International Congress of Mathematicians

726 C. Thieleis negative. If all of them are nonnegative, we say A is of type (pi,... ,p n ) If (1-10)holds. If one of them, say pj, is negative, then we define the dual operator T byA(/i,...,/„) = / T(fi,..., fj^i, f j+ i,..., f n )(x)fj(x) dx.We then say that A is of type (pi,... ,p n ) if\\T(fl,-- -,fj-l,fj+l,.. -,fn)\\ p >. < C|| WfiWpiwhere p'j = Pj/(pj — 1)- Observe 0 < p'j < 1. The following theorem is again dueto [6] (n = 3) and [16]:Theorem 2.2 Let Y and A be as in Theorem 2.1. Then A is of type (pi,... ,p n )if J2j Ì/Pj= 1; °'t most one of the pj is negative, none of the pj is in [0,1], andn — 2dim(F) + r1/p^ + • • • + l/p ir < ^for all 1 < ii < • • • < i r < n and 1 < r < n.A basic example of a modulation invariant form A is when n = 3 and m(£i, £2)is constant on both sides of a line F but not globally constant. With proper choiceof constants this form can be written asA a (/i,/2,/ 3 ) = /B a (fi,f 2 )(x)fi(x)dxwith the bilinear Hilbert transformB a = p.v. / fi(x - t)f 2 (x - at)- dtand a (projective) parameter a determining the direction of the line F. Theorems2.1 and 2.2 in this special case are due to [10] and [11].For the bilinear Hilbert transform nondegeneracy specializes to the conditiona $ {0,1,00}, and the conclusion of both theorems can be summarized to\\B a (fi,f2)\\ P 0M\{-e,e]also satisfy (2.4) provided a is not degenerate.f(x — t)g(x — at)- dtt