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.
Categories: math.AP