Let $\{u_n\}$ be a sequence of continuous functions $u_n:\mathbb{R} \to \mathbb{R}^+$ such that $$\Vert u_n\Vert_{L^1} < C$$ and $$\Vert \partial_x u_n \Vert_{L^2} < C.$$ $C$ does not depend on $n$. Suppose in addition that $u_n$ have compact supports. Can we use this information to get a uniform bound on $\Vert u_n \Vert_{L^p}$ with $p \ge 4$?

Also, do these assumptions imply $u_n \to u$ locally uniformly and $u$ continuous?

  • $\begingroup$ No uniform assumptions on support? $\endgroup$ – Calvin Khor Sep 28 '18 at 10:06

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