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

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 $$T_g:{\mathcal Z}\to{\mathcal B}_0$$, $$L_g:{\mathcal Z}\to{\mathcal B}_0$$.

MSC:

 45P05 Integral operators
Full Text:

References:

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