Higher radial derivative of Bloch type functions
Let \(B=\{z\in\mathbb{C}^n: |z|<1\}\) be the unit ball of \(\mathbb C^n\), let \(dv\) be the normalized Lebesgue measure of \(B\) and let \(d\lambda(z)=(1-|z|^2)^{-n-1}dv(z)\). The class of all holomorphic functions is denoted by \(H(B)\). For \(f\in H(B)\), its radial derivative at \(z\) is \(Rf(z)=\langle\bigtriangledown f(z),\bar z\rangle=\sum_{j=1}^{n}\frac{\partial f}{\partial z_j}z_j\), where the complex gradient of \(f\) is \(\bigtriangledown f(z)=\big(\frac{\partial f}{\partial z_1}(z),...,\frac{\partial f}{\partial z_n}(z)\big)\). For \(m\in\mathbb N\), \(R^mf(z)=R(R^{m-1}f)(z)\) denotes the higher radial derivative of \(f\) at \(z\) with order \(m\). For \(0<\alpha<\infty\), \(f\in H(B)\) is called an \(\alpha\)-Bloch function if \(\|f\|_{\mathcal B^{\alpha}}=\sup_{z\in B}|\bigtriangledown f(z)|(1-|z|^2)^{\alpha}<\infty\). The following notations are used: \(\varphi_a\in Aut(B)\) satisfying \(\varphi_a(0)=a\) and \(\varphi_a(a)=0\), the Möbius invariant Green function is \(G(z,a)=g(\varphi_a(z))\), where \(g(z)=\frac{n+1}{2n}\int_{|z|}^1(1-t^2)^{n-1}t^{-2n+1} dt\), for \(a\in B\), \(0<r<1\), denote \(E(a,r)=\{z\in B: |\varphi_a(z)|<r\}\) and the real Jacobian of \(\varphi_a(z)\) is \(J_R(\varphi_a(z))=\big(\frac{1-|a|^2}{|1-\langle z,a\rangle|^2}\big)^{n+1}\). The main result in the paper under review gives \(\alpha\)-Bloch functions some integral characterizations in terms of higher radial derivative as follows: Theorem: Suppose \(0<\alpha<\infty\), \(0<r<1\), \(0<p<\infty\), \(1<s<\frac{n}{n-1}\), \(\frac{n}{n+1}<\beta<\infty\), \(1<q<\infty\), \(m\in\mathbb N\), \(dv_{\beta}(z)=(1-|z|^2)^{-(n+1)(1-\beta)}dv(z)\) and \(f(z)\in H(B)\), then the following are equivalent (i) \(f(z)\in\mathcal B^{\alpha}(B)\), (ii) \(\sup_{a\in B}\frac{1}{v(E(a,r))}\int_{E(a,r)}|R^mf(z)|^p(1-|z|^2)^{p(\alpha+m-1)} dv(z)<\infty\), (iii) \(\sup_{a\in B}\int_B|R^mf(z)|^p(1-|z|^2)^{p(\alpha+m-1)}(1-|\varphi_a(z)|^2)^{nq} d\lambda(z)<\infty\), (iv) \(\sup_{a\in B}\int_B|R^mf(z)|^p(1-|z|^2)^{p(\alpha+m-1)}G^s(z,a) d\lambda(z)<\infty\), (v) \(\sup_{a\in B}\int_B|R^mf(z)|^p(1-|z|^2)^{p(\alpha+m-1)}(J_R\varphi_a(z))^{\beta} dv_{\beta}(z)<\infty\). The corresponding results are proved for little \(\alpha\)-Bloch functions.
scientific article; zbMATH DE number 1866501