On the Bekollé-Bonami condition
Let \(\mathfrak A\) be the system of the Carleson ``squares'' \[ Q=\{ re^{i\theta} \in {\mathbb D}: 1-| Q| \leq r\leq 1, | \theta -\theta_0| \leq | Q| /2\} \] in the unit disc \(\mathbb D\). The classes \({\mathcal B}_{p,q},\) \(0<p,q<\infty\), consist of non-negative functions \(w\) on \(\mathbb D\) such that \[ \sup_{Q\in {\mathfrak A}}\langle w^p\rangle ^{1/p}_Q \langle w^{-q}\rangle^{1/q}_Q<\infty, \] where \[ \langle w\rangle_E=\frac{1}{m_2(E)}\int_Ew(z)dm_2(z) \] and \(dm_2\) is Lebesgue area measure. The classes \( B_p={\mathcal B}_{1,1/(p-1)}\), \(1<p<\infty\), where introduced by \textit{D.~Bekollé} and \textit{A.~Bonami} [C. R. Acad. Sci. Paris 286, 775--778 (1978; Zbl 0398.30006)]. They proved that the Bergman projection operator \(T\) acts continuously on \(L^p({\mathbb D}, w\,dm_2)\) if and only if \(w\in B_p\). It is known that \[ {\mathcal B}_{p,q}\not \subset \bigcup_ {\varepsilon>0}{\mathcal B}_{p+\varepsilon,q+\varepsilon}. \] In the paper under review, the author considers additional conditions on the weight \(w\in {\mathcal B}_{p,q}\) which imply that \(w\in \bigcup_ {\varepsilon>0}{\mathcal B}_{p+\varepsilon,q+\varepsilon}\). For example, it is shown that, for \(0<p,q<\infty\), \[ {\mathcal A}\cap {\mathcal B}_{p,q}\subset \bigcup_ {\varepsilon>0}{\mathcal B}_{p+\varepsilon,q+\varepsilon}, \] where by \(\mathcal A\) denotes the class of functions \(| f| ^\alpha\), \(\alpha\in \mathbb{R}\), for \(f\) analytic in \(\mathbb D\).
scientific article; zbMATH DE number 2078021