On a classification of plane domains for BMOA
The space of analytic functions of bounded mean oscillation has been studied by many authors over the past few years. Here, using an idea due to the reviewer [Can. J. Math. 33, 1255-1260 (1981; Zbl 0442.30030)], the author studies BMOA(G) for arbitrary domains G. Specifically, an analytic function f(z) is said to belong to BMOA(G) if \[ \sup_{a\in G}\iint_{G}| f'(z)|^ 2g(z,a)dxdy<\infty, \] where \(g(\cdot,a)\) is the Green's function for G with pole at a. If u(z) is a subharmonic function on G then û(z) is defined to be the least harmonic majorant of u(z) in G. The author proves that f(z) belongs to BMOA(G) if and only if \(\sup_{a\in G}\hat u_ a(a)<\infty\) where for \(u_ a(z)\) one can take either \(| f(z)-f(a)|^ p\) for some \(p>0\) or \(\log^+| f(z)-f(a)|.\) In the second half of the paper it is shown that the classification results \(\cup_{p>0}O_ p\subset O_{BMOA}\subset O_{AB},\) originally proved by the reviewer (op. cit.) are all strict containments by constructing planar regions which satisfy the conditions needed to prove the stated conclusions.
scientific article; zbMATH DE number 3863551