On the boundedness of a family of oscillatory singular integrals

Let Omega is an element of H-1(Sn-1) with mean value zero, P and Q be polynomials in n variables with real coefficients and Q(0) = 0. We prove that vertical bar p.v. integral(Rn) e(i(P(x)+1/Q(x))) Omega(x/vertical bar x vertical bar)/vertical bar x vertical bar(n) dx <= A parallel to Omega parallel to(H1(Sn-1)) where A may depend on n, deg(P) and deg(Q), but not otherwise on the coefficients of P and Q. The above result answers an open question posed in [13]. Additional boundedness results of similar nature are also obtained.