Timeline for Pseudo-differential operators which are independent of lower order perturbations

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May 11, 2012 at 10:36	comment	added	Bazin		You name a property on the principal symbol, it becomes a geometric property because of the invariance of that principal symbol. That was my initial criticism of the formulation of the problem. Of course, I understand that the question is in fact what operator-theoretic property (e.g. estimate, propagation,...) is inherited from a property on the sole principal symbol. For your question on the iterated brackets, you can for instance assume $H_a^{2k+1}(b)>0$ at $a=b=H_a^j(b)=0$ for $1\le j\le 2k$ and add a geometric condition, not-so-easy to formulate.
May 10, 2012 at 20:43	comment	added	Deane Yang		The original question still makes sense for real-valued symbols. For systems or complex-valued symbols, it's a little more complicated. In particular, for your case, where the symbols is complex-valued, what happens if the Poisson brackets are not sufficiently non-degenerate?
May 10, 2012 at 20:28	history	edited	Bazin	CC BY-SA 3.0	added 1 characters in body; added 39 characters in body
May 10, 2012 at 20:00	history	answered	Bazin	CC BY-SA 3.0