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arXiv · 0902.2742

Subharmonicity of higher dimensional exponential transforms

Vladimir Tkachev
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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).

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Vladimir Tkachev. 2009-02-16. Subharmonicity of higher dimensional exponential transforms. https://doi.org/10.1007/3-7643-7316-4_13

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