Rate of convergence of mollified functions in $L^p$ norm

Question

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\supp}{\operatorname{supp}} $ Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a probability density function such that $\rho_n \in C_c^{\infty} (\bR^d)$ and $\supp \rho_n \subset \overline{B(0,1 / n)}$. We have from Brezis' Functional Anlysis that

Theorem 4.22. Assume $f \in L^p (\bR^d)$ with $1 \le p<\infty$. Then $\| \rho_n * f -f \|_{L^p} \to 0$ as $n\to \infty$.

Above, $*$ denotes the mollification operation.

Are there results about rate of above convergence?

References are appreciated. Thank you for your elaboration.

Answer 1

In general there is no uniform rate of convergence. In particular if one defines $T_n(f)=f\ast \rho_n$, it is not difficult to see that \begin{equation}\label{e:bound} \sup_{\|f\|_p\le 1} \|T_n(f)-f\|_p =2. \end{equation} Note that the operator norm of $T_n$ is always less than one, so the above bound is the worst possible one.

In general, assuming that all the function considered have fixed support (say $B_1$), if $K$ is a subset of the unit ball in $L^p(B_1)$, the following are equivalent

1. There exists a rate $\omega(n) \to 0$ such that
   $$ \sup_{f \in K} \|T_n(f)-f\|_p =\omega(n). $$
2. $K$ is compact in the strong $L^p(B_1)$ topology.

The proof of this equivalence is just a slight variation of the classical Fréchet-Kolmogorov criteria, https://en.wikipedia.org/wiki/Fréchet–Kolmogorov_theorem.