The higher order derivatives of \(Q_{K}\) type spaces
Let \(g(a,z)=\log(1/| \gamma_a(z)| )\) be the Green function on the unit disk \(\Delta\) in the complex plane \(\mathbb{C}\), where \(\gamma_a(z)=(a-z)/(1-{\overline a}z)\) is a Möbius transformation of \(\Delta\). Let \(K:[0,\infty)\to[0,\infty)\) be a right-continuous and nondecreasing function. For \(0<p<\infty\), \(-2<q<\infty\), the space \(Q_K(p,q)\) consists of all analytic functions \(f\) defined in \(\Delta\) satisfying \[ \sup_{a\in\Delta}\int_\Delta| f'(z)| ^{p}(1-| z| ^{2})^{q}\bigl(K(g(z,a)\bigr)\,dA(z)<\infty, \] where \(dA\) is the area measure on \(\Delta\) normalized so that \(A(\Delta)=1\). In the paper under review, the following assumptions are made: \(K\) is differentiable, \(K(2t)\approx K(t)\), and \[ \int_0^{1}(1-r^{2})^{q}K\left(\log\tfrac{1}{r}\right)r\,dr<\infty. \] Further, for the auxiliary function \(\varphi_K(s)=\sup_{0\leq t\leq1} K(st)/K(t)\), \(0<s<\infty\), it is assumed that (1) \(\int_0^{1}\varphi_K(s)\,\tfrac{ds}{s}<\infty\) and \[ \sup_{a\in\Delta}\int_\Delta\frac{(1-| z| ^{2})^{p-2}}{| 1-{\overline a}z| ^{p}}K\left(\log\tfrac{1}{| z| }\right)\,dA(z)<\infty.\tag{2} \] One of the main theorems is the following: Let \(K\) satisfy (2) and let \(0<p<\infty\), \(-1<q<\infty\). If \(n\) is a positive integer, then \(f\in Q_K(p,q)\) if and only if \[ \sup_{a\in\Delta}\int_\Delta| f^{(n)}(z)| ^{p}(1-| z| ^{2})^{np-p+q}K\bigl(g(z,a)\bigr)\,dA(z)<\infty. \] This result should be compared with \textit{J.\,Rättyä}'s dissertation [Ann.\ Acad.\ Sci.\ Fenn.\ Math.\ Diss.\ No.\ 124, 1--73 (2001; Zbl 0984.30019)]. Another main theorem, when \(K\) satisfies (1) and (2), characterizes \(f\) to belong to \(Q_k(p,q)\) if and only if \(| f^{n}(z)| ^{p}\bigl(1-| z| ^{2}\bigr)^{np-p+q}\,dA(z)\) is a \(K\)-Carleson measure.
scientific article; zbMATH DE number 5161768