Timeline for Commutator of pseudodifferential operator and multiplication operator
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 17, 2021 at 11:13 | comment | added | Benjamin | Thank you for the comment and for taking time to think about the problem, although it is of course not the answer I hoped for. ;) | |
| Aug 17, 2021 at 7:32 | comment | added | Giorgio Metafune | I do not think you can go far beyond a linear growth. Let $\hat g=\eta$ so that $g$ is in the Schwartz class and $\eta(-i\nabla) h=g*h$. Then the commutator is given by $$Ch(x)=\int_{\bf R^n}g(y) (f(x-y)-f(x))h(x-y)\, dy.$$ If $|f(x-y)-f(x)| \leq A(y)$ and $gA$ is in $L^1$, then $C$ is bounded by Young's inequality for convolutions. However this imples a linear growth for $f$, since the estimate is independent of $x$. If $f(x)=x^2$ (1d), the above difference is $2xy+y^2$. The $y^2$ term is treated as above, but the extra $x$ can make $C$ unbounded. I did not work the details, however. | |
| Aug 17, 2021 at 6:16 | history | edited | Benjamin | CC BY-SA 4.0 |
added 261 characters in body
|
| Aug 16, 2021 at 17:44 | history | asked | Benjamin | CC BY-SA 4.0 |