I have a problem and would like a hint or two if possible.

Problem: Let $f_0(x)\in L^P(\mathbb{R})$ for $1<p<\infty$ and for each natural number $n$ define $f_n(x)=f_0(x-n)$ for all $x\in\mathbb{R}$. Show that $\{f_n\}\rightharpoonup f$ where $f\equiv 0$ on $\mathbb{R}$. Is this true for $p=1$?

My thoughts so far: I had a couple of ideas, most turned out to be fruitless; however, my current idea is that I might be able to show that $f_n\rightarrow f$ point wise almost everywhere on $\mathbb{R}$ which would give me weak convergence (I have a theorem in my book to help with this part). Only problem is that I'm not sure how to show that. It would make sense that I can somehow use the fact that each $|f_n|^p$ is integrable over $\mathbb{R}$ but I'm not certain where to go from there.

Hopefully my thought is in the right direction. Thank you for you help/advice! :)

P.S. I'm using Royden Fitzpatrick 4th edition. Question is number 15 from chapter 8.


Let $1 < p < \infty$ be as stated. You are trying to show that $\lim_{n} f_{n}g\,dt = 0$ for each $g \in L^{q}(\mathbb{R})$, where $\frac{1}{p}+\frac{1}{q}=1$. Let $\epsilon > 0$ be given. For any $g \in L^{q}(\mathbb{R})$ for $1 < q < \infty$, one has $$ L^{q}-\lim_{N}\chi_{[-N,N]}g=g. $$ Therefore, for any $\epsilon > 0$, there exists $N$ large enough that $$ \|f\|_{p}\|g-\chi_{[-N,N]}g\|_{q} < \epsilon/2. $$ And there exists $K$ large enough that $\|f-\chi_{[-K,K]}f\|_{p}\|g\|_{q} < \epsilon/2$. Then write $$ \int f_{n}g\,dt = \int f_{n}(g-\chi_{[-N,N]}g)\,dt+\int f_{n}\chi_{[-N,N]}g\,dt\\ = \int f_{n}(g-\chi_{[-N,N]}g)\,dt +\int_{n-N}^{n+N}f(x)\chi_{[-N,N]}(t-n)g(t-n)\,dt. $$ If $n \ge K+N$, then $[-n+N,n+N]\subset [K,\infty)$, which gives the bound $$ \left|\int f_{n}g\,dt\right| \le \|f\|_{p}\|g-\chi_{[-N,N]}g\|_{q}+\|f-\chi_{[-K,K]}f\|_{p}\|g\|_{q} < \epsilon,\;\;\; n \ge N+K. $$ Therefore, if $f \in L^{p}$ for some $1 < p < \infty$, then $\{ f_{n}\}$ converges weakly to $0$. This is not true for $p=1$; to see this, let $f=\chi_{[0,1]}$ and let $g$ be the constant function $1$ on $\mathbb{R}$. Then $g \in L^{\infty}$ and $\int f_{n}g\,dt = 1$ for all $n =1,2,3,\cdots$.


Fix $g\in L^q$, the dual space. Given $\epsilon>0$, there is $M$ such that both $f \chi_{|x|\ge M}$ and $g \chi_{|x|\ge M}$ have norms (in their respective spaces) less than $\epsilon$. Use Hölder's inequality to show that for $n>2M$,
$$\int_{\mathbb R} |f(x-n)g(x)|\,dx \le \epsilon \|f\|_{L^p}+\epsilon \|g\|_{L^q} $$

Hint for $p=1$: the answer is negative, and the reason is that an $L^\infty$ function does not need to have a "small tail" at infinity.


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