A remark on a generalization of a logarithmic Sobolev inequality to the Holder class

TL;DR

This paper extends a parabolic Ogawa inequality to include Hölder continuous functions, broadening its applicability beyond Sobolev spaces and enhancing understanding of Sobolev embeddings in critical cases.

Contribution

The paper introduces a generalized parabolic Ogawa inequality that encompasses Hölder continuous functions, expanding the scope of previous Sobolev embedding results.

Findings

Extended Ogawa inequality to Hölder classes

Broadened applicability of Sobolev embedding inequalities

Improved understanding of critical Sobolev embeddings

Abstract

In a recent work of the author, a parabolic extension of the elliptic Ogawa type inequality has been established. This inequality is originated from the Brezis-Gallouet-Wainger logarithmic type inequalities revealing Sobolev embeddings in the critical case. In this paper, we improve the parabolic version of Ogawa inequality by allowing it to cover not only the class of functions from Sobolev spaces, but the wider class of Holder continuous functions.