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‖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