Anisotropic Caffarelli-Kohn-Nirenberg inequalities with higher-order fractional derivatives

We prove a family of anisotropic Caffarelli-Kohn-Nirenberg interpolation inequalities involving higher-order fractional derivatives and weights of the form $\mathrm{\mid }{x}^{\mathrm{\prime }}{\mathrm{\mid }}^{{\theta }_1}\mathrm{\mid }x{\mathrm{\mid }}^{{\theta }_2}\mathrm{\mid }{x}_n{\mathrm{\mid }}^{{\theta }_3}$.