EIGENFUNCTION EXPANSIONS AND SPACETIME ESTIMATES FOR GENERATORS IN DIVERGENCE-FORM

Abstract

Let [Formula: see text] be a formally self-adjoint (elliptic) operator in L2 (ℝn), n ≥ 2. The real coefficients aj, k(x) = ak, j(x) are assumed to be bounded and to coincide with -Δ outside of a ball. The paper deals with two topics: (i) An eigenfunction expansion theorem, proving in particular that H is unitarily equivalent to -Δ, and (ii) Global spacetime estimates for the associated inhomogeneous wave equation, proved under suitable ("nontrapping") additional assumptions on the coefficients. The main tool used here is a Limiting Absorption Principle (LAP) in the framework of weighted Sobolev spaces, which holds also at the threshold.