On the weak type (1,1) inequality for conjugate functions

Abstract

A theorem of Kolmogorov states that there is a positive constant K K such that if f ~ \tilde f is the conjugate function of an integrable real valued function f f on the unit circle then m { | f ~ | ≧ λ } ≦ K | | f | | 1 / λ , λ > 0 m\{ |\tilde f| \geqq \lambda \} \leqq K||f|{|_1}/\lambda ,\lambda > 0 . It is shown that the smallest possible value for K K in this theorem, the so called weak type (1, 1) norm of the conjugate function operator, is ( 1 + 3 − 2 + 5 − 2 + ⋯ ) / ( 1 − 3 − 2 − 5 − 2 − ⋯ ) ≈ 1.347 (1 + {3^{ - 2}} + {5^{ - 2}} + \cdots )/(1 - {3^{ - 2}} - {5^{ - 2}} - \cdots ) \approx 1.347 . This number is also shown to be the weak type (1, 1) norm of the Hilbert transform operator on functions defined on the real line. The proof uses P. Levy’s result that Brownian motion in the plane is conformally invariant.