The best constant for 𝐿^{∞}-type Gagliardo-Nirenberg inequalities

In this paper we derive the best constant for the following L ∞ L^{\infty } -type Gagliardo-Nirenberg interpolation inequality ‖ u ‖ L ∞ ≤ C q , ∞ , p ‖ u ‖ L q + 1 1 − θ ‖ ∇ u ‖ L p θ , θ = p d d p + ( p − d ) ( q + 1 ) , \begin{equation*} \|u\|_{L^{\infty }}\leq C_{q,\infty ,p} \|u\|^{1-\theta }_{L^{q+1}}\|\nabla u\|^{\theta }_{L^p},\quad \theta =\frac {pd}{dp+(p-d)(q+1)}, \end{equation*} where parameters q q and p p satisfy the conditions p > d ≥ 1 p>d\geq 1 , q ≥ 0 q\geq 0 . The best constant C q , ∞ , p C_{q,\infty ,p} is given by C q , ∞ , p = θ − θ p ( 1 − θ ) θ p M c − θ d , M c ≔ ∫ R d u c , ∞ q + 1 d x , \begin{equation*} C_{q,\infty ,p}=\theta ^{-\frac {\theta }{p}}(1-\theta )^{\frac {\theta }{p}}M_c^{-\frac {\theta }{d}},\quad M_c≔\int _{\mathbb {R}^d}u_{c,\infty }^{q+1} dx, \end{equation*} where u