Several analytic inequalities in some $Q-$spaces

TL;DR

This paper establishes various inequalities in $Q$-spaces, including John-Nirenberg, Gagliardo-Nirenberg, Trudinger-Moser, and Brezis-Gallouet-Wainger inequalities, advancing the understanding of function spaces and their properties.

Contribution

It introduces necessary and sufficient John-Nirenberg inequalities for $Q_{eta}^{eta}$ spaces and derives several important inequalities in related $Q$-spaces, expanding theoretical knowledge.

Findings

Established John-Nirenberg inequalities for $Q_{eta}^{eta}$ spaces.

Derived Gagliardo-Nirenberg inequalities in $Q_{eta}$ spaces.

Obtained Trudinger-Moser and Brezis-Gallouet-Wainger inequalities in $Q$-spaces.

Abstract

In this paper, we establish separate necessary and sufficient John-Nirenberg (JN) type inequalities for functions in $Q_{\alpha}^{\beta}(\mathbb{R}^{n})$ which imply Gagliardo-Nirenberg (GN) type inequalities in $Q_{\alpha}(\mathbb{R}^{n}).$ Consequently, we obtain Trudinger-Moser type inequalities and Brezis-Gallouet-Wainger type inequalities in $Q_{\alpha}(\mathbb{R}^{n}).$