Improved Caffarelli–Kohn–Nirenberg inequalities in unit ball and sharp constants in dimension three

Abstract

We show that the Caffarelli–Kohn–Nirenberg inequalities in unit ball Bn can be improved by subtraction of Hardy term. In three dimension and 0≤a<12, we show that the sharp constant coincides with that in R3. This is an analogous result to that of the sharp constant in the n−12-th order Hardy-Sobolev-Maz’ya inequality in the unit ball of dimension n when n is odd. As an application, we obtain a sharp Sobolev inequality on hyperbolic Caffarelli–Kohn–Nirenberg space introduced by L. Dupaigne, I. Gentil and S. Zugmeyer.