On construit par voie géométrique une classe de symboles classiques en dehors d’une sous-variété. La classe d’opérateurs pseudodifférentiels associée contient les paramétrix d’opérateurs tels que ${\sum }_{i=1}^{n-1}{\left(\frac{\partial }{\partial x_i}\right)}^4+{\left(\frac{\partial }{\partial x_n}\right)}^3$ ou ${\left(\frac{\partial }{\partial x_n}\right)}^3+{\sum }_{i=1}^{n-1}{\left(\frac{\partial }{\partial x_i}\right)}^2.$

The aim of this work is to give a Gårding inequality for pseudodifferential operators acting on functions in $L^2\left({ℝ}^n\right)$ supported in a closed regular region $F\subset {ℝ}^n$. A natural idea is to suppose that the symbol is non-negative in $F\times {ℝ}^n$. Assuming this, we show that this result is true for pseudo-differential operators of order one, when $F$ is the half-space, and under a supplementary weak hypothesis of degeneracy of the symbol on the boundary.