Timeline for Reference request - Fourier multiplier of vector valued function
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| 2 days ago | comment | added | Daniele Tampieri | This is another question, and it is not so easy to find an answer. I can only suggest a path: interpreting $f$ as $f=(f_1, f_2,f_3)$ and $-\Delta ^{-1}$ as the convolution operator with the fundamental solution to the laplacian, you may try to express the vector operators $\nabla \wedge$ and $\nabla$ as integral operators like what is done by Claus Müller (1969)[1957], Foundations of the Mathematical Theory of Electromagnetic Waves, MR0253638, Zbl 0181.57203. | |
| 2 days ago | comment | added | Rundasice | Say then that I want to prove an $L^p$ estimate of something of the type $\nabla (\nabla \wedge (-\Delta)^{-1} f)$. Is there any hope ? How one should then proceed ? | |
| Dec 10 at 18:59 | comment | added | Daniele Tampieri | ... Continue from the previous comment. I saw some indication on how to do this in Ehrenpreis' Fourier analysis in several complex variables, MR285849, Zbl 0195.10401, chapter VI, §VI.2, pp. 179-187. | |
| Dec 10 at 18:56 | comment | added | Daniele Tampieri | If you are trying to construct a fundamental solution for the rotational by just using the Fourier transform, you'll fail: I know since I tried this years ago. Nevertheless this is not due to the fact that this is impossible, but just to the fact that this is not the right way to proceed for system of constant coefficients PDEs (in other words this means the there's no multiplier theorem in this setting). Continue in the following comment... | |
| Dec 10 at 17:03 | review | Close votes | |||
| Dec 10 at 19:04 | |||||
| S Dec 10 at 16:00 | review | First questions | |||
| Dec 10 at 16:45 | |||||
| S Dec 10 at 16:00 | history | asked | Rundasice | CC BY-SA 4.0 |