Timeline for Extension of a compactly supported pseudo-differential operator
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Nov 11, 2019 at 23:25 | comment | added | Math | So using a density argument and the fact that the spaces involved are Fréchet, the result seems to follow. | |
| Nov 11, 2019 at 22:56 | comment | added | Math | Taking $\phi=\varphi$ we obtain $||Pu||_{s-m}=||\phi Pu||_{s-m}\leq C_{\phi,s}||\psi u ||_s$, for all $u \in C_0^\infty(\Omega)$ and some $\psi \in C_0^\infty(\Omega)$. Is this enough to guarantee extension? | |
| Nov 11, 2019 at 22:52 | comment | added | Math | I am following the definition of Petersen's book. Introduction To The Fourier Transform And Pseudo-differential Operators. He gives this definition of a compactly supported operator on page 250 and states the existence of the function $\phi$ verifying $\varphi P=P$. Perhaps the result of the Folland book may help: For every $\phi \in C_0^\infty(\Omega)$ there is a $\psi \in C_0^\infty(\Omega)$ such that $||\phi Pu||_{s-m}\leq C_{\phi,s}||\psi u ||_s$. | |
| Nov 11, 2019 at 22:44 | comment | added | Bombyx mori | I am not entirely following. I thought a compactly supported PsiDO is the one whose action on functions that support outside a compact set vanishes. The kernel of P may not be finite dimensional and may not be compact. | |
| Nov 11, 2019 at 22:38 | history | asked | Math | CC BY-SA 4.0 |