A surprising threshold for the validity of the method of singular projection

Given a compact manifold $\mathcal{N}$ embedded into ${\mathbb{R}}^{\nu }$ and a projection $P$ that retracts ${\mathbb{R}}^{\nu }$ except a singular set of codimension $\mathrm{\ell }$ onto $\mathcal{N}$, we investigate the maximal range of parameters $s$ and $p$ such that the projection $P$ can be used to turn an ${\mathbb{R}}^{\nu }$-valued $W^{s,p}$ map into an $\mathcal{N}$-valued $W^{s,p}$ map. Devised by Hardt and Lin with roots in the work of Federer and Fleming, the method of projection is known to apply in $W^{1,p}$ if and only if $p<\mathrm{\ell }$, and has been extended in some special cases to more general values of the regularity parameter $s$. As a first result, we prove in full generality that, when $s\ge 1$, the method of projection can be applied in the whole expected range $sp<\mathrm{\ell }$. When $0<s<1$, the method of projection was only known to be applicable when $p<\mathrm{\ell }$, a more stringent condition than $sp<\mathrm{\ell }$. As a second result, we show that, somehow surprisingly, the condition $p<\mathrm{\ell }$ is optimal, by constructing, for every $0<s<1$ and $p\ge \mathrm{\ell }$, a bounded $W^{s,p}$ map into ${\mathbb{R}}^{\mathrm{\ell }}$ whose singular projections onto the sphere ${\mathbb{S}}^{\mathrm{\ell }-1}$ all fail to belong to $W^{s,p}$. As a byproduct of our method, a similar conclusion is obtained for the closely related method of almost retraction, devised by Hajłasz, for which we also prove a more stringent threshold of applicability when $0<s<1$.