Integral type operators from mixed-norm spaces to \(\alpha\)-Bloch spaces. (English) Zbl 1131.47031

Let \(B=\{z\in\mathbb C^n:| z| <1\}\) and \(H(B)\) be the class of all holomorphic functions on the unit ball. For \(f\in H(B)\) with the Taylor expansion \(f(z)=\sum_{| \beta| \geq0}a_\beta z^\beta\), let \(\operatorname{Re} f(z)=\sum_{|\beta|\geq0}|\beta| a_\beta z^\beta\) be the radial derivative of \(f\), where \(\beta=(\beta_1, \beta_2,\dots,\beta_n)\) is a multi-index and \(z^\beta=z_1^{\beta_1}\ldots z_n^{\beta_n}\). Let \(\alpha>0\). The \(\alpha\)-Bloch space \(\mathcal{B}^\alpha=\mathcal{B}^\alpha(B)\) is the space of all holomorphic functions \(f\) on \(B\) such that \(b_\alpha(f)=\sup_{z\in B}(1-| z| ^2)^\alpha|\operatorname{Re} f(z)|<\infty\). Let \(\mathcal{B}_0^\alpha\) be the subspace of \(\mathcal B\) consisting of those \(f\in{\mathcal B}^\alpha\) for which \((1-| z|^2)^\alpha| \operatorname{Re} f(z)| \to0\) as \(| z|\to1\). This space is called the little \(\alpha\)-Bloch space. Let \(p,q\in[0,\infty)\) and \(\gamma>-1\). The mixed norm space \(H_{p,q,\gamma}(B)\) consists of all holomorphic functions \(f\) on the unit ball such that \[ \| f\| ^q_{H_{p,q,\gamma}}=\displaystyle\int_0^1 M_p^q(f, r)(1-r)^\gamma\,dr<\infty, \] where \(M_p(f, r)=(\int_{\partial B}| f(r\xi)| ^p\,d\sigma(\xi))^{1/p}\) and \(\partial B=\{z\in\mathbb C^n:| z| =1\}\). Let \(g: B\mapsto\mathbb C^1\) be a holomorphic map of \(B\). For any holomorphic function \(f: B\mapsto \mathbb C^1\) and \(z\in B\), define \[ T_gf(z)=\displaystyle\int_0^1 f(tz)\operatorname{Re} g(tz)\,\frac{dt}{t} \quad\text{and}\quad L_gf(z)=\displaystyle\int_0^1 \operatorname{Re} f(tz)g(tz)\frac{dt}{t}. \] In this paper, the authors obtain the boundedness and compactness of the operators \(T_g\) and \(L_g\) from \(H_{p,\,q,\,\gamma}\) to the \(\alpha\)-Bloch space \(\mathcal B^\alpha(B)\) and little \(\alpha\)-Bloch space \(\mathcal B_0^\alpha(B)\).


47B38 Linear operators on function spaces (general)
30H05 Spaces of bounded analytic functions of one complex variable
47G10 Integral operators
Full Text: DOI


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