A critical parabolic Sobolev embedding via Littlewood-Paley decomposition

TL;DR

This paper establishes a new parabolic Sobolev embedding inequality relating the supremum norm of functions to their parabolic BMO norm, utilizing Littlewood-Paley decomposition and logarithmic Sobolev estimates.

Contribution

It introduces a parabolic Sobolev embedding inequality of Ogawa type, extending classical results to the parabolic setting with novel analytical techniques.

Findings

Derived a parabolic $L^{ abla}$ norm estimate in terms of parabolic BMO

Utilized Littlewood-Paley decomposition for the proof

Characterized parabolic BMO spaces in the process

Abstract

In this paper, we show a parabolic version of the Ogawa type inequality in Sobolev spaces. Our inequality provides an estimate of the $L^{\infty}$ norm of a function in terms of its parabolic $BMO$ norm, with the aid of the square root of the logarithmic dependency of a higher order Sobolev norm. The proof is mainly based on the Littlewood-Paley decomposition and a characterization of parabolic $BMO$ spaces.