Asymptotic bounds on the quality of the nonparametric regression estimation in $\mathscr L_p$

We consider the asymptotic set-up of the following nonparametric problem. Choose the points of measurement $X_1,\dots,X_N$ and estimate the unknown function $f$ on the base of observations
$$ Y_i=f(X_i)+G_i(X_i,\omega),\quad i=1,\dots,N, $$
where noise variables $G_1,\dots,G_N$ are independent when $X_1,\dots,X_N$ are fixed. We suppose that the deviation of estimator from regression function $f$ is measured in $\mathscr L_p$ metrix, $1\le p<\infty$. The case $p=\infty$ we consider in [1].