# Compactness properties of a sequence of continuous functions such that $\Vert u_n\Vert_{L^1} , \Vert \partial_x u_n \Vert_{L^2} < C$

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?

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