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