zbMATH — the first resource for mathematics

On analytic and meromorphic functions and spaces of \(Q_K\)-type. (English) Zbl 1048.30017
Let \(K\) be a right-continuous nonnegative nondecreasing function on \([0,\infty).\) The authors define \(Q_K\) to be the set of functions \(f\) analytic in the unit disk \(\Delta\) for which \[ | | f| | _K^2 :=\sup \int_{\Delta} K(g(z,a)) | f'(z)| ^2\,dxdy < \infty, \] where \(g\) is the Green function of \(\Delta\) and the sup is over \(a\in \Delta.\) The set \(Q_K^{\#}\) is defined to be the set of meromorphic functions in \(\Delta\) for which the supremum above is finite when \(f'\) is replaced by the spherical derivative \(f^{\#}\) of \(f.\) The sets \(Q_K\) and \(Q_K^{\#}\) are invariant under Möbius transformations mapping \(\Delta\) onto itself, and the \(Q_K\) are Banach spaces. For \(K(t) = t^p, 0<p<\infty,\) these sets, denoted by \(Q_p\) and \(Q_p^{\#}\), have been the subject of a good deal of recent work. It is known, for example, that \(Q_1\) coincides with the space \(BMOA\) of analytic functions of bounded mean oscillation, and that for \(p>1\) \(Q_p\) coincides with the Bloch space. Moreover, \(Q_p^{\#}\) coincides with the set of normal functions when \(1<p<\infty.\) In the paper under review the authors develop some theory for general \(K.\) For example: If \(K_1, K_2\) satisfy the conditions for \(K,\) if \(K_1(r) = o(K_2(r))\) as \(r\rightarrow 0\) and if a certain integral involving \(K_2\) diverges, then \(Q_{K_2}\) is properly contained in \(Q_{K_1}.\)

30D45 Normal functions of one complex variable, normal families
30D50 Blaschke products, etc. (MSC2000)
Full Text: Link