Brezis-Gallouet-Wainger type inequality with critical fractional Sobolev space and BMO

TL;DR

This paper establishes a Brezis-Gallouet-Wainger type inequality connecting BMO, fractional Sobolev, and logarithmic norms, extending classical inequalities to critical fractional spaces.

Contribution

It introduces a new inequality involving BMO, fractional Sobolev, and logarithmic norms for critical fractional spaces, broadening the scope of existing inequalities.

Findings

Proved a new inequality linking BMO, fractional Sobolev, and logarithmic norms.

Extended classical inequalities to critical fractional Sobolev spaces.

Provided mathematical tools for analyzing functions in these spaces.

Abstract

In this paper, we prove the Brezis-Gallouet-Wainger type inequality involving the BMO norm, the fractional Sobolev norm, and the logarithmic norm of $\mathcal{\dot{C}}^\eta$, for $\eta\in(0,1)$.