Precompact set in L2 space?

Question

Let A be a bounded interval in R. Suppose we have a collection of functions, such that

1. Each function is $\in$ $L^r(A)$, where r is any number $\in$ $[1, \infty]$,

2. The fractional derivative of order 1/2 of the functions are bounded set in $L^2(A)$.

I want to know whether the set is precompact in $L^2(A)$ in the norm topology?

Actually, I was reading a paper, it seems to me that the author assumed that this is correct. I think it might be a well-know fact in distribution theory. But I do not know the reference for the statement.

Answer 1

Apparently, your assumption is that your collection is bounded in the Sobolev space $H^{1/2}(A)$. More generally, an $L^2$-function is in $H^s(A)$ with $s>0$ (not necessarily an integer) if its derivatives of order $s$ are in $L^2$. And a subset of $H^s(A)$ is bounded if it is bounded in $L^2$ and the $s$-derivatives form a bounded set in $L^2$.

Rellich-Kondrachov Theorem. If $s>0$ and $A$ is bounded, then a bounded set in $H^s(A)$ is precompact in $L^2(A)$.