Sharp $L^{p} \to L^{q,\infty}$ estimates for the Hilbert transform

Abstract

For any $1 < q < p < \infty$, we identify the best constant $K_{p,q}$ with the following property. If $\mathcal{H}$ is the Hilbert transform on the unit circle $\mathbb{T}$ and $A \subset \mathbb{T}$ is an arbitrary measurable set, then $\int_{A} | \mathcal{H}f | \mathrm{d}m \leq K_{p,q} \| f \|_{L^{p}(\mathbb{T},m)} m(A)^{1-1/q}$. The proof rests on the construction of certain special superharmonic functions on the plane, which are of independent interest.