Anisotropic Caffarelli-Kohn-Nirenberg type inequalities

Caffarelli, Kohn and Nirenberg considered in 1984 the interpolation inequalities $$\||x|^{\gamma_1}u\|_{L^s(\mathbb{R}^n)}\le C\||x|^{\gamma_2}\nabla u\|_{L^p(\mathbb{R}^n)}^a\||x|^{\gamma_3}u\|_{L^q(\mathbb{R}^n)}^{1-a}$$ in dimension $n\ge 1$, and established necessary and sufficient conditions for which to hold under natural assumptions on the parameters. Motivated by our study of the asymptotic stability of solutions to the Navier-Stokes equations, we consider a more general and improved anisotropic version of the interpolation inequalities $$\||x|^{\gamma_1}|x'|^{\alpha}u\|_{L^s(\mathbb{R}^n)}\le C\||x|^{\gamma_2}|x'|^{\mu}\nabla u\|_{L^p(\mathbb{R}^n)}^{a}\||x|^{\gamma_3}|x'|^{\beta}u\|_{L^q(\mathbb{R}^n)}^{1-a}$$ in dimensions $n\ge 2$, where $x=(x', x_n)$ and $x'=(x_1, ..., x_{n-1})$, and give necessary and sufficient conditions for which to hold under natural assumptions on the parameters. Moreover we extend the Caffarelli-Kohn-Nirenberg inequalities from $q\ge 1$ to $q>0$. This extension, together with a nonlinear Poincar\'{e} inequality which we obtain in this paper, has played an important role in our proof of the above mentioned anisotropic interpolation inequalities.