On the Lp norms of the Hilbert transform of a characteristic function

The Lp norm of the Hilbert transform of the characteristic function of a set is invariant with respect to the structure of the set: it can be written as an explicit function of its Lebesgue measure and of the exponent p. We show that more is true: there is a fixed ratio, only dependent on p, between the Lp norms of such a Hilbert transform computed on the given set and on the whole line.