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 $\ell$ 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 $p < \ell$, 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 $s \geq 1$, the method of projection can be applied in the whole expected range $sp < \ell$. When $0 < s < 1$, the method of projection was only known to be applicable when $p < \ell$, a more stringent condition than $sp < \ell$. As a second result, we show that, somehow surprisingly, the condition $p < \ell$ is optimal, by constructing, for every $0 < s < 1$ and $p \geq \ell$, a bounded $W^{s,p}$ map into $\mathbb{R}^{\ell}$ whose singular projections onto the sphere $\mathbb{S}^{\ell-1}$ 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 Haj\l asz, for which we also prove a more stringent threshold of applicability when $0 < s < 1$.