Subharmonicity of higher dimensional exponential transforms

Abstract

Our main result is an extension of the classical Cauchy inequality for the case of bounded densities. In particular, this implies subharmonicity of the function $M_n(E)$, where $V_n(x)$ is the critical Riesz potential in $R^n$ ($\alpha=n$) of a density $0\leq \rho\leq 1$ and $M_n(t)$ is the profile function: the solution of $y'(t)=1-y^{n/2}(t)$, $y(0)=0$. We show thath this result is optimal (in the sense that $M_n(E)$ is harmnoic for characteristic functions of a ball) and give thereby an affirmative answer to one question posed by B. Gustafsson and M. Putinar (Ind. Univ. Math. J., 52(2003), 527-568).