critical case in $L^p$ convergence Let $f_n$ be a bounded sequence in $L^1\cap L^{p}$ with $1<p<\infty$. Assume that $f_n\to f$ strongly in $L^1$. Then basically we have $f_n\to f$ strongly in $L^q$ for all $1\leq q<p$. But we lose the case $q=p$.
My question is: In which supplement condition do we have the strong convergence in $L^p$?. Here I am interested in the following suplement condition 
$$
\int f_n(x)f^{p-1}(x){\rm d}x\to\int f^{p}.
$$
My attempt is to improve the above convergence like
$$
\int f_n^{1+\epsilon}f^{p-1-\epsilon}\to\int f^{p}
$$
and then repeat this process many times to obtain the strong convergence in $L^p$.
Does anyone have an idea?.
 A: No. Consider the standard counterexample for the limit case where we take $X = (0,1)$ and,
$$f_n = n^{\frac1p} 1_{(0,1/n)}.$$
Then $\lVert f_n \rVert_{L^p((0,1))} = 1$ for all $n$ and $\lVert f_n \rVert_{L^1(0,1))} = n^{\frac1p - 1} \rightarrow 0$ as $n \rightarrow 0.$ But as the limit $f \equiv 0,$ the supplementary condition is vacuously satisfied.

Added later: In fact, your supplementary condition actually follows from the given assumptions and moreover we can say more; namely that $f_n \rightharpoonup f$ weakly in $L^p.$
To see this, note that since $f_n \rightarrow f$ in $L^1,$ for any $g \in L^{\infty}$ we have,
$$ \left|\int_X (f_n -f)g \,\mathrm{d}\mu\right| \leq \lVert f_n-f\rVert_{L^1}\lVert g \rVert_{L^{\infty}} \rightarrow 0.$$
Now given $g \in L^{p'}$ ($p'$ is the Hölder conjugate of $p$) and $\varepsilon > 0,$ we can find $g_m \in L^{\infty}$ such that $g_m \rightarrow g$ in $L^{p'}.$ Then using Hölder's inequality,
\begin{align*}
  \left|\int_X (f_n-f)g \,\mathrm{d}\mu \right| &\leq  \left|\int_X (f_n-f)(g-g_m) \,\mathrm{d}\mu \right| + \left|\int_X (f_n-f)g_m \,\mathrm{d}\mu \right| \\
&\leq \lVert f_n - f \rVert_{L^p} \lVert g - g_m \rVert_{L^{p'}} + \left|\int_X (f_n-f)g_m \,\mathrm{d}\mu \right|.
\end{align*}
Now as $f_n$ is bounded in $L^p,$ we have $f \in L^p$ by Fatou also, and there is $M>0$ such that 
$$\lVert f_n - f \rVert_{L^p} \leq \lVert f_n \rVert_{L^p} + \lVert f \rVert_{L^p} \leq M$$
 for all $n.$ So we choose $m$ such that,
$$ \lVert g_m -g \rVert_{L^{p'}} \leq \frac{\varepsilon}{2M}, $$
and $N$ such that for all $n \geq N$ we have,
$$ \left|\int_X (f_n-f)g_m \,\mathrm{d}\mu \right| < \frac{\varepsilon}2. $$
This gives for all $n \geq N,$
$$ \left|\int_X (f_n-f)g \,\mathrm{d}\mu \right| \leq \varepsilon, $$
so $f_n \rightharpoonup f$ in $L^{p'}.$ In particular, taking $g = f^{p-1} \in L^{p'}$ gives the supplementary condition.
