Preparing for an exam, I was wondering about general statements about the convergence of products.
1) Let $p, q \in ]1, \infty[$ such that $\frac{1}{p}+\frac{1}{q} = 1$ and
- $a_n \rightharpoonup a$ in $L^p(\mathbb{R}^n)$,
- $b_n \rightharpoonup b$ in $L^q(\mathbb{R}^n)$.
What can we say about the convergence of $a_nb_n$?
2) What if $p$ and $q$ don't satisfy $\frac{1}{p}+\frac{1}{q}=1?$ Is there anything you can say about the convergence?
3) What if one (or both) of the sequences converge strongly?
The attempts:
1) I think $a_nb_n$ converge weakly in $L^1$.
To show: $\int{a_nb_nc} \rightarrow \int{abc}$ for all $c \in L^\infty$.
Using the trick $|\int{a_nb_nc} – \int{abc}| \leq |\int{a_nb_nc} – \int{a_nbc}| + |\int{a_nbc} – \int{abc}| = |\int{(b_n-b)a_nc}| + |\int{(a_n-a)bc}|$
you see that both summands converge to 0 because $a_nc \in L^p$ and $bc \in L^q$ respectively (a function in $L^\infty$ multiplied by a function in $L^p$ is in $L^p$).
Is that correct? What about strong convergence?
In all examples I have studied so far, one of the two sequences converges strongly. So I wondered what I was missing in my calculations above.
Hint: In the following example the sequence $a_n$ converges strongly in $L^2$ and $b_n$ weakly in $L^2$. The accepted poster suggests that $a_nb_n$ converge strongly:
weak convergence of product of weakly and strongly convergent $L^{2}$ sequences in $L^{2}$
2) I don't think you can say anything about convergence in general? Only if $p$ or $q$ are $\infty$, as seen here:
Product of weak/strong converging sequences
If I understand right, you need the strong convergence of the $L^\infty$ sequence because $2$ isn't dual to $\infty$.
3) Clearly, having strong convergence for both sequences does not harm, but does it change anything? I think it doesn't.
(I already asked above if we require one of them to converge strongly.)
Thanks a lot for your input!
Best Answer
Saying $b_n\to b$ weakly in $L^q$ says that $\int(b_n-b)\phi\to0$ if $\phi$ is an element of $L^p$. But your $a_nc$ is not an element of $L^p$, it's a sequence of element$s$ of $L^p$.
Example on $\Bbb R$: Say $a_n=b_n=\chi_{[n,n+1]}$. Then $a_n\to0$ weakly in $L^q$ and $b_n\to$ weakly in $L^p$ (if $1<p<\infty$) but $\int a_nb_n$ does not converge to $0$.