The Dunford-Pettis theorem states that a family $\mathcal{F}\subset L^1$ is relatively weakly compact if and only if $\mathcal{F}$ is bounded in norm and uniformly integrable, i.e. $\sup_{f\in\mathcal{F}}\lVert f \rVert_1<\infty$ and $\forall \varepsilon>0$ there exists some $\delta>0$ such that

$$ \int_A \vert f(x)\vert dx<\varepsilon $$

for every $f\in\mathcal{F}$ whenever $\vert A\vert \leq \delta$.

In the other hand, the de la Vallée-Poussin theorem says that $\mathcal{F}$ is uniformly integrable if and only if there exists a non-negative increasing convex function $\phi(t)$ such that

$$ \lim_{t\rightarrow \infty} \frac{\phi(t)}{t}=\infty \qquad \text{and} \qquad \sup_{f\in\mathcal{F}} \int \phi(\vert f(x)\vert) dx<\infty. $$

See as reference the Uniform Integrability Wiki.

With this theorems we can get a lot weakly compact set in $L^1$. For example, when $\phi(t)=t^p$ with $1<p<\infty$, we get that every set $\mathcal{F}$ bounded in the $L^p$-norm is weakly compact in $L^1$, such as the Rademacher sequence or the unit-ball of $L^2$.

So here is my question: Is there any example of a weakly compact sequence $(f_n)$ (or set $\mathcal{F}$) that is not norm-bounded in any $L^p$ space ($1<p<\infty$)? Where there can be a reference for such example? suppose that there must be (in Orlicz spaces and such) but I am not finding anyone.

  • 1
    $\begingroup$ Find functions $f_n$ with $\Vert f_n \Vert_1=1$ and $\Vert f_n\Vert_{1+1/n}=n^2$. Then take $g_n=f_n/n$. (Or take $f$ in $L_1$ but not in any $L_p$, $p>1$ and set $g_n=f$.) $\endgroup$ – David Mitra May 6 at 11:57
  • $\begingroup$ $\sup_n \int |f_n|(\log |f_n|)^{+}<\infty$ is also a sufficient condition for uniform integrability, so it is easy to construct examples you are looking for. $\endgroup$ – Kavi Rama Murthy May 6 at 12:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.