Cesàro-type operators on some spaces of analytic functions on the unit ball. (English) Zbl 1166.45009

Li, Songxiao; Stević, Stevo

Appl. Math. Comput. 208, No. 2, 378-388 (2009).

Consider the unit ball \(B\) in \(\mathbb{C}^n\), the space \(H(B)\) of the holomorphic functions \(f: B\to\mathbb{C}\), the radial derivative
\[ {\mathcal R}f(z)= \sum^n_{j=1} z_j{\partial f\over\partial z_j}(z),\quad z= (z_1,\dots, z_n)\in B, \]
the Bloch space \({\mathcal B}={\mathcal B}(B)\subset H(B)\) with the norm
\[ \| f\|_{{\mathcal B}}= |f(0)|+ \sup_{z\in B}(1-|z|^2)|\nabla f(z)|<+\infty, \]
the little Bloch space \({\mathcal B}_0(B)\) containing the functions \(f\in H(B)\) such that \[ \lim_{|z|\to 1} (1-|z|^2)|\nabla f(z)|= 0, \] the Banach space
\[ {\mathcal Z}={\mathcal Z}(B)= \{f\in H(B);\,\sup_{z\in B} (1-|z|^2)|{\mathcal R}^2 f(z)|<+\infty\}, \]
where \({\mathcal R}^2f={\mathcal R}({\mathcal R} f)\), with the norm
\[ \| f\|_{{\mathcal Z}}= |f(0)|+ \sup_{z\in B}(1-|z|^2)|{\mathcal R}^2f(z)|, \]
the closure \({\mathcal Z}_0\) in \({\mathcal Z}\) of the set of polynomials, and for a holomorphic map \(g: B\to \mathbb{C}\) the operators \(T_g: H(B)\to\mathbb{C}\) and \(L_g: H(B)\to \mathbb{C}\) defined by
\[ T_gf(z)= \int^1_0 f(tz){\mathcal R}g(z) t^{-1}\,dt,\qquad L_gf(z)= \int^1_0{\mathcal R}f(tz)g(tz)t^{-1} \,dt \]
for \(z\in B\) and \(f\in H(B)\).
The authors study the boundedness and compactness of the operator \(T_g\) and \(L_g\) using \({\mathcal Z}\), \({\mathcal Z}_0\), \({\mathcal B}\) and \({\mathcal B}_0\).
For example, if \(g\in H(B)\) then \(T_g:{\mathcal Z}\to{\mathcal Z}\) is bounded or compact if and only if \(g\in{\mathcal Z}\), \(L_g:{\mathcal Z}\to {\mathcal Z}\) or \(L_g:{\mathcal Z}_0\to{\mathcal Z}_0\) is bounded if and only if \(g\in H^\infty\cap{\mathcal B}_{\log}\), \(L_g:{\mathcal Z}\to{\mathcal Z}\) or \(L_g:{\mathcal Z}_0\to{\mathcal Z}_0\) is compact if and only if \(g= 0\), \(T_g:{\mathcal Z}_0\to{\mathcal Z}_0\) is bounded if and only if \(T_g:{\mathcal Z}_0\to{\mathcal Z}_0\) is compact and if and only if \(g\in{\mathcal Z}_0\).
There exist results of boundedness and compactness of the operators \(T_g:{\mathcal Z}\to{\mathcal B}\), \(L_g:{\mathcal Z}\to{\mathcal B}\) and \(T_g:{\mathcal Z}\to{\mathcal B}_0\), \(L_g:{\mathcal Z}\to{\mathcal B}_0\).

Reviewer: Dan-Mircea Borş (Iaşi)

MSC:

45P05

Integral operators

Keywords:

extended Cesàro operator; unit ball; Zygmund space; Bloch space; boundedness; compactness; spaces of analytic functions

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