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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:

$Re \langle a(x, D) u, u \rangle \geq - c \|u\|_{L^2}^2$

?

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I used to know the answer to this. You might want to take a look at the appendix of the Duke Math Journal paper by Bryant, Griffiths, and me on isometric embeddings. I believe the discussion there can be adapted to answer your question. But I also believe that there are discussions elsewhere, for example by Michael Taylor, about pseudodifferential operators and nonlinear PDE, where this is addressed directly. –  Deane Yang May 13 '11 at 1:37
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2 Answers

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

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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 $a^w+C\ge 0$.

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