Let $X$ be a random vector in $\mathbb R^d$, with "sufficiently smooth" probability density function on $\rho$. For unit-vectors $w$ and $u$ in $\mathbb R^d$, and a scalar $b \in \mathbb R$, define $$ F(w,u,b;\rho) := \int_{\mathbb R^d}\delta(b-x^\top w) (x^\top u)\rho(x)\, dx = \int_{H_{w,b}}(x^\top u)\rho(x)\, d\sigma(x), $$ where $d\sigma$ is the area element on the hyperplane $H_{w,b}:= \{x \in \mathbb R^d \mid x^\top w = b\}$.
Note that $F(w,u,b;\rho)$ is nothing but the Radon transform of the function $x \mapsto (x^\top u)\rho(x)$.
Now, $X_1,\ldots,X_n$ be iid copies of $X$, and for a bandwidth parameter $h>0$, consider the kernel density estimate (KDE) for $\rho$, namely $$ \widehat{\rho}_{n,h} := \frac{1}{n}\sum_{i=1}^n \varphi_{X_i,h I_d}, $$ where $\varphi_{\mu,\Sigma}$ is the pdf of the multivariate Gaussian distribution $N(\mu,\Sigma)$. I'm interested in upper-bounding the difference $|F(w,u,b;\rho)-F(w,u,b;\widehat{\rho}_{n,h})|$ by a vanishing quantity (for large $n$), uniformly over $w$, $u$, and $b$.
Question. Under what minimalistic conditions on $\rho$ and $h$ does one have bound like $\mathbb E_{X_1,\ldots,X_n} \sup_{w,u,b}|F(w,u,b;\rho)-F(w,u,b;\widehat{\rho}_{n,h})| = \widetilde{O}(n^{-\beta})$ for some constant $\beta>0$ ?
Observation
If we make the following "sloppy" assumptions
(1) the true density $\rho$ is bounded away from zero, i.e., for some constant $\alpha>0$, it holds that $$ \inf_{x \in \mathbb R^d}\rho(x) \ge \alpha, $$
(2) $F$ is uniformly bounded, i.e., for some constant $C>0$, it sholds that $$ \sup_{w,u,b}\int_{\mathbb R^d}\delta(b-x^\top w)|x^\top u|\rho(x)\,dx \le C, $$
then, we may compute $$ \begin{split} |F(w,u,b;\rho)-F(w,u,b;\widehat{\rho}_{n,h})| &\le \int_{\mathbb R^d}\delta(b - x^\top w)|x^\top u|\cdot |\rho(x)-\widehat{\rho}_{n,h}(x)|\,dx \\ &= \int_{\mathbb R^d}\delta(b - x^\top w)|x^\top u|\cdot |\rho(x)-\widehat{\rho}_{n,h}(x)|\,dx\\ &\le \frac{1}{\alpha}\int_{\mathbb R^d}\delta(b - x^\top w)|x^\top u|\rho(x)\cdot |\rho(x)-\widehat{\rho}_{n,h}(x)|\,dx\\ & \le \frac{1}{\alpha}\|\widehat{\rho}_{n,h}-\rho\|_\infty \int_{\mathbb R^d}\delta(b-x^\top w)|x^\top u| \rho(x)\,dx\\ % &\le \|\widehat{\rho}_{n,h}-\rho\|_\infty \int_{\mathbb R^d}\delta(b-x^\top w)\|x\| \rho(x),dx\\ &\lesssim \frac{1}{\alpha}\|\widehat{\rho}_{n,h}-\rho\|_\infty\\ &\lesssim \frac{1}{\alpha}\left(\frac{\log n}{n}\right)^{s/(2s+d)}, \end{split} $$ for an appropritate choice of the bandwidth $h$, and under the condition that all partial derivatives of order upto and including $s$ are uniformly bounded, for some fixed $s \ge 1$.
