We prove a sharp general inequality estimating the distance of two probability measures on a compact Lie group in the Wasserstein metric in terms of their Fourier transforms. We use a generalized form of the Wasserstein metric, related by Kantorovich duality to the family of functions with an arbitrarily prescribed modulus of continuity. The proof is based on smoothing with a suitable kernel, and a Fourier decay estimate for continuous functions. As a corollary, we show that the rate of convergence of random walks on semisimple groups in the Wasserstein metric is necessarily almost exponential, even without assuming a spectral gap. Applications to equidistribution and empirical measures are also given.

我们证明了一个尖锐的一般不等式，根据Wasserstein度量的一个紧凑Lie群，通过其傅立叶变换来估计两个概率度量的距离。我们使用Wasserstein度量的广义形式，通过Kantorovich对偶性将其与具有任意规定的连续模数的函数族相关联。该证明是基于使用合适内核进行的平滑处理以及对连续函数的傅立叶衰减估计。作为推论，我们证明了在Wasserstein度量中，半简单组上随机游动的收敛速度几乎必须是指数级的，即使不假设存在光谱间隙也是如此。还给出了均值分配和经验测度的应用。