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Let $D \subset \mathbb{R}^n$ be a bounded domain with Lipschitz boundary, and let $\Delta$ be the Laplace operator with the Dirichlet boundary condition on $D$. Let $e^{t\Delta}$ be the corresponding heat semigroup.

My question is, what are updated upper bounds on $\| e^{t\Delta}\|_{L^\infty \to L^\infty}$? I guess it is realistic to expect very good bounds in the regime of small $t$. The literature in this general area is enormous, and it is rather difficult to find the state of the art. This is mainly a reference request.

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  • $\begingroup$ What are "updated bounds"? Small $t$ or large $t$? Upper or lower bounds? $\endgroup$
    – Mateusz Kwaśnicki
    Commented Mar 2, 2022 at 11:43
  • $\begingroup$ @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. $\endgroup$
    – SMS
    Commented Mar 2, 2022 at 11:48
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    $\begingroup$ 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? $\endgroup$
    – Mateusz Kwaśnicki
    Commented Mar 2, 2022 at 13:08

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