Sharp 𝐿^𝑝 → 𝐿^𝑞,∞ estimates for the Hilbert transform

Abstract

For any 1 < 𝑞 < 𝑝 < ∞, we identify the best constant 𝐾_𝑝,𝑞 with the following property. If ℋ is the Hilbert transform on the unit circle 𝕋 and 𝐴 ⊂ 𝕋 is an arbitrary measurable set, then ∫_𝐴 | ℋ𝑓 | d𝑚 ≤ 𝐾_𝑝,𝑞 ‖ 𝑓 ‖_{𝐿^𝑝(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.