Remarks on Gagliardo–Nirenberg type inequality with critical Sobolev space and BMO

We consider the generalized Gagliardo–Nirenberg inequality in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}^n$$\end{document} in the homogeneous Sobolev space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{H}^{s, r}({\mathbb{R}}^n)$$\end{document} with the critical differential order s = n/r, which describes the embedding such as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p({\mathbb{R}}^n) \cap \dot{H}^{n/r,r}({\mathbb{R}}^n) \subset L^q({\mathbb{R}}^n)$$\end{document} for all q with p ≦ q < ∞, where 1 < p < ∞ and 1 < r < ∞. We establish the optimal growth rate as q → ∞ of this embedding constant. In particular, we realize the limiting end-point r = ∞ as the space of BMO in such a way that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$||u||_{L^{q}({\mathbb{R}}^n)} \leqq C_n q||u||_{L^{p}({\mathbb{R}}^n)}^{\frac{p}{q}}||u||_{BMO}^{1-\frac{p}{q}}$$\end{document} with the constant Cn depending only on n. As an application, we make it clear that the well known John–Nirenberg inequality is a consequence of our estimate. Furthermore, it is clarified that the L∞-bound is established by means of the BMO-norm and the logarithm of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{H}^{s, r}$$\end{document} -norm with s > n/r, which may be regarded as a generalization of the Brezis–Gallouet–Wainger inequality.