Hilbert transform of logbfb
Javad Mashreghi · Proceedings of the American Mathematical Society · 2002
There are two general ways to evaluate the Hilbert transform of a function of real variable u(x). We can extend u to a harmonic function in the upper half plane by the Poisson integral formula. Non-tangential limit of its harmonic conjugate exists almost everywhere and is defined to be the Hilbert transform of u(x). There is also a singular integral formula for the Hilbert transform of u(x). It is fairly difficult to directly evaluate the Hilbert transform of u(x). In this paper we give an explicit formula for the Hilbert transform of log |f|, where f is a function in the Cartwright class.