Timeline for $L^\infty-L^\infty$ bounds for heat semigroups constructed from the Dirichlet Laplacian
Current License: CC BY-SA 4.0
7 events
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| Mar 2, 2022 at 13:08 | comment | added | Mateusz Kwaśnicki | For large $t$, by intrinsic ultracontractivity, the norm is roughly $e^{-\lambda_1 t} \sup \varphi_1$, where $\varphi_1$ is the eigenfunction of $\Delta$ in $D$ with least eigenvalue. I believe explicit bound on the error of this approximation can be given in terms of higher eigenvalues $\lambda_n$. For small $t$, this should be roughly $1$, with the error term bounded by some explicit (rapidly decaying) function of (say) the inradius of $D$ and $t$. Is that what you are looking for? | |
| Mar 2, 2022 at 12:01 | history | edited | SMS | CC BY-SA 4.0 |
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| Mar 2, 2022 at 11:53 | history | edited | SMS |
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| Mar 2, 2022 at 11:48 | comment | added | SMS | @MateuszKwaśnicki Any reference on any regime of $t$ would be highly welcome! But you are right, I should have mentioned "upper bounds", my bad. I have edited the question now. | |
| Mar 2, 2022 at 11:46 | history | edited | SMS | CC BY-SA 4.0 |
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| Mar 2, 2022 at 11:43 | comment | added | Mateusz Kwaśnicki | What are "updated bounds"? Small $t$ or large $t$? Upper or lower bounds? | |
| Mar 2, 2022 at 11:36 | history | asked | SMS | CC BY-SA 4.0 |