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

Abstract

Given a compact manifold $ \mathcal{N} $ embedded into $ \mathbb{R}^ν $ and a projection $ P $ that retracts $ \mathbb{R}^ν $ 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}^ν $-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 Hajłasz, for which we also prove a more stringent threshold of applicability when $ 0 < s < 1 $.