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Let $\{f_n\}\subset L^1(\Omega,\mu)$, where $\mu$ is the Lebesgue measure, and $\Vert f_n\Vert_1\leq M$ and $\Vert Df_n\Vert_{1/2}\leq C$ uniformly in $n$.

Question. Is there a subsequence $\{f_{n_k}\}$ of $\{f_n\}$ such that $f_{n_k}\rightarrow f$ in the $L^1$-norm, for some $f\in L^1$?

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    $\begingroup$ What does 1/2 norm mean ? $\endgroup$
    – ABMath
    Commented Feb 16, 2017 at 3:55
  • $\begingroup$ No. If $f_n$ is $\delta$-like (in $d=1$), of height $n$ and width $1/n$, say, then you can keep $\|f'_n\|_{1/2}$ bounded. $\endgroup$
    – Christian Remling
    Commented Feb 16, 2017 at 4:09

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