We study interpolation inequalities between Hölder Integral Probability Metrics (IPMs) in the case where the measures have densities on closed submanifolds. Precisely, it is shown that if two probability measures $μ$ and $μ^\star$ have $β$-smooth densities with respect to the volume measure of some submanifolds $\mathcal{M}$ and $\mathcal{M}^\star$ respectively, then the Hölder IPMs $d_{\mathcal{H}^γ_1}$ of smoothness $γ\geq 1$ and $d_{\mathcal{H}^η_1}$ of smoothness $η>γ$, satisfy $d_{ \mathcal{H}_1^γ}(μ,μ^\star)\lesssim d_{ \mathcal{H}_1^η}(μ,μ^\star)^\frac{β+γ}{β+η}$, up to logarithmic factors. We provide an application of this result to high-dimensional inference. These functional inequalities turn out to be a key tool for density estimation on unknown submanifold. In particular, it allows to build the first estimator attaining optimal rates of estimation for all the distances $d_{\mathcal{H}_1^γ}$, $γ\in [1,\infty)$ simultaneously.