Wasserstein and weighted metrics for multidimensional Gaussian distributions

We present a number of low and upper bounds for Lévy – Prokhorov, Wasserstein, Frechét, and Hellinger distances between probability distributions of the same or different dimensions. The weighted (or context-sensitive) total variance and Hellinger distances are introduced. The upper and low bounds for these weighted metrics are proved. The low bounds for the minimum of different errors in sensitive hypothesis testing are proved.