Subcritical compactness for compactly supported Slobodeckij functions

Given: the Axiom of Choice, $d\ge 1$, $0<\theta <1$, $1\le p<\mathrm{\infty }$ with $p\theta <d$, $p_{\star }=dp\mathrm{/}(d-p\theta )$, a family $\mathcal{F}\subseteq W^{\theta ,p}({\mathbb{R}}^d)$ supported in one fixed bounded set and bounded in the norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_p+[\cdot {]}_{\theta ,p}$ by $M<\mathrm{\infty }$, and $1\le q<p_{\star }$.

Fractional Sobolev inequality. For real compactly supported $g$, $\mathrm{\parallel }g{\mathrm{\parallel }}_{L^{p_{\star }}}^p\le C_1[g{]}_{\theta ,p}^p$; hence $\mathrm{\parallel }g-h{\mathrm{\parallel }}_{p_{\star }}\le C_1^{1\mathrm{/}p}([g]+[h])$ for compactly supported $g,h$. For complex $g$, apply the real inequality to its real and imaginary parts, whose seminorms are at most $[g{]}_{\theta ,p}$, and use the $L^{p_{\star }}$ triangle inequality, enlarging the constant by at most $2$. (The critical fractional Sobolev inequality on ${\mathbb{R}}^d$)

Mollification rates. With the radial mollifier at scale $\delta$, $\mathrm{\parallel }g-g_{\delta }{\mathrm{\parallel }}_{L^p}\le C_2{\delta }^{\theta }[g{]}_{\theta ,p}$ and $\mathrm{\parallel }{g}_{\delta }{\mathrm{\parallel }}_{W^{1,p}}\le C_2(\mathrm{\parallel }g{\mathrm{\parallel }}_p+{\delta }^{\theta -1}[g{]}_{\theta ,p})$; the mollified functions are supported in the $\delta$-neighbourhood of the fixed support set, and $[g_{\delta }{]}_{\theta ,p}\le [g{]}_{\theta ,p}$ because $\mathrm{\mid }{g}_{\delta }(x)-g_{\delta }(y)\mathrm{\mid }\le \int {\eta }_{\delta }(z)\mathrm{\mid }g(x-z)-g(y-z)\mathrm{\mid }dz$ and Minkowski's inequality applies in the weighted $L^p$-space of the seminorm. (Mollification rates for compactly supported Slobodeckij functions, The Gagliardo--Slobodeckij space on Euclidean space)

First-order compactness on a ball. For fixed $\delta >0$, the mollified family is bounded in $W^{1,p}$ on a smooth ball containing all its supports. Its closure in $L^p$ is compact by the first-order Rellich theorem, including dimension one. (Compactness of $W^{1,p}(\mathrm{\Omega })\hookrightarrow L^p(\mathrm{\Omega })$ on bounded extension domains)

Interpolation, completeness and compactness. Strict interpolation between $p$ and $p_{\star }$ is supplied by Lyapunov interpolation inequality for $L^p$ norms; finite-measure inclusion follows from Holder's inequality for integrals, including the endpoint cases. Under Countable Choice, $L^q$ is complete and total boundedness passes to closures, so a totally bounded family has compact closure. Under Countable and Dependent Choice the closure is sequentially compact. (Riesz-Fischer completeness of $L^p$ for $1\le p\le \mathrm{\infty }$, A totally bounded metric space is bounded, every subspace of a totally bounded space is totally bounded, and the closure of a totally bounded subset is totally bounded, A complete, totally bounded metric space is compact, proved from countable choice used exactly once, For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice, Finite $\epsilon$-net and totally bounded metric space)