Fractional and anisotropic Gagliardo--Nirenberg inequalities with applications

In this article, we first investigate the necessary and sufficient conditions on the ranges of $s,s_1,s_2\in R$, $1\leq p_1,p_2,q\leq \infty$, $θ_1,θ_2\geq 0$ for the validity of the fractional Gagliardo--Nirenberg inequalities $$\|D^su\|_{L^q(R^d)}\lesssim \|D^{s_1}u\|_{L^{p_1}(R^d)}^{θ_1}\|D^{s_2}