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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:

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

?

edited May 13 2011 at 10:12

Willie Wong
6,987●20●43

asked May 13 2011 at 0:01

Necro
21●1

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 2011 at 1:37

Piero D'Ancona · Answer 1 · 2011-05-13 15:44:56Z UTC

2

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.

answered May 13 2011 at 15:44

Piero D'Ancona
3,657●1●6●18

Bazin · Answer 2 · 2012-05-09 12:46:32Z UTC

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 $a^w+C\ge 0$.

Seminorms in sharp Garding’s inequality

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