It is well-known that the bounded Lipschitz metric for probability measures induce the weak convergence, such as the question: Metrizability of weak convergence by the bounded Lipschitz metric.
If we substitute the Hölder norm, can we define a corresponding metric, and does it induce the weak convergence? Is there any studies about this so-called "bounded Hölder metric"?
Let $(X,d)$ be a compact complete metric space. The Hölder norm with parameter $0 < \alpha<1$ is equal to the Lipschitz norm with respect to the metric given by $d_\alpha(x,y) = d(x,y)^\alpha$. That metric induces the same topology on $X$, so you can pretty much reduce your question to the general case.