Investigations on the behaviour of the Hardy-Littlewood maximum operator (Q1977650)

By a clever construction the author proves the following remarkable result: Let \(E\subset [-\pi,\pi)\) be a set of Lebesgue measure zero. There exists a function \(f\) of vanishing mean oscillation such that \[ \lim_{r \to 1}{1\over 2\pi} \int^\pi_{-\pi} f(\tau){1-r^2 \over 1-2r\cos(t-\tau)+ r^2}d\tau =\infty \] and \[ \lim_{h\to 0}{1\over 2\pi} \int^{t+h}_{t-h}f (\tau) d\tau= \infty, \] for any \(t\in E\).