Pseudo-differential operators which are independent of lower order perturbations - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T01:18:45Z http://mathoverflow.net/feeds/question/96504 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96504/pseudo-differential-operators-which-are-independent-of-lower-order-perturbations Pseudo-differential operators which are independent of lower order perturbations Uday 2012-05-09T20:03:00Z 2013-05-14T21:22:00Z <p>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 $E^{-1}(P+R)E=q(x,D).$ </p> <p>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. </p> <p>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? </p> <p>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, <a href="http://annals.math.princeton.edu/wp-content/uploads/annals-v163-n2-p02.pdf" rel="nofollow">http://annals.math.princeton.edu/wp-content/uploads/annals-v163-n2-p02.pdf</a></p> http://mathoverflow.net/questions/96504/pseudo-differential-operators-which-are-independent-of-lower-order-perturbations/96610#96610 Answer by Bazin for Pseudo-differential operators which are independent of lower order perturbations Bazin 2012-05-10T20:00:07Z 2012-05-10T20:28:48Z <p>More a comment than an answer.</p> <p>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.</p> <p>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.</p>