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-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.