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

We consider the generalized Gagliardo-Nirenberg inequality in ℝⁿ in the homogeneous Sobolev space H^{s, rn} with the critical differential order s = n/r, which describes the embedding such as L^pℝⁿ∩ H^n/r,rℝⁿ L^qℝⁿ 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 ||u|| _L^qℝⁿC_nq||u||_L ^pℝⁿp}{q}}||u||BMO¹p}{q}} with the constant C _n 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 H^{s, r} -norm with s > n/r, which may be regarded as a generalization of the Brezis-Gallouet- Wainger inequality.