Hilbert transforms and maximal functions along variable flat curves

Abstract

Abstract. We study certain Hilbert transforms and maximal functions along variable flat curves in the plane. We obtain their L 2 (R 2 ) boundedness by considering the oscillatory singular integrals which arise from an application of a partial Fourier transform.

The non-resonant bilinear Hilbert--Carleson operator

“…• incorporates both approaches (I) and (II) above, and • requires modulation invariance techniques for both components in decomposition (5). These features reflect the hybrid nature of our operator T := BC a which is designed as a mixture between the bilinear Hilbert transform and a Carleson-type operator:…”

“…The next step was the study of (34) for smooth dependence of γ on x, y. Thus in [16] and [5] the authors prove the expected L p range for the situation γ(x, y, t) = P (x)γ(t), where P is a polynomial and γ is smooth and obeys some suitable nonvanishing curvature condition. In a different direction, this time analyzing the behavior of H Γ under the assumption that γ(x, y, t) obeys (x, y, t)-smoothness and non-zero curvature in t hypotheses, we have: in the nilpotent setting the work in [18], and in the context of singular Radon transforms (and their maximal analogues) i) along differentiable submanifolds the work in [22], or ii) along variable curves in a diffeomorphism invariant setting, the work in [92].…”

“…From the above we then deduce that h1 Aj , φ2u,p am term, applying (301) and a change of variable, we claim that 152 for a large fraction of the possible values of u ∼ 2 am 151 Or some smooth variant of it which is still equal to 1 on[1,2].152 Recall our earlier convention e iy = e 2πiy .Indeed, for this we first notice that while u runs through the set Z∩[], the expression 2 − am 2 +1 u + 1 covers the set 2 − am 2 +1 Z + 1 ∩[3,5]. Thus, for m ∈ N large enough, (say am ≥ 10), there exists a value of u = u 0 in the range u 0 ∼ 2am 2 such that dist(2 − am 2 +1 u 0 + 1, Z) ≥ 1 Then an easy argument shows that for u ∈ Z ∩ [one still has dist(2 − am 2 +1 u + 1, Z) ≥ 1 4 which implies the lower bound in (304) Hence, for u belonging to Z ∩ [∩ [u 0 − 1 100 one can now verify the validity of the statement (I) ≥ 10 −10 2 − 3am For the second part (II) we are using (302) for, say, a choice of L ≥ 100.…”

“…Finally, for simplicity we will appeal to a notational abuse and write e iA as a shortcut for the expression e 2πiA , where here A is always assumed to be real. 5 For a discussion on the necessity of imposing the restriction a / ∈ {1, 2} please see the items 1) and 2) in the open question Section 1.4 below. 6 Due to space limitation concerns we make no attempt to provide a proof of this result in our present paper.…”

$L^2$ Boundedness of Hilbert Transforms along Variable Flat Curves

“…where P : R → R is a real polynomial. Bennett in [3] pointed out that the prime interest in the general curve study for the operator above is to include some γ which vanish to infinite order at the origin. One will see that the curve (2.1) is such kind of curve and satisfies all of conditions in Theorem 1.1.…”