Sharp Moser-Onofri inequality on the CR sphere under zero-moment constraints and its application to the stability of the CR Moser-Onofri inequality

In this paper, we establish the sharp Moser–Onofri inequality in CR sphere under zero-moment constraints. By combining a decomposition into low and high spherical harmonics with the known CR localization estimates, we first prove the existence of extremals of the subcritical Moser–Onofri inequality under zero-moment constraints. We then derive their Euler–Lagrange equation, establish their smoothness, and eliminate the moment multiplier by means of the infinitesimal CR conformal action. Using the summation identity (4.9) on the Hardy space $H^2(B^n)$, we show that the extremals of subcritical Moser-Onofri inequality in CR sphere under zero-moment constraints are constants. By a limiting argument, we deduce the sharp Moser–Onofri inequality in CR sphere under zero-moment constraints. As an application, we establish the stability of Moser–Onofri inequality in CR sphere with the best stability constant.