一个维数无关不等式（英文）

A Dimension Free Inequality

为了得到p-调和映照的正则性,Giaquinta-Modica与Acerbi-Fusco使用了如下具有基本作用的不等式:对任意γ>-1/2,都只存在只依赖于γ和n的正数c_1,c_2> 0,使得对任意的x,y∈R^n与μ≥0,都有下述不等式成立:c_2|x-y|≤|(μ~2+|x|~2)~γx-(μ~2+|y|~2)~γy|/(μ~2+|x|~2+|y|~2)~γ≤c_1|x-y|证明了上述不等式在任意的希尔伯特空间中都成立,并且常数c_1,c_2只依赖于γ,从而证明上述不等式与空间维数无关。

To deduce regularity of p-harmonic mappings,the following important inequality plays a fundamental role given by Giaquinta-Modica and Acerbi-Fusco:For anyγ>-1/2,there exist constants c 1,c2>0 depending only onγand n,such that for any x,y∈R^n andμ≥0,there holds:c2|x-y|≤|(μ^2+|x|^2^γx-(μ^2+|y|^2)^γy|/(μ^2+|x|^2+|y|^2)^γ≤c1|x-y|In this paper this inequality is improved by showing that the above inequality holds in any Hilbert space with c 1,c 2 depending only onγ.This implies that the above inequality is dimension free.