We study interpolation inequalities between H\"older Integral Probability Metrics (IPMs) in the case where the measures have densities on closed submanifolds. Precisely, it is shown that if two probability measures $\mu$ and $\mu^\star$ have $\beta$-smooth densities with respect to the volume measure of some submanifolds $\mathcal{M}$ and $\mathcal{M}^\star$ respectively, then the H\"older IPMs $d_{\mathcal{H}^\gamma_1}$ of smoothness $\gamma\geq 1$ and $d_{\mathcal{H}^\eta_1}$ of smoothness $\eta>\gamma$, satisfy $d_{ \mathcal{H}_1^{\gamma}}(\mu,\mu^\star)\lesssim d_{ \mathcal{H}_1^{\eta}}(\mu,\mu^\star)^\frac{\beta+\gamma}{\beta+\eta}$, 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 $d_{\mathcal{H}_1^\gamma}$, $\gamma \in [1,\infty)$ simultaneously.

翻译：我们研究了在测度具有闭子流形上密度的情况下，Hölder积分概率度量（IPMs）之间的插值不等式。具体而言，研究表明，若两个概率测度$\mu$和$\mu^\star$分别关于某子流形$\mathcal{M}$和$\mathcal{M}^\star$的体积测度具有$\beta$阶光滑密度，则光滑度$\gamma\geq 1$的Hölder IPM $d_{\mathcal{H}^\gamma_1}$与光滑度$\eta>\gamma$的IPM $d_{\mathcal{H}^\eta_1}$满足$d_{ \mathcal{H}_1^{\gamma}}(\mu,\mu^\star)\lesssim d_{ \mathcal{H}_1^{\eta}}(\mu,\mu^\star)^\frac{\beta+\gamma}{\beta+\eta}$（忽略对数因子）。我们给出了该结果在高维推断中的应用。这些函数不等式被证明是未知子流形上密度估计的关键工具。特别地，该理论使得我们能够构建首个同时对所有距离$d_{\mathcal{H}_1^\gamma}$（$\gamma \in [1,\infty)$）达到最优估计速率的估计器。