Sharp weighted Carleman and Huber isoperimetric inequalities on the unit ball in higher dimensions

In this paper, using a limiting approach, we establish a new type of weighted Carleman inequality in all dimensions $n\geq 2$ and classify all extremal functions. In particular, when $n=2$, we prove that our inequality is equivalent to a sharp norm inequality in the Bergman space. In even dimensions, we further establish a sharp weighted Huber isoperimetric inequality on the unit ball, which generalizes Huber's original result \cite[Ann. Math., 1954]{Huber} and may be regarded as a sharp counterpart of Y. Wang's isoperimetric inequality in the unit ball \cite[Adv. Math., 2015]{Wang}.