Uniform Sobolev inequalities for second order non-elliptic differential operators

TL;DR

This paper establishes uniform Sobolev inequalities for second order non-elliptic differential operators, identifying precise conditions on exponents and extending unique continuation results for such operators.

Contribution

It proves new uniform Sobolev inequalities for non-elliptic operators with explicit exponent conditions and endpoint estimates, broadening the class of functions for unique continuation.

Findings

Sobolev inequalities hold under specific exponent relations

Endpoint estimates are established for critical cases

Results extend unique continuation properties for non-elliptic operators

Abstract

We study uniform Sobolev inequalities for the second order differential operators $P(D)$ of non-elliptic type. For $d\ge3$ we prove that the Sobolev type estimate $\|u\|_{L^q(\mathbb{R}^d)}\le C \|P(D)u\|_{L^p(\mathbb{R}^d)}$ 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.