Strict power concavity of a convolution

Abstract

We give a sufficient condition for the strict parabolic power concavity of the convolution in space variable of a function defined on ${\mathbb{R}}^n\times (0,+\infty )$ and a function defined on ${\mathbb{R}}^n$. Since the strict parabolic power concavity of a function defined on ${\mathbb{R}}^n\times (0,+\infty )$ naturally implies the strict power concavity of a function defined on ${\mathbb{R}}^n$, our sufficient condition implies the strict power concavity of the convolution of two functions defined on ${\mathbb{R}}^n$. As applications, we show the strict parabolic power concavity and strict power concavity in space variable of the Gauss--Weierstass integral and the Poisson integral for the upper half-space.

19 pages