Über die mittlere Oszillation der logarithmischen Funktion. (On the mean oscillation of the logarithmic function) (Q1089126)

We study the mean oscillation of the real logarithm, as well as of the complex logarithm on the unit circle, and we evaluate the usual BMO norm for both functions. It is shown, that the BMO norm of log\(| t|\), \(t\in {\mathbb{R}}\setminus \cup 0\}\), is equal to the mean oscillation of this function on a certain family of intervals, which are unsymmetrical in respect to the origin. In the case of the complex logarithm \(\log (1-e^{it})\), \(0<t<2\pi\), the situation is quite different. The square BMO norm here is equal to the supremum of the square mean oscillation on intervals of vanishing length centered in the singular point \(t=0\). The value of this supremum is equal to \(\sqrt{1+\frac{\pi^ 2}{4}}\).