On Carleman Estimates with Two Large Parameters

A Carleman estimate for a differential operator P is a weighted energy estimate with a weight of exponential form exp(tau phi) that involves a large parameter, tau > 0. The function phi and the operator P need to fulfill some sub-ellipticity properties that can be achieved by, for instance, choosing phi = 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-Hormander 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.