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

TL;DR

This paper establishes an optimal $L^{ abla}( ext{R}^2)$ estimate for the heat semigroup related to the critical Sobolev embedding, highlighting the precise singularity behavior as time approaches zero.

Contribution

It provides a novel $L^{ abla}( ext{R}^2)$ estimate for the heat semigroup connected to the critical Sobolev embedding and proves the optimality of the time-singularity.

Findings

The estimate is closely related to the failure of $H^1( ext{R}^2)$ to embed into $L^{ abla}( ext{R}^2)$.

Multiple approaches are provided to establish the estimate.

The time-singularity as $t o 0^+$ is shown to be optimal.

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.