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–Weierstrass integral and the Poisson integral for the upper half-space.