I have tried to solve this problem for almost a week and did not manage to, so I figured to ask it here:
Let $(u_n)\to u$ in $L^1(0,1)$ strongly and let $\{u_n\}_{n\in\mathbb{N}}$ be bounded in $L^p(0,1)$ for some $p>1$. Show that $u_n\to u$ in $L^q(0,1)$ strongly for all $1 \leq q < p$.
Thanks in advance.
The case $q = 1$ is done by the premises, so let's suppose $1 < q < p$. The idea is to use Hölder's inequality to get an estimate
$$\int_0^1 \lvert u_n(t) - u_m(t)\rvert^q\,dt \leqslant \lVert u_n - u_m\rVert_{L^1}^\alpha \cdot \lVert u_n-u_m\rVert_{L^p}^\beta,$$
which then shows that $(u_n)$ is a Cauchy sequence in $L^q(0,1)$, since $\lVert u_n - u_m\rVert_{L^p} \leqslant \lVert u_n\rVert_{L^p} + \lVert u_m\rVert_{L^p}$ is bounded, and $(u_n)$ is an $L^1$-Cauchy sequence.
Thus we write
$$\int_0^1 \lvert u_n(t) - u_m(t)\rvert^q\,dt = \int_0^1 \lvert u_n(t)-u_m(t)\rvert^\gamma\cdot \lvert u_n(t) - u_m(t)\rvert^{q-\gamma}\,dt,$$
and it remains to determine $\gamma$ so that Hölder's inequality gives the exponent $1$ in the first and $p$ in the second factor, i.e., since for the first factor we raise to the $\frac{1}{\gamma}$-th power, we need $$\frac{q-\gamma}{1-\gamma} = p.$$
I trust the remaining details are not hard to find.
