Semi-classical Estimates for Non-selfadjoint Operators

Abstract

This is a survey paper on the topic of proving or disproving a priori $L^2$ estimates for non-selfadjoint operators. Our framework will be limited to the case of scalar semi-classical pseudodifferential operators of principal type. We start with recalling the simple conditions following from the sign of the first bracket of the real and imaginary part of the principal symbol. Then we introduce the geometric condition $(\psi)$ and show the necessity of that condition for obtaining a weak $L^2$ estimate. Considering that condition satisfied, we investigate the finite-type case, where one iterated bracket of the real and imaginary part does not vanish, a model of subelliptic operators. The last section is devoted partly to rather recent results, although we begin with a version of the 1973 theorem of R. Beals and C. Fefferman on solvability with loss of one derivative under condition $P$; next, we present a 1994 counterexample by N. L. establishing that $(\psi)$ does not ensure an estimate with loss of one derivative. Finally, we show that condition $(\psi)$ implies an estimate with loss of 3/2 derivatives, following the recent papers by N. Dencker and N. L. Our goal is to provide a general overview of the subject and of the methods; we do not enter in the details of the proofs, although we provide some key elements of the arguments, in particular in the last section.