Sharp analytic version of Fefferman's inequality

Abstract

Let $T$ be a unit circle. Assume further that f is an element of the Hardy space $H^1\left(T\right)$ and g belongs to the analytic BMO space on $T$. The paper contains the identification of the optimal universal constant C in the estimate$\left|\frac{1}{2\pi }\underset{T}{\int }\overline{f\left(\zeta \right)}g\right(\zeta \left)\text{d}\zeta \right|\le C{\Vert f\Vert }_{H^1\left(T\right)}{\Vert g\Vert }_{BMO\left(T\right)}.$ Actually, the inequality is studied in the stronger form, involving the Littlewood-Paley function on the left and the sharp maximal function of g on the right. The proof rests on the construction of an appropriate plurisuperharmonic function on a parabolic domain and the application of probabilistic techniques.