Unimodular bilinear Fourier multipliers on Lp spaces

In this paper we investigate the boundedness properties of bilinear multiplier operators associated with unimodular functions of the form m = eif(.-.). We prove that if f 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 Holder's condition 1 p + 1 q = 1 r, the bilinear multiplier norm Mp, q,r (Rn).8,.. R, |.|. 8. For exponents in the local L2-range, we give examples of unimodular functions of the form eif, which do not give rise to bilinear multipliers. Further, we also discuss the essential continuity property of bilinear multipliers for exponents outside local L-2-range.