Timeline for How to prove that the Laplace oparator of the Heisenberg group is sub-elliptic?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 2, 2016 at 20:03 | comment | added | Nate Eldredge | Subellipticity is mostly useful as a substitute for elliptic regularity, and those definitions don't imply anything of the kind. Even if we strengthen them by adding a bracket-generating type condition, so that they do imply subellipticity in the sense of Folland, the proof of this implication is not trivial. | |
| Feb 2, 2016 at 20:02 | comment | added | Nate Eldredge | That talk shows two (inequivalent) definitions of "subelliptic", both of which are satisfied by the Heisenberg Laplacian, which is very easy to check. If you're using one of those definitions, then I don't understand what your question is. But those are strange definitions, since they don't lead directly to any useful analytic properties; and they're satisfied by truly degenerate operators like $\partial_x^2 + \partial_y^2$ on $\mathbb{R}^3$, or the 0 operator. | |
| Feb 2, 2016 at 18:37 | comment | added | Z. Alfata | For example, using the definition in people.emich.edu/ocalin/Contact_files/Subelliptic_Talk.pdf, we can prouve that $\Delta $ is sub-elliptic, because: $$ \Delta = X^2 +Y^2 + T^2$$ where $X,Y $ and $T$ are the vector fields on $H^3$ | |
| Feb 2, 2016 at 18:05 | comment | added | Nate Eldredge | So you mean this definition (as in Folland 1972): there exists $\epsilon > 0$ such that for each bounded open set $V$ there exists a constant $C$ such that for all $f \in C^\infty_c(V)$ we have $$\|f\|_{W^{\epsilon,2}}^2 \le C(|\langle \Delta f, f \rangle| + \|f\|_{L^2}^2)$$ If not, please state the definition you are using. | |
| Feb 2, 2016 at 17:44 | comment | added | Z. Alfata | I would just use the definition of sub-elliptic of a operator | |
| Feb 2, 2016 at 17:39 | comment | added | Nate Eldredge | Also, for subellipticity, do you just want the "qualitative" statement "if $\Delta u \in C^\infty$ then $u \in C^\infty$", or do you want the "quantitative" subelliptic estimates on Sobolev norms? | |
| Feb 2, 2016 at 17:37 | comment | added | Z. Alfata | Hello, @NateEldredge, thank you for your comment. with a direct argument (for example by calculation) will be better. Thank you | |
| Feb 2, 2016 at 17:30 | comment | added | Nate Eldredge | Non-ellipticity is easy, just check that the principal symbol is degenerate (compute determinant or whatever you like). For subellipticity, are you happy with the "big gun" of Hormander's theorem, or do you want a direct argument? | |
| Feb 2, 2016 at 16:57 | history | asked | Z. Alfata | CC BY-SA 3.0 |