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Weak Bloch property and weight estimates for elliptic operators. Appendix: On the equality between weak and strong extensions (by M. A. Shubin and J. Sjöstrand). (English) Zbl 0716.35052
Sémin. Équations Dériv. Partielles 1989-1990, No.5, 30 p. (1990).

The paper deals with the following problem.

Let A denote a differential operator on a non-compact Riemannian manifold M, and assume that A defines an unbounded operator in the Hilbert space ${L}^{2}\left(M\right)$. Assume that for some complex $\lambda$ we know a solution u of the equation $Au=\lambda u$, when can we conclude that $\lambda$ is in the spectrum $\sigma$ (A) of the operator A in ${L}^{2}\left(M\right)?$ After an introduction with the main definitions and some preliminary lemmata, one considers weight estimates and decay of Green functions; then the third section deals with uniform properly supported pseudo-differential operators and structural inverse operators; and the last section contains some results related to the spectral properties of uniformly elliptic operators on manifolds of subexponential growth.

Reviewer: G.Jumarie
##### MSC:
 35P05 General topics in linear spectral theory of PDE 35J15 Second order elliptic equations, general 35S05 General theory of pseudodifferential operators