Uniform Sobolev inequalities for second order non-elliptic differential operators

We study uniform Sobolev inequalities for the second order differential operators P(D) of non-elliptic type. For d >= 3 we prove that the Sobolev type estimate parallel to u parallel to(Lq(Rd)) <= C parallel to P(D)u parallel to(Lp(Rd)) holds with C independent of the first order and the constant terms of P(D) if and only if 1/p - 1/q = 2/d and 2d(d-1)/d(2)+2d-4 < p < 2(d-1)/d. We also obtain restricted weak type endpoint estimates for the critical (p, q) = (2(d-1)/d, 2d(d-1)/(d-2)(2)), (2d(d-1)/d(2)+2d-4, 2(d-1)/(d-2)). As a consequence, the result extends the class of functions for which the unique continuation for the inequality vertical bar P(D)u vertical bar <= vertical bar Vu vertical bar holds. (C) 2016 Elsevier Inc. All rights reserved.