An optimal time-singularity of the estimate for the heat semigroup related to the critical Sobolev embedding

Abstract

We give a certain $L^{\infty}(\mathbb{R}^2)$-estimate for the heat semigroup $\{e^{t\Delta}\}_{t \ge 0}$ that is closely related to the fact $H^1(\mathbb{R}^2) \not\subset L^{\infty}(\mathbb{R}^2)$, i.e., the critical Sobolev (non-)embedding and the standard Brezis-Gallou\"et inequality. While we provide several approaches to show such an assertion, we also reveal that the time-singularity of our estimate as $t \to 0^+$ is indeed optimal.