7
$\begingroup$

In the area of pseudo-differential operators, we know that for elliptic type or real principal type operators reductions are independent of lower order terms. For example, if $P$ is zeroth order real principal type operator with principal symbol $q(x,\xi),$ then for a lower order perturbation $R$ we can always find a zeroth order elliptic operator $E$ such that $E^{-1}(P+R)E=q(x,D).$

The reason for this independence of lower terms in elliptic type operator is because the principal term can be inverted and in real principal type operator is there exists a non-degenerate Hamiltonian flow.

Are there any other class of pseudo-differential operators which have this property? If not, is there any work in microlocal analysis which tells us that elliptic and real principal type operators form the largest such class?

Note: As stated in the question, my interest is in 'reductions of operators', i.e., transforming to simpler forms. I am not interested in regularity or solvability problems of pseudo-differential operators. I am aware of examples(not stated in the question) of operators for which the solvability issues are independent of lower order terms. For instance, http://annals.math.princeton.edu/wp-content/uploads/annals-v163-n2-p02.pdf

Improve this question
$\endgroup$
1
  • 2
    $\begingroup$ I believe that when symbols that are not real principal type or elliptic, then most if not all solvability and regularity results require assumptions on lower order terms. So there is at least a lot of circumstantial evidence for your hypothesis. $\endgroup$
    – Deane Yang
    Commented May 10, 2012 at 7:23

1 Answer 1

Reset to default
1
$\begingroup$

More a comment than an answer.

I am not quite sure to understand the property you are looking for. After all the principal symbol of a pseudodifferential operator $P$ of order $m$ is a positively homogeneous function $p_m$ with degree $m$ on the pointed cotangent bundle and you may ask for some property of that principal symbol, indeed such as ellipticity or principal type.

Let me give you what I believe is a significant example, not included in your classification. Consider a pseudodifferential operator $P$ of order $m$ with a complex-valued principal symbol $p_m=a+ib$ such that $$ a=b=0\Longrightarrow\text{ {$a,b$}>0}, $$ where {$a,b$} is the Poisson bracket. Then for $R$ of order $m-1$, $P+R$ is subelliptic with loss of $1/2$ derivative in the following sense $$ (P+R)u\in H^s_{loc} \Longrightarrow u\in H^{s+m-\frac12}_{loc}. $$ There are more examples with more Poisson brackets.

Improve this answer
$\endgroup$
2
  • 2
    $\begingroup$ 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? $\endgroup$
    – Deane Yang
    Commented May 10, 2012 at 20:43
  • $\begingroup$ 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. $\endgroup$
    – Bazin
    Commented May 11, 2012 at 10:36

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .