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 $Q_{\alpha}(\mathbb{R}^{n})$.
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