Lpestimates for the Hilbert transforms along a one-variable vector field

Lp ESTIMATES FOR THE HILBERT TRANSFORMS ALONG A ONE-VARIABLE VECTOR FIELD

We prove L estimates on the Hilbert transform along a measurable, non-vanishing, one-variable vector field in R. Aside from an L 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 ([2]).

Bilinear Hilbert Transforms along Curves I. the Monomial Case

We establish an L2×L2 to L estimate for the bilinear Hilbert transform along a curve defined by a monomial. Our proof is closely related to multilinear oscillatory integrals.

Double Hilbert Transforms along Polynomial Surfaces in R3

where P(s, t) is a polynomial in s and t with P(0,0)= 0, and ∇P(0,0)= 0. We call H the (local) double Hilbert transform along the surface (s, t,P (s, t)). The operator may be precisely defined for a Schwartz function f by integrating where ≤ |s| ≤ 1 and η ≤ |t | ≤ 1, and then taking the limit as ,η→ 0. The corresponding 1-parameter problem has been extensively studied (see [RS1], [RS2], and [S]...

Uniform Bounds for the Bilinear Hilbert Transforms

It is shown that the bilinear Hilbert transforms Hα,β(f, g)(x) = p.v. ∫ R f(x− αt)g(x− βt) dt t map Lp1(R) × Lp2(R) → Lp(R) uniformly in the real parameters α, β when 2 < p1, p2 < ∞ and 1 < p = p1p2 p1+p2 < 2. Combining this result with the main result in [9], we deduce that the operators H1,α map L2(R)×L∞(R) → L2(R) uniformly in the real parameter α ∈ [0, 1]. This completes a program initiated...

Two-dimensional Hilbert Transforms'