Investigations on the truncated Hilbert transform of BMO functions and VMO functions (Q1125266)

For \(\varepsilon\in (0,\pi]\) let the truncated Hilbert transform \(H_\varepsilon\) of a function \(f\) on \([-\pi,\pi)\) be defined by \[ (H_\varepsilon f)(t)={1\over 2\pi} \int_{\varepsilon\leq |\tau|\leq\pi} {f(t+ \tau)\over \tan(\tau/2)} d\tau \] and let \[ v(r,t)= {1\over 2\pi} \int^\pi_{-\pi} f(\tau) {r\sin(t- \tau)\over 1- 2r\cos(t- \tau)+ r^2} d\tau \] be the conjugated potential function for \(f\in L^1[-\pi,\pi)\). The author proves that for all functions \(f\) of vanishing mean oscillation (VMO) the equation \[ \lim_{\varepsilon\to 0} \Biggl(\max_{t\in[-\pi, \pi)} |(H_\varepsilon f)(t)- v(1- \varepsilon, t)|\Biggr)= 0 \] is valid. This enables him to give a new characterization of VMO functions.