Rate of $L_1$ loss in estimating density on $[0,1]$ Let $f$ be a density on $[0,1]$ and let $X_1,X_2,\ldots$ be $\textit{iid}$ $f$-distributed. Also, let $f_n$ denote the kernel density estimator, i.e.
$$f_n(x) = \frac{1}{nh_n} \sum_{i=1}^n K\left(\frac{x-X_i}{h_n}\right),$$
where $K$ is a kernel function (non-negative and integrates to 1) and $h_n$ is an appropriate bandwidth depending on $n$. I'm interested in finding the rate of convergence of the $L_1$ loss
$$\mathbb{E}||f-f_n||_1 = \int_0^1|f(x)-f_n(x)| \text{d}x.$$
The existing literature is quite technical and I'm trying to understand what we need to impose on $f$ in order to get a result like
$$\mathbb{E}||f-f_n||_1 = \mathcal{O}(n^{-s/(2s+1)})$$
when $f$ is $s$ times differentiable (among more restrictions). The literature I have been looking at does not seem to apply to my specific case or makes very technical assumptions (involving Besov norms). Searching for more available literature has not yet brought the desired result. So my question is, does there exists a book or article that clearly states a usable result for this setting? Or is it a matter of digging through a very dense topic?
These are the books I have been reading:
$\textbf{Devroye, L. and Györfi, L.}\textit{ Nonparametric density estimation: The }L_1\textit{View}\\$
$\textbf{Giné, E. and Nickl, R.}\textit{ Mathematical Foundations of Infinite-dimensional Statistical Models}$
NB. Any smoothness assumption on $f$ is allowed.
 A: As the support of $f$ is known to be $[0,1]$, the $L_1$-loss is bounded by the $L_2$-loss, e.g. the square root of the Mean Integrated Square Error $\mathbb{E}[\|f_n-f\|_2^2].$
This quantity is controlled (for instance) at the beginning of Tsybakov's book Introduction to nonparametric estimation under strong constraints on the choice of the kernel. I describe now their assumptions:


*

*Assume that $\int K(u)^2du < \infty$ and that $\int |u|^s|K(u)|^s du <\infty$. Also assume that $K$ is a kernel of order $l = \lfloor s \rfloor$: $\int u^jK(u)du=0$ for $j=1,\dots,l$.

*Assume that $f$ is a density in the Nikol’ski class $\mathcal{H}(s,L)$: its $l$-derivative $f^{(l)}$ satisfies
$$ \left(\int (f^{(l)}(x+t)-f^{(l)}(x))^2dx \right)^{1/2} \leq L|t|^{s-l}\ , \ \forall t\in \mathbb{R}.$$
(This contains the Sobolev class of order $s$.)

*Choose $h_n = c n^{-1/(2s+1)}$.


Then, $E[\|f-f_n\|_1] \leq E[\|f-f_n\|_2] = O(n^{-s/(2s+1)})$. (Theorem 1.2, Chapter 1)
Of course, having a kernel of order $l$ is very restrictive (in particular, it produces kernel estimators $f_n$ which might be negative in some point, which is weird), but this theorem is a reasonable trade-off between simplicity of proof and efficiency of the result.
