Timeline for Is every continuous microlocal operator a pseudo-differential operator?
Current License: CC BY-SA 3.0
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| Jan 13, 2018 at 0:56 | history | edited | Ilya Zakharevich | CC BY-SA 3.0 |
Argument “in the other direction”: why this an example indeed.
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| Jan 7, 2018 at 20:36 | comment | added | Ilya Zakharevich | The “usual conditions” on ΨDO are a (very weak) flavor of the condition of “Regular singularities”. As I said, I’m ready to omit this condition in many cases (this being a millennium of “multisummability”, after all) — but with a lot of reservations. | |
| Jan 7, 2018 at 20:32 | comment | added | Ilya Zakharevich | In more than 1-dimensional case, all hell breaks loose. Def: a ΨD kernel K(x,y)dy is a kernel with a singularity on (conormal bundle to) Diag≔{x=y} such that after blowing up Diag, it has a singularity in codimension 1. (In other words, K is a direct image of a generalized function on Z with such a property; here Z → X×X is the blowup map.) This is my mental picture for ΨDO in dim>1. Omitting the condition of codim=1 gives the “preservation of wavefront” condition. One can see how far these notions are from each other. | |
| Jan 7, 2018 at 20:24 | comment | added | Ilya Zakharevich | Now this is a very productive question! In 1-dimensional case the answer is kind of tricky: I’m mentally ready to treat any such operator as a ΨDO, but I’m not ready to do this for two such operators simultaneously. In my mental picture, different operators may require different, incompatible, generalizations. | |
| Jan 7, 2018 at 6:32 | comment | added | მამუკა ჯიბლაძე | So would you say that any microlocal operator is, in some generalized sense, pseudodifferential? | |
| Jan 6, 2018 at 22:57 | comment | added | Ilya Zakharevich | x(xᵏF(x)) is a product of x and of xᵏF(x). ;-) [And I said it many times: I do not see how it may make sense to have a notion of ΨDO with a “well-defined boundary”. With (ρ,δ), it is easy to say in which sense the composition is defined as “a sum” of a series. However, if a need arises, one could use other notions of “summation of a series” to handle more general classes.] | |
| Jan 6, 2018 at 9:03 | comment | added | მამუკა ჯიბლაძე | Question: what do you mean by x(xᵏF(x))? Comment: maybe a sensible question is whether there is at all a rigorously formulable property that distinguishes a proper subclass of microlocal operators which are definitely not pseudodifferentiable in any possible sense, then | |
| Jan 6, 2018 at 7:37 | history | edited | Ilya Zakharevich | CC BY-SA 3.0 |
mispring
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| Jan 4, 2018 at 22:30 | review | Late answers | |||
| Jan 4, 2018 at 23:08 | |||||
| Jan 4, 2018 at 22:12 | history | answered | Ilya Zakharevich | CC BY-SA 3.0 |