Timeline for Is every continuous microlocal operator a pseudo-differential operator?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Nov 4, 2017 at 22:13 | comment | added | Ilya Zakharevich | @მამუკაჯიბლაძე: I’m under impression that I did check it some time in the past that matching ∞-jets is enough for microlocality in dim=1. | |
| Nov 4, 2017 at 16:10 | comment | added | Joonas Ilmavirta | Thanks for the remark! I know that there are various classes of $\Psi$DOs, and the question is really whether a microlocal operator is a $\Psi$DO of any kind. I updated my question to reflect this. | |
| Nov 4, 2017 at 11:33 | comment | added | მამუკა ჯიბლაძე | Why exactly is it pseudolocal at the origin? | |
| Nov 4, 2017 at 9:21 | comment | added | Ilya Zakharevich | I now believe that this may indeed give a necessary example. The width of Gaussian measure should decay quicker than any power of x (when x→0); then the operator would send a smooth decaying function on [0,∞⟧ to a similar function, and the ∞-jet at 0 would not change. This allows gluing with id operator on ⟦-∞,0]. (In other words: replace xˢ above to, say, x-ˢ ˡᵒᵍ ˣ near 0.) The “symbol” of this operator is 1 on negative semiaxis, and 0 on the positive. | |
| Nov 4, 2017 at 5:32 | comment | added | მამუკა ჯიბლაძე | Welcome to MO, Ilya! Great to have you here! | |
| Nov 4, 2017 at 4:19 | comment | added | Ilya Zakharevich | Oups: the example above is acting in 𝒮, not in 𝒮 ′. Dualize! | |
| Nov 4, 2017 at 3:58 | comment | added | Ilya Zakharevich | A possible candidate: start with multiplicative convolution with a Gaussian kernel on ℝ₊⊂ℝ (extended as id onto ℝ₋), and tune up the parameters. Something like exp(-(x-y)²/x²ˢ)/xˢ dy for an appropriate s (or an appropriate substitute for xˢ). | |
| Nov 4, 2017 at 3:50 | review | First posts | |||
| Nov 4, 2017 at 8:54 | |||||
| Nov 4, 2017 at 3:49 | history | answered | Ilya Zakharevich | CC BY-SA 3.0 |