FULL RANGE BOUNDEDNESS OF BILINEAR HILBERT TRANSFORM ALONG CERTAIN POLYNOMIALS

Let P and Q be two polynomials without constant term. Assume that the opera- tor B-P,B-Q (f,g)(x) = integral f(x - P(t))g(x - Q(t))dt/t is bounded from L-p1 x L-p2 into L-r, p(1),p(2) is an element of (1,infinity), 1/p(1) + 1/p(2) = 1/r. It is proved that if P'(t)> 0 for all t not equal 0, then r >= d/d+1. Here d is the correlation degree of P and Q which is defined as the largest multiplicity of non-zero real mots of P' - Q'.