Denote $\mathbb{S}_n(\sqrt{n})$ the sphere in $\mathbb{R}^n$ with radius $\sqrt{n}$. Recall Stein's lemma (Gaussian integration by parts), which reads for every function $f_n$ such that the expectations below are well-defined: $$\mathbb{E}_{Z \sim \mathcal{N}(0,Id)}[Z_1 f_n(Z)] = \mathbb{E}_{Z \sim \mathcal{N}(0,Id)}[\partial_1 f_n(Z)].$$
My goal is to prove that this relation holds also asymptotically for large $n$ when the measure is no longer Gaussian but is uniform over $\mathbb{S}_n(\sqrt{n})$. This is natural since we expect the Gaussian measure to concentrate on $\mathbb{S}_n(\sqrt{n})$. So I would like to have a bound for $$\left|\int_{\mathbb{S}_n(\sqrt{n})}[Z_1 f_n(Z) - \partial_1f_n(Z)] \, \mu_n(\mathrm{d}Z)\right|$$ when $\mu_n$ is the uniform measure over $\mathbb{S}_n(\sqrt{n})$. This bound should depend on the function $f_n$ (via Lipschitz constants for example) and on $n$. Does there exists such bounds ?