Strong convergence in $L^p(\mathbb{R}^N)$

Question

I would like to ask your help about this:

Let $q>1$ fixed.

Consider $\{m_n\}$ a sequence of non-negative functions in $L^s(\mathbb{R}^N)$ for every $s\in[1,q)$ such that $\int_{\mathbb{R}^N}m_n=M$ and $m_n\to m$ in $L^1(\mathbb{R}^N)$ assume also boundedness of $m_n$ in $L^s$ for $1\le s<q$

I want to prove that $m_n\to m$ in $L^s(\mathbb{R}^N)$ for every $1\le s<q$

maybe it is easy but I do not remember very well this topic

Thanks

Answer 1

Lyapunov's inequality gives, for $1\leqslant p<r<q<\infty$ and each $g\in L^p\cap L^q$ that $$ \lVert g\rVert_r\leqslant \lVert g\rVert_p^\alpha \lVert g\rVert_q^{1-\alpha}, $$ where $\alpha\in (0,1)$ depends only on $p,r$ and $q$ (there is actually an explicit expression, but we do not need it here).

Pick $1\leqslant s<q$ and let $s'$ in the interval $(s,q)$. Apply the previous inequality to $g=m_n-m$, $p=1$, $r=s$ and $q=s'$ in order to control $\lVert m_n-m\rVert_s$ by a constant independent of $n$ times $\lVert m_n-m\rVert_1$.