Let $B$ denote the open unit ball in $\mathbb{R}^3$. I want to either prove or disprove that a sequence of functions $u_m$ in the Sobolev space $W^{1,2}(B)$ which is uniformly bounded in the $W^{1,2}(B)$ norm and which is convergent in $L^p(B)$ for all $1 \leq p < 6$ must be convergent in $L^6(B)$.
Can someone please point me in the right direction?
I know (cf. Chapter 5 of Evans PDE book) that the bound $ \| u \|_{L^q(U)} \leq C(k,p,n,U) \| u \|_{W^{k,p}(U)} $ holds when $U$ is a subset of $\mathbb{R}^n$ having smooth boundary, $u \in W^{k,p}$, $k < \frac{n}{p}$, and $\frac{1}{q} = \frac{1}{p} - \frac{k}{n}$. The constant $C = C(k,p,n,U)$ is independent of $u$.
It follows from this bound that the $u_m$ are uniformly bounded in $L^6(B)$.
Since $W^{1,2}(B)$ is a Hilbert Space, uniform boundedness of $u_m$ in $W^{1,2}(B)$ also implies existence of a subsequence $u_{m_j}$ that converges weakly in $W^{1,2}(B)$, hence weakly in $L^6(B)$.
Thanks!