Lᵖ estimates for the Hilbert transforms along a one-variable vector field

الأصل · EN

Stein conjectured that the Hilbert transform in the direction of a vector field is bounded on, say, L² whenever v is Lipschitz. We establish a wide range of Lᵖ estimates for this operator when v is a measurable, non-vanishing, one-variable vector field in ². Aside from an L² estimate following from a simple trick with Carleson's theorem, these estimates were unknown previously. This paper is closely related to a recent paper of the first author (B2).