Characterizations of \(Q_p\) spaces in the unit ball of \(\mathbb C^n\)
By writing \(B\), resp., \({\mathcal A}\) for the Euclidean unit ball of the complex \(n\)-space \(\mathbb C^n\) and the family of its holomorphic automorphisms, a holomorphic function \(f:B\to \mathbb C\) belongs to \(Q_p\) if \(\sup_{\varphi\in{\mathcal A}} \int_B |\nabla f(z)|^2 (1-| z|^2)^2 (1-|\varphi(z)|^2)^{np} \;d\lambda(z) <\infty\) where \(d\lambda(z):= (1-| z|^2)^{-n-1}\) in terms of the normalized Lebesgue measure \(d\nu\) on \(B\). Similarly, by definition, \(f\in Q_{p,0}\) if \(\lim_{| \varphi(0)| \to 1, \varphi\in{\mathcal A}} \int_B |\nabla f(z)|^2 (1-| z|^2)^2 (1-|\varphi(z)|^2)^{np} \;d\lambda(z) =0\). The paper is aimed to give various derivative free and oscillation type characterizations of these relationships. A typical result, for instance Theorem 3, asserts that, in case of \(1-n^{-1} < p\leq 1\), we have \(f\in Q_p\) if and only if \(\sup_{\varphi\in{\mathcal A}} \int_B [\text{MO}(f)(z)]^2 (1-| \varphi(z)|^2)^{np} \;d\lambda(z) < \infty\) where \(\text{MO}(f)(z):= \big[ \int_B | f(w)-f(z)|^2 (1-| z|^2 )^{n+1} | 1 - \langle z,w\rangle^{-2(n+1)} \;d\nu(w) \big]^{1/2}\). To give more insight, the following part of Theorem 5 is also worth to be mentioned: given \(r>0\) and \(1-n^{-1}< p\leq 1\), we have \(f\in Q_p\) if and only if \(\sup_{\varphi\in{\mathcal A}} \int_B \big( \sup_{w\in E(z,r)} | f(z)-f(w)| (1-| \varphi(z)|^2)^{np/4} (1-| \varphi(w)|^2)^{np/4} \big)^2 \;d\lambda(z) < \infty\) where \( E(z,r):= \{ w\in B:\;|\varphi(w)| < r \;\text{for some} \;\varphi\in{\mathcal A} \;\text{with} \;\varphi(0)=z\}\). The corresponding descriptions for \(f\in Q_{p,0}\) are obtained in both cases by replacing the terms \(\sup_{\varphi\in{\mathcal A}}\) and \(<\infty\) with \(\lim_{| \varphi(0)|\to 1 , \varphi\in{\mathcal A}}\) and \ \(=0\), respectively.
scientific article; zbMATH DE number 5626943
scientific article