Let $I=(0, 1)$. Suppose we are given $1<p<+\infty$ and a sequence of function $f_n$ which is bounded in $L^p(I)$. I was thinking.. if we also assume that $f_n$ converges to $0$ in $L^1(I)$, can we conclude that $f_n$ converges to $0$ in $L^r(I)$, at least for $1\leq r<p$?
Many thanks
Guido