } [\delta, 1) \text{ and } \lim_{r\rightarrow 1}\frac{\mu(r)}{(1-r)^b}=\infty, \] the authors obtain a complete characterization of the boundedness and compactness of the products of the radial derivative and the multiplication operator \(\mathcal{R}M_u,\) \(u\) holomorphic, from \(F(p,q,s)\) to the weighted-type spaces of the unit ball in terms of conditions on \(u\) and \(\mu\). Observe that, for every holomorphic function \(f\), \[ \mathcal{R}M_uf(z)=M_u\mathcal{R}f(z)+\mathcal{R}u(z)f(z). \] The boundedness and compactness of the operator \(M_u\mathcal{R}\) between various spaces of holomorphic functions on the unit ball can be found in [\textit{Y.-M. Liu} and \textit{J. Zhou}, Complex Anal. Oper. Theory 7, No.~3, 593--606 (2013; Zbl 1295.47034)] and [\textit{S. Stević}, Appl. Math. Comput. 218, No. 18, 9241--9247 (2012; Zbl 1246.32005)], and some related operators between \(F(p,q,s)\) spaces and various spaces on the unit ball were treated previously by many authors, cf.\ the references of the article.
scientific article; zbMATH DE number 6157596