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

We give a certain $L^{\mathrm{\infty }}({\mathbb{R}}^2)$-estimate for the heat semigroup $\{e^{t\mathrm{\Delta }}{\}}_{t\ge 0}$ that is closely related to the fact $H^1({\mathbb{R}}^2)\subset ̸L^{\mathrm{\infty }}({\mathbb{R}}^2)$, i.e., the critical Sobolev (non-)embedding and the standard Brezis-Gallouët 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.