Subharmonicity of higher dimensional exponential transforms

TL;DR

This paper extends the classical inequality to bounded densities, proving subharmonicity of a profile function related to Riesz potentials, and confirms the optimality of this result in higher dimensions.

Contribution

It introduces a new subharmonicity result for a profile function associated with Riesz potentials, answering an open question in the field.

Findings

Proves subharmonicity of the function $M_n(E)$ for bounded densities.

Shows the result is optimal by demonstrating harmonicity for characteristic functions of a ball.

Provides an affirmative answer to a question posed by Gustafsson and Putinar.

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).