A sharp one-sided bound for the Hilbert transform

Abstract:

Let $\mathcal {H}^{\mathbb {T}}$, $\mathcal {H}^{\mathbb {R}}$ denote the Hilbert transforms on the circle and real line, respectively. The paper contains the proofs of the sharp estimates \[ |\{\zeta \in \mathbb {T}:\mathcal {H}^{\mathbb {T}}f(\zeta )\geq 1\}|\leq 2\pi ||f||_1, \qquad f\in L^1(\mathbb {T}),\] and \[ |\{x\in \mathbb {R}:\mathcal {H}^{\mathbb {R}}f(x)\geq 1\}|\leq ||f||_1, \quad \qquad f\in L^1(\mathbb {R}).\] A related estimate for orthogonal martingales is also established.