Uniform Sobolev inequalities for second order non-elliptic differential operators

Abstract

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‖Lq(Rd)≤C‖P(D)u‖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)d2+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)d2+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 |P(D)u|≤|Vu| holds.