Analytic properties of elliptic and conditionally elliptic operators.

Abstract:

In this note we give a short proof of a theorem of Kotake and Narasimhan to the effect that if $A$ is a strongly elliptic operator of order $2m$ with analytic coefficients and $||{A^j}u|| \leqq {C^{j + 1}}(2mj)!$, where $||\;||$ is some suitable norm, then $u$ is analytic. (Actually Kotake and Narasimhan prove the theorem when $A$ is elliptic, but the trick we use here requires some specialization.) This is applied to derive a short proof of a theorem proved by Gårding and Malgrange, in the constant coefficients case, concerning conditionally elliptic operators.