In this paper we obtain uniform upper bounds for the $L_1$ error of kernel estimators in estimating monotone densities and densities of bounded variation. The bounds are nonasymptotic and optimal in $n$, the sample size. For the bounded variation class, it is also optimal wrt an upper bound of the total variation. The proofs employ a one-sided kernel technique and are extremely simple.
"Some Nonasymptotic Bounds for $L_1$ Density Estimation using Kernels." Ann. Statist. 20 (3) 1658 - 1667, September, 1992. https://doi.org/10.1214/aos/1176348791