Uniform convergence for operator of translation

Question

For $a\in R^d$, let $T_af(x)=f(x-a),$ for all $f\in L^p(R^d), 1\leq p<\infty$ and all $x\in R^d$. I need example that this operator doesn't converge uniformly when $a\rightarrow 0$. I know that this operator has strong (norm) limit.

Answer 1

Let $d=1,p=1$ and $f_n(x)=nx^{n}$ for $0<x<1$, $0$ for $x \notin (0,1)$. Then $\|f_n\| <1$ for all $n$ . Suppose $\int_{\mathbb R}|f_n(x-a)-f_n(x)|\, dx \to 0$ uniformly in $n$ as $a\to 0$. Then, given $\epsilon \in (0,e^{-1}-e^{-2}) $ there exists $b>0$ such that $\int_{\mathbb R}|f_n(x-a)-f_n(x)|\, dx <\epsilon $ for all $a \in (0,b)$ for all $n$. Put $a=\frac 1 n$ where $n$ is so large that $\frac 1 n <b$. After a little computation we get $\frac n {n+1} ((1-\frac 1n)^{1+n}-(\frac 1 n)^{1+n}-(1-\frac 2 n)^{1+n}) <\epsilon $. Letting $n \to \infty$ we get $e^{-1}-e^{-2}<\epsilon$. This is a contradiction.