When working with symbols of limited regularity, what is exactly the number of seminorms of the symbol $Re (a) \geq 0$ that one needs in the sharp Garding's inequality:
$Re \langle a(x, D) u, u \rangle \geq  c \u\_{L^2}^2$
?
When working with symbols of limited regularity, what is exactly the number of seminorms of the symbol $Re (a) \geq 0$ that one needs in the sharp Garding's inequality: $Re \langle a(x, D) u, u \rangle \geq  c \u\_{L^2}^2$ ? 


Probably the sharpest result on this is due to Tataru, he proved sharp Garding for $C^2$ symbols (and even FeffermanPhong for $C^4$ symbols). The paper is available here. 


The paper by A. Boulkhemair (MR2427955 in mathscinet) gives the sharpest result: (1) If $0\le a$ satisfies the estimates of $S^1_{1,0}$ for $\frac n2+2+\epsilon$ derivatives, then $a^w+C\ge 0$. (2) If $0\le a$ satisfies the estimates of $S^2_{1,0}$ for $\frac n2+4+\epsilon$ derivatives, then $a^w+C\ge 0$. 

