Timeline for Positive square roots of inverse operators on different Sobolev spaces
Current License: CC BY-SA 3.0
13 events
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| Dec 14, 2017 at 3:41 | comment | added | geometricK | @Deane: thank you for that. One query I had about Taylor's chapter 12 he studies functional calculus of pseudodifferential operators when the manifold is compact, but do the results hold in the non-compact setting also (without bounded geometry)? Another point I am unclear on is whether a pseudodifferential operator of order $m$ on a general non-compact manifold is continuous $H^s\rightarrow H^{s-m}$... | |
| Dec 13, 2017 at 15:18 | comment | added | Michael Renardy | @ougoah: If, for instance, $M$ is $R^2$, and $D=i\partial_x$, then the domain of $D$ is not $H^1$. | |
| Dec 13, 2017 at 14:57 | comment | added | Deane Yang | I was thinking implicitly of $D$ as a first order matrix differential operator. A good book on pseudodifferential operators is the one by Michael Taylor. There is also the first volume of the books by Francois Treves. The one I liked best (when I studied this many years ago) is the one by Chazarain and Piriou. | |
| Dec 13, 2017 at 4:54 | comment | added | geometricK | I'm treating $H^i$ as the domain of the self-adjoint closure of $D^i$, with inner product $\langle x,y\rangle_i = \langle x,y\rangle_0 + \langle Dx,Dy\rangle_0 + ... + \langle D^i x,D^i y\rangle$. | |
| Dec 13, 2017 at 3:32 | comment | added | Michael Renardy | Also, how do you conclude D is self-adjoint in the $H^i$ norm? | |
| Dec 13, 2017 at 3:16 | history | edited | geometricK | CC BY-SA 3.0 |
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| Dec 13, 2017 at 3:13 | comment | added | geometricK | Incidentally I don't really know of resources about functional analysis of pseudodifferential operators (I am told that pseudodifferential calculus does not always agree with functional calculus, though they may agree to first order), so any suggestions on this front would be great. | |
| Dec 13, 2017 at 3:05 | comment | added | geometricK | Hi Michael, I'm thinking of complex coefficients. | |
| Dec 13, 2017 at 3:05 | comment | added | geometricK | Hi Deane, thanks for that. Do you happen to know of a reference for Questions 0 and 1? For Question 2, I'm intending for the range to be $H^i$. | |
| Dec 13, 2017 at 2:52 | comment | added | Michael Renardy | Is D supposed to have real coefficients? Then there are not that many first order differential operators around which are self-adjoint. | |
| Dec 13, 2017 at 2:49 | comment | added | Deane Yang | I suggest finding an introduction to pseudodifferential operators and studying it thoroughly. | |
| Dec 13, 2017 at 2:47 | comment | added | Deane Yang | Question 0: Only if $D^2$ is hypoelliptic (of which the most important case is when $D$ is elliptic). Otherwise, the inverse "loses derivatives". Question 1: Yes. Question 2: Depends on what you want to allow for the range to be. If you are willing to live with distributions, then the domain can be much larger. Question 3: Yes, because everything commutes. | |
| Dec 13, 2017 at 2:32 | history | asked | geometricK | CC BY-SA 3.0 |