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 $d\ge 3$ we prove that the Sobolev type estimate $\Vert u{\Vert }_{L^q({\mathbb{R}}^d)}\le C\Vert P(D)u{\Vert }_{L^p({\mathbb{R}}^d)}$ 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 $\frac{2d(d-1)}{d^2+2d-4}<p<\frac{2(d-1)}{d}$. We also obtain restricted weak type endpoint estimates for the critical $(p,q)=(\frac{2(d-1)}{d},\frac{2d(d-1)}{(d-2{)}^2})$, $(\frac{2d(d-1)}{d^2+2d-4},\frac{2(d-1)}{d-2})$. As a consequence, the result extends the class of functions for which the unique continuation for the inequality $|P(D)u|\le |Vu|$ holds.

23 pages, 1 figure. To appear in Advances in Math

References in corpus (1)

• Fourier transform and regularity of characteristic functions