Timeline for Leibniz rule bound for the inverse of the Laplacian?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 29, 2023 at 12:58 | comment | added | Isaac | Yes I just wanted to check for sure. Thank you. | |
| Jul 29, 2023 at 12:57 | comment | added | fedja | Of course, that is true. The point is that it still doesn't help much... | |
| Jul 29, 2023 at 12:45 | comment | added | Isaac | Thank you for your comment. At least isn't it true that the zeroth mode of the Fourier expansion must be eliminated if we want to act on the inverse Laplacian? | |
| Jul 25, 2023 at 22:18 | comment | added | fedja | It is hopeless even on $\mathbb T$ and for $(-\Delta)^{-1}$. Take $f,g$ with Fourier coefficients $1/\sqrt{N}$ on $[-2N,-N]\cup[N,2N]$ and $0$ elsewhere. Their $L^2$ norms are about $1$ but the $L^2$-norm of their inverse Laplacians are like $N^{-2}$. However the product $fg$ has Fourier coefficients of size $1$ on the entire interval $[-N/2,N/2]$, say, so just killing the zeroth Fourier coefficient will not bring the $L^1$ (or any other) norm of $(-\Delta)^{-1}(fg)$ down. | |
| Jul 22, 2023 at 20:01 | history | asked | Isaac | CC BY-SA 4.0 |