A kind of integral operator on normal weight Dirichlet type space in the unit ball. (Chinese. English summary) Zbl 07802325

Guo, Yuting; Zhang, Xuejun

Chin. Ann. Math., Ser. A 44, No. 3, 255-266 (2023).

Summary: Let \(\mu\) be a normal function on \([0,1)\) and Bn be the unit ball in \(n\) dimensions complex space \(\mathbb{C}^n\). Suppose that \(\psi\) is a holomorphic function on Bn and \(\varphi\) is a holomorphic self-map of Bn. The authors consider a kind of integral operator as follows: \[ T_{\varphi, \psi} (f)(z) = \int^1_0 f[\varphi (tz)] R\psi (tz) \frac{\mathrm{d}t}{t}, \quad z\in B_n. \] The authors mainly characterize the boundedness and compactness of \(T_{\varphi, \psi}\) on the normal weight Dirichlet type space \(\mathcal{D}^p_\mu (B_n) (0 < p \leqslant 1)\). At the same time, the authors discuss the same problem from the normal weight Bergman space \(\mathcal{A}^p_\mu (B_n)\) to \(\mathcal{D}^p_\mu (B_n) (p > 0)\) by measures on Carleson square and Bergman ball. Necessary and sufficient conditions are given for all the cases discussed.

MSC:

47B38	Linear operators on function spaces (general)
32A37	Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))

Keywords:

composition Cesáro operator; Bergman type space; Dirichlet type space; boundedness; compactness

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[27]	Abstract Let µ be a normal function on [0, 1) and B n be the unit ball in n dimensions complex space C n . Suppose that ψ is a holomorphic function on B n and ϕ is a holomorphic