标题： 该方法有效性的令人惊讶的阈值 显示英文标题

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

摘要： 给定一个嵌入到$ \mathbb{R}^{\nu} $中的紧致流形$ \mathcal{N} $和一个投影$ P $，该投影将$ \mathbb{R}^{\nu} $除了一个余维数为$ \ell $的奇异集外，重新映射到$ \mathcal{N} $，我们研究参数$ s $和$ p $的最大范围，使得投影$ P $可以用来将一个取值为$ \mathbb{R}^{\nu} $的$ W^{s,p} $映射转换为一个取值为$ \mathcal{N} $的$ W^{s,p} $映射。 由Hardt和Lin提出，其根源在于Federer和Fleming的工作，投影方法在$ W^{1,p} $时已知仅当$ p < \ell $适用，并且在一些特殊情况下已被扩展到更一般的正则性参数$ s $的值。 作为第一个结果，我们证明了在$ s \geq 1 $时，投影方法可以在整个预期范围$ sp < \ell $中应用。 当$ 0 < s < 1 $时，投影方法仅知在$ p < \ell $时适用，这是一个比$ sp < \ell $更严格的条件。 作为第二个结果，我们证明了，某种意义上令人惊讶的是，条件$ p < \ell $是最优的，通过为每一个$ 0 < s < 1 $和$ p \geq \ell $构造一个有界$ W^{s,p} $映射到$ \mathbb{R}^{\ell} $，其在球面$ \mathbb{S}^{\ell-1} $上的奇异投影都不属于$ W^{s,p} $。 作为我们方法的副产品，对于Haj\l asz设计的类似几乎收缩的方法，也得到了类似的结论，当$ 0 < s < 1 $时，我们还证明了更严格的适用性阈值。 显示英文摘要

摘要： 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 $.