$L_p$ space,convergence

Question

Let $1<p<\infty$ and $h\in L_p(\mathbb{R})$,that is,$\left(\displaystyle\int_{\mathbb{R}}|h|^p\right)^{1/p}<\infty$. Define a sequence $(f_n)_{n\in\mathbb{N}}$ by $f_n(x):=h(x-n)$. How to prove that $$\lim_{n\to\infty}\displaystyle\int_{\mathbb{R}}f_n g=0$$ for all $g\in L_q(\mathbb{R})$ with $\dfrac{1}{p}+\dfrac{1}{q}=1$?

Answer 1

1. The sequence $\{f_n\}$ is bounded in $L^p$, hence it's enough to prove the result for $g$ lying in a dense set of $L^q$?
2. We can choose simple functions, and by linearity, the characteristic functions of sets of finite measure.
3. Such sets can be approximated by bounded sets. So actually we just have to deal with the case $g=\chi_B$, where $B$ is a bounded set.
4. Assuming that $B\subset [-R,R]$ and $h\geqslant 0$, show that $\lim_{n\to +\infty}\int_{[-R+n,R+n]}h(x)dx=0$.

Answer 2

First, I think you are able to show that $\|f_n\|_p=\|h\|_p<\infty$. We need to use the following two points:

• $g\in L^q(\mathbb{R})\ \Rightarrow\ \exists\ M_1>0\ s.t.\ \int_{|x|>M_1} |g|^q \le \epsilon^{\ q}$
• $h\in L^p(\mathbb{R})\ \Rightarrow\ \exists\ M_2>0\ s.t.\ \int_{|x|>M_2} |h|^p \le \epsilon^{\ p}$

Now we are ready to show the result: $$ |\int_{\mathbb{R}} f_n\cdot g|\le \int_{|x|\le M_1}|f_n\cdot g|+\int_{|x|> M_1}|f_n\cdot g| $$

• $H\ddot{o}lder$'s inqequalty: $\int_{|x|>M_1}|f_n\cdot g|\le\|f_n\|_p\cdot (\int_{|x|>M_1} |g|^q)^{\frac{1}{q}}\le \epsilon\cdot \|h\|_p$
• \begin{eqnarray} \int_{|x|\le M_1}|f_n\cdot g|&\le& \|g\|_q\cdot (\int_{|x|\le M_1}|f_n|^p)^{\frac{1}{p}}\\ &\le& \|g\|_q\cdot (\int_{|x|\le M_1}|h(x-n)|^p\mathrm{d}x)^{\frac{1}{p}}\\ &\le& \|g\|_q\cdot (\int_{x\in (-M_1-n,\ M_1-n)}|h|^p)^{\frac{1}{p}} \end{eqnarray} since $x-n\in(-M_1-n,\ M_1-n)$, we can choose large $n$ such that $ M_1-n<-M_2 $, then $\int_{|x|\le M_1}|f_n\cdot g|\le \epsilon\cdot \|g\|_q$

Form the above, we can conclude that $$ \int_{\mathbb{R}}f_n\cdot g\to 0$$