L p Estimates on the Bilinear Hilbert Transform for 2 < p < &#8734

Abstract

For the bilinear Hilbert transform given by:

The Hörmander multiplier theorem for multilinear operators

“…We now introduce a nonnegative smooth bump ψ supported in the interval [(10m) −1 , 2] and equal to 1 on the interval [(5m) −1 , 12 10 ], and we decompose the identity on (R n ) m \ {0} as follows…”

“…In this article, we provide a version of the Hörmander multiplier theorem in the case of multilinear operators. The study of such operators originated in the work of Coifman and Meyer [2], [3], [4] and was later revived by the groundbreaking work of Lacey and Thiele's on the bilinear Hilbert transform [12], [13]. The multilinear Fourier multiplier operator T σ associated with a symbol σ is defined by…”

Minimal regularity conditions for the end-point estimate of bilinear Calderón-Zygmund operators

“…Under regularity assumptions analogous to (1.1) or (1.2), the theory has been developed through works of Coifman-Meyer [2], [3], [4], Christ-Journé [6], Kenig-Stein [15] and Grafakos-Torres [10], [11]. Some of the most recent work in the subject was motivated in part by the results of Lacey-Thiele [16], [17] on the bilinear Hilbert transform and a search for the optimal range of exponents where boundedness in Lebesgue spaces can be obtained. Unlike the case of the bilinear Hilbert transform, a more singular operator not covered by the Calderón-Zygmund theory, the boundeness of Calderón-Zygmund operators on the full range of exponents is known.…”

Translation-Invariant Bilinear Operators with Positive Kernels

“…The study of bilinear operators within the context of harmonic analysis was initiated by Coifman and Meyer [2,3] in the late seventies but recent attention in the subject was rekindled by the breakthrough work of Lacey and Thiele [9,10] on the bilinear Hilbert transform. The behavior of this operator is still not understood on spaces near L 1 × L 1 .…”