Compactness of a bounded $W^{1,p}$ sequence on an interval

Given: the Axiom of Choice, $I=(0,1)$, $1<p\le \mathrm{\infty }$, and a sequence $(u_j)$ bounded in $W^{1,p}(I)$, with $M:={\sup }_j\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{W^{1,p}(I)}<\mathrm{\infty }$. For each $j$ let $u_j^{\ast }$ be the continuous absolutely continuous representative of One-dimensional $W^{1,p}$ functions have unique absolutely continuous representatives.

One-dimensional ACL representatives. There is exactly one continuous representative $u_j^{\ast }$ of $u_j$ that is absolutely continuous on $[0,1]$, and $u_j^{\ast }(x)-u_j^{\ast }(y)={\int }_y^{x}D{u}_j$ for all $x,y\in [0,1]$. (One-dimensional $W^{1,p}$ functions have unique absolutely continuous representatives, Absolute continuity on almost every coordinate line)

H"older's inequality on an interval. For $1<p<\mathrm{\infty }$, $\mid {\int }_y^{x}D{u}_j\mid \le \mathrm{\mid }x-y{\mathrm{\mid }}^{1-1\mathrm{/}p}\mathrm{\parallel }D{u}_j{\mathrm{\parallel }}_{L^p(I)}$; for $p=\mathrm{\infty }$, $\mid {\int }_y^{x}D{u}_j\mid \le \mathrm{\mid }x-y\mathrm{\mid }\mathrm{\parallel }D{u}_j{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(I)}$. (Holder's inequality for integrals, including the endpoint cases)

The sup bound. Because $I$ has measure $1$, some point $x_0\in I$ has $\mathrm{\mid }{u}_j^{\ast }(x_0)\mathrm{\mid }\le \mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{L^p(I)}$, and then [F1] and [F2] give $\mathrm{\parallel }{u}_j^{\ast }{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{L^p(I)}+\mathrm{\parallel }D{u}_j{\mathrm{\parallel }}_{L^p(I)}$. (Integer-order Sobolev spaces and their norms, The space $L^p(\mu )$ as the quotient by null functions)

Arzel`a--Ascoli. A uniformly bounded equicontinuous family of real functions on a compact metric space has a uniformly convergent subsequence; for a complex-valued family apply this to the real and imaginary parts. (Arzelà--Ascoli for real $C(K)$ under Countable Choice and Dependent Choice: compact closure iff equicontinuous and pointwise bounded, The space $C(K,\mathbb{R})$ of continuous real-valued functions on a nonempty compact metric space)

Uniform convergence gives $L^q$ convergence. If $g_j\to g$ uniformly on the finite-measure set $I$, then $\mathrm{\parallel }{g}_j-g{\mathrm{\parallel }}_{L^q(I)}\le \mathrm{\parallel }{g}_j-g{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }I{\mathrm{\mid }}^{1\mathrm{/}q}\to 0$ for every finite $q$. (Holder's inequality for integrals, including the endpoint cases, The space $L^p(\mu )$ as the quotient by null functions)