Timeline for Existence of a partition of unity with uniformly bounded derivatives.
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 29, 2023 at 6:17 | comment | added | AlexE | Consider an integral operator on $L^2(M)$ given by a kernel function $k(x,y)$. If this operator should extend to Sobolev spaces $H^s(M)$, then one needs bounds on the derivatives of $k$. Now I want to force this operator to have finite propagation. To this end I choose such a partition of unity and define a new integral operator by the kernel $\sum_n g_n^{1/2}(y) k(x,y) g_n^{1/2}(x)$. We need the bounds on the derivatives of $g_n^{1/2}$ to deduce that this operator still extends to the Sobolev spaces. | |
| Aug 29, 2023 at 0:31 | comment | added | shuhalo | I am curious about the application of such properties. Can you give more context or point out a reference? I am particularly curious about the fact that the square root of the partition of unity functions needs an upper bound. Thank you. | |
| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
replaced http://math.stackexchange.com/ with https://math.stackexchange.com/
|
|
| Mar 23, 2011 at 8:15 | vote | accept | AlexE | ||
| Mar 22, 2011 at 20:31 | answer | added | Anonymous | timeline score: 4 | |
| Mar 22, 2011 at 16:30 | answer | added | Deane Yang | timeline score: 3 | |
| Mar 22, 2011 at 16:16 | history | edited | AlexE | CC BY-SA 2.5 |
added 87 characters in body
|
| Mar 22, 2011 at 15:28 | comment | added | Deane Yang | I believe this can be done, if there is a uniform lower bound on either the injectivity radius or the volume of small geodesic balls, as well as the appropriate bounds on the Ricci or sectional curvature. You might want to look up the paper by Bemelmans, Min-Oo, Ruh on smoothing Riemannian metrics. The paper by Shi in JDG on the Ricci flow on a complete Riemannian manifold might also have some relevant results. | |
| Mar 22, 2011 at 14:48 | answer | added | Sergei Ivanov | timeline score: 7 | |
| Mar 22, 2011 at 13:39 | history | edited | AlexE | CC BY-SA 2.5 |
added a note concerning the metric
|
| Mar 22, 2011 at 13:30 | comment | added | Łukasz Grabowski | If your manifold is an open subset of a compact manifold, and $g$ is a restriction of a metric from this compact manifold, then it's true; but I suspect this isn't very useful to you. | |
| Mar 22, 2011 at 13:17 | comment | added | Orbicular | In what norm do you measure the derivatives? More precisely, what Riemannian metric do you use? Is it fixed? | |
| Mar 22, 2011 at 12:58 | history | asked | AlexE | CC BY-SA 2.5 |