On the boundedness of the bilinear Hilbert transform along “non-flat” smooth curves

Abstract

We are proving L 2 (R) × L 2 (R) → L 1 (R) bounds for the bilinear Hilbert transform HΓ along curves Γ = (t, −γ(t)) with γ being a smooth "non-flat" curve near zero and infinity.

Bilinear Hilbert Transforms Associated with Plane Curves

“…In his proof, he combined results and tools from both time-frequency analysis and oscillatory integral theory and used ingeniously a uniformity concept (the so-called σ-uniformity; see [11,Section 6]). Lie [13] improved Li's results both qualitatively, by extending monomials to more general curves (certain "slow-varying" curves with extra curvature assumptions), and quantitatively, by improving the estimates. Instead of using Li's method of σ-uniformity, Lie used a Gabor frame decomposition to discretize certain operators in a smart way and then worked with the discretized operators which have variables separated on the frequency side and preserve certain main characteristics (see the appendix of [13] for a detailed comparison between their methods).…”

“…We use the methods that are used in Li [11] (with modifications). Although we use different methods from Lie [13], the curves we consider are similar and the estimates we get are the same.…”

“…Our criterion, motivated by results in Lie [13] and Nagel, Vance, Wainger, and Weinberg [15], mainly asks one to check whether certain bounds of various expressions involving derivatives of a quotient are satisfied. In [15] a simple necessary and sufficient condition is provided (among other results) for the L 2 -boundedness of the Hilbert transform along the curve Γ with an odd function γ and it is expressed in terms of an auxiliary function h(t) = tγ ′ (t) − γ(t).…”

A polynomial Roth theorem on the real line

“…If P (t) = 2t, this is the classical bilinear Hilbert transform, which is the subject of Lacey and Thiele's breakthrough papers [LT97], [LT98] and has since been studied extensively. For certain nonlinear P the operator in (1.2) has recently been studied in [Li13], [Lie11], [LX16], [GX16], [Lie15]. We invite the reader to consult these papers to learn about the development of this subject.…”

On the bilinear Hilbert transform along two polynomials

“…For example, when P and Q are distinct linear polynomials, B P,Q is in fact the famous bilinear Hilbert transform, whose boundedness was proved by Lacey and Thiele in a pair of breakthrough papers ( [11,12]). Xiaochun Li [15] first studied the case P (t) = t, Q(t) = t d , d ∈ N, and showed that B P,Q is bounded from L 2 × L 2 to L 1 (see also [9,16] for some generalizations). Together with Lechao Xiao, Li later ( [17]) obtained the L p estimates in full range when P (t) = t and Q is any polynomial without linear term.…”