Subharmonicity of higher dimensional exponential transforms

Vladimir Tkachev

doi:10.1007/3-7643-7316-4_13

paper

Subharmonicity of higher dimensional exponential transforms

arXiv:0902.2742 · doi:10.1007/3-7643-7316-4_13

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 $R^{n}$ ( $α = n$ ) of a density $0 \leq ρ \leq 1$ 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).

21 pages