Timeline for Literature on the product of two distributions satisfying the Hörmander condition
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Feb 25, 2021 at 19:13 | comment | added | Abdelmalek Abdesselam | Unfortunately, I don't of precise references I can point you to. I think however the comments give you enough to be able to work it out. | |
| Feb 25, 2021 at 18:05 | comment | added | B K | Dear Abdelmalek, what you have just explained in the comments is a great example of the kind of interesting content which I hope can be found in the literature that my question is asking for. Do you have references where the statements/explanations you gave can be found? I am totally fine with references predating Hörmander. | |
| Feb 25, 2021 at 16:18 | comment | added | Abdelmalek Abdesselam | For proving properties satisfied by the product, the definition $\lim u_n v_n$ may be all you need. | |
| Feb 25, 2021 at 16:15 | comment | added | Abdelmalek Abdesselam | ...of course $u_n v_n$ is the ordinary product. Let $P\subset \mathscr{D}'\times\mathscr{D}'$ be the set of pairs $(u,v)$ where for all approximating sequences the limit exists. Then by interlacing and the Hausdorff property of $\mathscr{D}'$, the limit is uniquely defined. What Hormander does is just to tell you there is subset $H$ of pairs which satisfy the wavefront condition and $H\subset P$. | |
| Feb 25, 2021 at 15:56 | comment | added | Abdelmalek Abdesselam | Sorry my answer was not quite to your satisfaction, but I will leave it since it could be useful to someone else. You may be able to answer the questions you have, e.g., the one about $uv$, by yourself as follows. The definition of Hormander's product is not whatever is produced by Hormander's Theorem black box. The definition (which predates Hormander) is: let $u_n$ be a sequence in $\mathscr{E}$ (smooth functions) which converges to $u$ in the $\mathscr{D}'$ topology, and likewise for $v_n$ converging to $v$, then $uv:=\lim_{n\rightarrow\infty} u_n v_n$ in $\mathscr{D}'$... | |
| Feb 25, 2021 at 0:22 | comment | added | B K | Thank you for pointing out that my question was perhaps not clear enough. I made an edit. I know the paper by Brouder, Dang, and Hélein which is very nice and has interesting examples but which does not go into details concerning the properties of the Hörmander product (apart from the product rule). | |
| Feb 24, 2021 at 23:50 | history | edited | Abdelmalek Abdesselam | CC BY-SA 4.0 |
added 1 character in body
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| Feb 24, 2021 at 23:04 | history | answered | Abdelmalek Abdesselam | CC BY-SA 4.0 |