# Weakly compact set in $L^1$ not bounded in $L^p$

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 $$1, we get that every set $$\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)? 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.

• 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 $g_n=f$.) – David Mitra May 6 at 11:57
• $\sup_n \int |f_n|(\log |f_n|)^{+}<\infty$ is also a sufficient condition for uniform integrability, so it is easy to construct examples you are looking for. – Kavi Rama Murthy May 6 at 12:05