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Isoperimetric inequalities for the least harmonic majorant of $ \vert x\vert \sp p$


Author: Makoto Sakai
Journal: Trans. Amer. Math. Soc. 299 (1987), 431-472
MSC: Primary 31B05; Secondary 30C85, 30D55, 60J65
DOI: https://doi.org/10.1090/S0002-9947-1987-0869215-0
MathSciNet review: 869215
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Abstract: Let $ D$ be an open set in the $ d$-dimensional Euclidean space $ {{\mathbf{R}}^d}$ containing the origin 0 and let $ {h^{(p)}}(x,D)$ be the least harmonic majorant of $ \vert x{\vert^p}$ in $ D$, where $ 0 < p < \infty $ if $ d \geqslant 2$ and $ 1 \leqslant p < \infty $ if $ d = 1$. We shall be concerned with the following isoperimetric inequalities $ {h^{(p)}}{(0,D)^{1/p}} \leqslant cr(D)$, where $ r(D)$ denotes the volume radius of $ D$, namely, a ball with radius $ r(D)$ has the same volume as $ D$ has and $ c$ is a constant dependent on $ d$ and $ p$ but independent of $ D$. We fix $ d$ and denote by $ c(p)$ the infimum of such constants $ c$. As a function of $ p$, $ c(p)$ is nondecreasing and satisfies $ c(p) \geqslant 1$. We shall show

(1) there are positive constants $ {C_1}$ and $ {C_2}$ such that $ {C_1}{p^{(d - 1)/d}} \leqslant c(p) \leqslant {C_2}{p^{(d - 1)/d}}$ for $ p \geqslant 1$,

(2) $ c(p) = 1$ if $ p \leqslant d + {2^{1 - d}}$. Many estimations of $ {h^{(p)}}(0,D)$ and their applications are also given.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1987-0869215-0
Keywords: Harmonic majorants, harmonic measures, the Hardy norms, exit times of Brownian motion, the Poisson equation
Article copyright: © Copyright 1987 American Mathematical Society

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