IUMJ

Title: On Carleman estimates with two large parameters

Authors: Jerome Le rousseau

Issue: Volume 64 (2015), Issue 1, 55-113

Abstract:

A Carleman estimate for a differential operator $P$ is a weighted energy estimate with a weight of exponential form $\exp(\tau\varphi)$ that involves a large parameter, $\tau>0$. The function $\varphi$ and the operator $\mathrm{P}$ need to fulfill some sub-ellipticity properties that can be achieved by, for instance, choosing $\varphi=\exp(\alpha\psi)$, involving a second large parameter, $\alpha>0$, with $\psi$ satisfying some geometrical conditions. The purpose of this article is to give the framework to keep explicit the dependency upon the two large parameters in the resulting Carleman estimates. The analysis is based on the introduction of a proper Weyl-H\"ormander calculus for pseudo-differential operators. Carleman estimates of various strengths are considered: specifically, estimates under the (strong) pseudo-convexity condition, and estimates under the simple-characteristics property. In each case, the associated geometrical conditions for the function $\psi$ is proven necessary and sufficient. In addition, some optimality results with respect to the power of the large parameters are proven.