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After discussion about linearization of differential operators here, I found another problem. That is, the uniqueness of extension of nonlinear operator.

Here, I consider especially nonlinear differential operator $L: \Gamma(E) \rightarrow \Gamma (F)$ where $E, F$ is vector bundles of compact Riemannian manifold $M$

T. Sunada for example says in Sunada that we can extend differential operators of order $m$ (naturally?) to $ H^{k,p}$ if $k> \frac{n}{2} + m$
When I ask Gemini , it said that has something to do with "Sobolev algebra". Does anyone have any idea about this?

Thank you.

G.W.
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  • Do you have any information about $L$ like regularity or some estimates? Is suspect that any sort of uniqueness relies on those. Also, I doubt any LLM can give you anything close to verifiably accurate information about a topic this niche and technical – whpowell96 Jun 22 '25 at 14:22
  • Did you mean $H^k$ instead of $H^{k,p}$? – Severin Schraven Jun 22 '25 at 14:22
  • I mainly consider Ricci flow. It's quasilinear degenerate parabolic equation. I thought this extension is the key of linearization, so if this topic is too niche, it seems I misunderstood the definition of linearization. – G.W. Jun 22 '25 at 14:47
  • I think whether $p=2$ or not is not important thing here. You can fix p if you want. Thank you. – G.W. Jun 22 '25 at 14:50
  • @G.W. I have the feeling that some sort of Sobolev inequality should give you the result. Namely, once you know that the $C^m$ norm is bounded by the $H^{k}$ norm, you should be good to go (the individual terms are just continuous multilinear forms). – Severin Schraven Jun 22 '25 at 15:02
  • Maybe a similar argument as here https://math.stackexchange.com/questions/314820/sobolev-space-hs-mathbbrn-is-an-algebra-with-2sn could help. – Severin Schraven Jun 22 '25 at 15:36

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