The subcritical W0 one p embedding into L2

If $\mathrm{\Omega }=\mathrm{\varnothing }$, then $W_0^{1,p}(\mathrm{\Omega })=L^2(\mathrm{\Omega })=\{0\}$ and the conclusion holds. Assume $\mathrm{\Omega }\mathrm{\ne }\mathrm{\varnothing }$. First consider $d=1$. Choose a bounded open interval $I=(a,b)$ with $\bar{\mathrm{\Omega }}\subset I$ and let $L=b-a$. By [F4] and the closure definition [F8], each $v\in W_0^{1,p}(\mathrm{\Omega })$ has zero extension $E_{0}v{\mathrm{\mid }}_I\in W_0^{1,p}(I)$: choose ${\phi }_m\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ converging to $v$ in $W^{1,p}$ and extend each by zero into $C_c^{\mathrm{\infty }}(I)$. Countable Choice selects this approximating sequence. For every such smooth extension ${\phi }_m$, [F6] gives $\mathrm{\mid }{\phi }_m(x)-{\phi }_m(y)\mathrm{\mid }\le \mathrm{\mid }x-y{\mathrm{\mid }}^{1-1\mathrm{/}p}\mathrm{\parallel }{\phi }_m^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^p(I)}$ and $\mathrm{\parallel }{\phi }_m{\mathrm{\parallel }}_{L^{\mathrm{\infty }}([a,b])}\le L^{1-1\mathrm{/}p}\mathrm{\parallel }{\phi }_m^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^p(I)}$, because it vanishes near both endpoints. Applying the second inequality to differences shows that $({\phi }_m)$ is uniformly Cauchy; its continuous uniform limit $w$ represents $E_{0}v{\mathrm{\mid }}_I$ since ${\phi }_m\to E_{0}v$ in $L^p(I)$. For every $\zeta \in C_c^{\mathrm{\infty }}(I)$, uniform convergence and ${\phi }_m^{\mathrm{\prime }}\to E_0(v^{\mathrm{\prime }})$ in $L^p(I)$ give ${\int }_{I}w{\zeta }^{\mathrm{\prime }}=-{\int }_I{E}_0(v^{\mathrm{\prime }})\zeta$. Thus $w^{\mathrm{\prime }}=E_0(v^{\mathrm{\prime }})$ weakly and [F4] gives $\mathrm{\parallel }{w}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^p(I)}=\mathrm{\parallel }{v}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}$. Passing to the limit in the two displayed inequalities gives the same Hölder and supremum bounds for $w$.

Now let $d\ge 2$. Put $s_0=\max (1,2d\mathrm{/}(d+2))$ and $s_1=\min (p,d)$. The strict hypothesis gives $s_0<s_1$; set $r=(s_0+s_1)\mathrm{/}2$. Then $1<r<p$, $r<d$, and $2<r^{\ast }=dr\mathrm{/}(d-r)$. By finite-measure Hölder [F3], $W_0^{1,p}(\mathrm{\Omega })\hookrightarrow W_0^{1,r}(\mathrm{\Omega })$ continuously: the function and gradient $L^r$ norms are bounded by their $L^p$ norms, and a $C_c^{\mathrm{\infty }}$ approximating sequence in $W^{1,p}$ also converges in $W^{1,r}$. Therefore [F1] gives compactness of $W_0^{1,p}(\mathrm{\Omega })\to L^2(\mathrm{\Omega })$.

For a bounded sequence $(v_j)$ in $W_0^{1,p}(\mathrm{\Omega })$, the representatives $w_j$ from step 1.1 are uniformly bounded by $L^{1-1\mathrm{/}p}{\sup }_j\mathrm{\parallel }\mathrm{\nabla }{v}_j{\mathrm{\parallel }}_p$ and share the modulus $\mathrm{\mid }{w}_j(x)-w_j(y)\mathrm{\mid }\le \mathrm{\mid }x-y{\mathrm{\mid }}^{1-1\mathrm{/}p}{\sup }_j\mathrm{\parallel }\mathrm{\nabla }{v}_j{\mathrm{\parallel }}_p$. Arzelà–Ascoli [F7] gives a uniformly convergent subsequence on the fixed compact interval $[a,b]$. Its restrictions converge in $L^2(\mathrm{\Omega })$ by [F5], proving compactness. For each $v\in W_0^{1,p}(\mathrm{\Omega })$ the same supremum estimate gives $\mathrm{\parallel }v{\mathrm{\parallel }}_2\le \mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{1\mathrm{/}2}{L}^{1-1\mathrm{/}p}\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_p\le L^{3\mathrm{/}2-1\mathrm{/}p}\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_p$, proving continuity.

In the $d\ge 2$ branch, for $v\in W_0^{1,p}(\mathrm{\Omega })$, [F2] and [F3] yield the continuous estimate $\mathrm{\parallel }v{\mathrm{\parallel }}_2\le \mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{1\mathrm{/}2-1\mathrm{/}{r}^{\ast }}\mathrm{\parallel }v{\mathrm{\parallel }}_{r^{\ast }}\le C\mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{1\mathrm{/}2-1\mathrm{/}{r}^{\ast }+1\mathrm{/}r-1\mathrm{/}p}\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_p$, using the Sobolev inequality [F2] with exponent $r$. Thus the inclusion is continuous on the $W_0^{1,p}$ norm. Finally [F5] and the closure definition [F8] give $C_c^{\mathrm{\infty }}(\mathrm{\Omega })\subset W_0^{1,p}(\mathrm{\Omega })$ dense in $L^2(\mathrm{\Omega })$. The preceding arguments prove the complete $d=1$ and $d\ge 2$ branches; [F9] then gives the claimed compact embedding in its continuous/sequential sense.