Hilbert transform along measurable vector fields constant on Lipschitz curves: $L^p$ boundedness

Abstract

We prove the L p (p > 3/2) boundedness of the directional Hilbert transform in the plane relative to measurable vector fields which are constant on suitable Lipschitz curves, extending the L 2 bounds in [15]. Statement of the main resultIn [15] we proved that the Hilbert transforms along measurable vector fields which are constant on a suitable family of Lipschitz curves are bounded in L 2 . The main goal of this paper is to generalize the above L 2 bounds to L p for p other than 2 in the same setting.Theorem 1.1 (Main Theorem). For vector fields v : R 2 → R 2 of the form (1, u(h)) where h : R 2 → R is a Lipschitz function such thatand u : R → R is a measurable function such thatthe associated Hilbert transform, which is defined asis bounded in L p for all p > 3/2.

The non-resonant bilinear Hilbert--Carleson operator

“…Indeed, Bateman -in the single annulus case, [1], and then Bateman and Thiele -for the general, non-localized frequency case, [2], proved that H Γ is L p bounded for p > 3 2 . A few years later, in [46], [47], the same boundedness range was established by Guo for the situation when a is constant along suitable families of Lipschitz curves.…”

“…In particular, it is worth noticing that BC 1 encapsulates both the behavior of the bilinear Hilbert transform and that of the classical Carleson operator. 39 Indeed, if we assume by contradiction that the strategy in our present paper remains true for a = 2, we see that by choosing the linearizing function λ(x) = 2 2m+1 for some fixed large m ∈ N, relation (46) would imply in particular that ´|t| 1 f (x − t)g(x + t)e i2 2m+1 t 2 dt t 2 −mδ f L 2 g L 2 for some absolute constant δ > 0 and any f, g ∈ L 2 (R). However, this latter inequality is trivially false, as can be seen by taking…”

“…where here χ m (•, •) is a smooth function such that χ m (λ, t) = 0 iff |λt a | ∼ 2 am . Thus, our goal -formulated below as Theorem 3.1 -will be to show that each such piece T m obeys an exponentially decaying bound of the form (46) T m (f, g) L r a,p,q 2 −δam f L p g L q ,…”

Hilbert transforms along double variable fractional monomials

“…Proof of Theorem 1.2. From [7] and [9], we know that the single annulus L p (R 2 ) estimate plays a important role in obtaining the L p (R 2 ) boundedness of the Hilbert transform along variable curve. There are many other works about single annulus L p (R 2 ) estimate; see, for example, [1,13].…”

Square functions for bi-Lipschitz maps and directional operators

“…It is a folklore conjecture and discussed by several authors, for example [Ste93], [LL10], [Guo17], that Lipschitz is the critical regularity assumption on a direction field to yield L p boundedness of some associated directional operators. Possibly at the heart of positive results in this direction appears to be a one dimensional Littlewood-Paley diagonalization estimate for bi-Lipschitz maps, which is our first main theorem.…”