Hilbert transforms and maximal functions along variable flat curves

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.

Maximal cluster sets of L-analytic functions along arbitrary curves

Let Ω be a domain in the N -dimensional real space, L be an elliptic differential operator, and (Tn) be a sequence whose members belong to a certain class of operators defined on the space of L-analytic functions on Ω. It is proved in this paper the existence of a dense linear manifold of L-analytic functions all of whose nonzero members have maximal cluster sets under the action of every Tn al...

On the Boundedness of the Bilinear Hilbert Transform along “non-flat” Smooth Curves

We are proving L(R) × L(R) → L(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.

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]).

Surface Measures - Maximal Functions and Fourier Transforms