Timeline for Integral kernel of resolvent of Sub-Laplacian?
Current License: CC BY-SA 4.0
4 events
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| Aug 5, 2021 at 23:54 | comment | added | Christian Remling | Next, in the continuous case it is well known that (up to constants) the FT of $|x|^{-\alpha}$ is $|t|^{\alpha-d}$. What you want to show looks essentially like a discrete version of this statement. | |
| Aug 5, 2021 at 23:52 | comment | added | Christian Remling | First of all, since $(1-\Delta)^{-\delta}$ is multiplication on the Fourier side, it is convolution by the FT $\sum (1+n^2)^{-2\delta} e^{ixt}$ (in your formula for $K$, one of the signs of $x,y$ is wrong). | |
| Aug 5, 2021 at 22:23 | comment | added | Mateusz Kwaśnicki | One way to proceed — perhaps the most elementary one — is via the heat kernel and Bochner's subordination: $(I-\Delta)^{-\delta}=\frac{1}{\Gamma(\delta)} \int_0^\infty t^{-1+\delta} e^{-t(I-\Delta)} dt$, and we have good bounds for the kernel of $e^{-t(I-\Delta)}=e^{-t} e^{t\Delta}$. | |
| Aug 5, 2021 at 20:11 | history | asked | scroo0ooge | CC BY-SA 4.0 |