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 ‖u‖_L^q(R^d)≤C‖P(D)u‖_L^p(R^d) 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) d2+2d−4<p<2 (d−1) d. We also obtain restricted weak type endpoint estimates for the critical (p,q)=2. As a consequence, the result extends the class of functions for which the unique continuation for the inequality |P(D)u|≤|Vu| holds.