Unimodular bilinear Fourier multipliers on Lp spaces

Abstract

In this paper we investigate the boundedness properties of bilinear multiplier operators associated with unimodular functions of the form m(ξ, η) = eiϕ(ξ-η). We prove that if ϕ is a C1(Rn) real-valued non-linear function, then for all exponents p, q, r lying outside the local L2-range and satisfying the Hölder’s condition 1p+1q=1r, the bilinear multiplier norm ‖eiλϕ(ξ-η)‖Mp,q,r(Rn)→∞,λ∈R,|λ|→∞.For exponents in the local L2-range, we give examples of unimodular functions of the form eiϕ(ξ-η), which do not give rise to bilinear multipliers. Further, we also discuss the essential continuity property of bilinear multipliers for exponents outside local L2-range.