Given $u \in L^1$, is there approximating sequence $u_n \in L^\infty$ uniformly bounded in $L^p$?

Question

Let $u \in L^1(U)$ where $U$ is a bounded domain. Is it possible to find a sequence $u_n \in L^\infty $ converging to $u$ in $L^1$ such that the $u_n$ are uniformly bounded for all $n$ in some $L^p$ space for $p > 1$?

I thought mollification might do the trick but it does not because the mollifier blows up so we cannot simply estimate the absolute value in that way.

Answer 1

• If $u\in\mathbb L^p$ for some $p>1$, then take $u_n:=u\chi_{\{|u|\leqslant n\}}$.

• If $u$ does not belong to any $\mathbb L^p$ space for any $p>1$, then it is not possible: if $\lVert u_n-u\rVert_1\to 0$ and $(u_n)_n$ is bounded in $\mathbb L^p$, then extract a subsequence $(u_{n_k})_{k\geqslant 1}$ which converges almost everywhere to $u$. Then using Fatou's lemma, we would have $\lVert u\rVert_p\leqslant \liminf_{k\to\infty}\lVert u_{n_k}\rVert_p\lt+\infty$.