Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. (English) Zbl 0644.35012

The main goal of this paper is to prove “Sobolev inequalities” for constant coefficient second order differential operators with bounds which do not depend on the lower order terms. More precisely, let Q(\(\xi)\) denote a non singular real quadratic form on \({\mathbb{R}}^ n\), \(n\geq 3\), which, for some \(2\leq j\leq n\), is given by \[ Q(\xi) = =\xi^ 2_ 1-...-\xi^ 2_ j\quad +\quad \xi^ 2_{j+1}+...+\xi^ 2_ n. \] Then, if dual exponents p and p’ enjoy the relationship \(1/p- 1/p'=2/n\), there is an absolute constant C such that whenever P(D) is a constant coefficient operator with complex coefficients and principal part Q(D) one has: \[ \| u\|_{L^{p'}({\mathbb{R}}^ n)}\leq C\| P(D)u\|_{L^ p({\mathbb{R}}^ n)},\quad u\in H^{2,P}({\mathbb{R}}^ n). \] The main motivation behind proving these uniform Sobolev inequalities is that they imply certain local or global unique continuation theorems for operators P(D) as above. More specifically, they imply certain Carleman inequalities which in turn give unique continuation results for solutions to Schrödinger equations of the form \(P(D)u+Vu=0\), when certain conditions are placed on the solution u and the potential V.
Reviewer: P.Bolley


35B45 A priori estimates in context of PDEs
35G05 Linear higher-order PDEs
35B60 Continuation and prolongation of solutions to PDEs
35E20 General theory of PDEs and systems of PDEs with constant coefficients
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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