We extend the well-known result that any \(f \in W^{1,n}({\Omega }, \mathbb {R}^{n})\), \({\Omega } \subset \mathbb {R}^{n}\) with strictly positive Jacobian is actually continuous: it is also true for fractional Sobolev spaces \(W^{s,\frac {n}{s}}({\Omega })\) for any \(s \geq \frac {n}{n+1}\), where the sign condition on the Jacobian is understood in a distributional sense. Along the way we also obtain extensions to fractional Sobolev spaces \(W^{s,\frac {n}{s}}\) of the degree estimates known for W^1,n-maps with positive or non-negative Jacobian, such as the sense-preserving property.

我们扩展了众所周知的结果，即任何\（f \ in W ^ {1，n}（{\ Omega}，\ mathbb {R} ^ {n}）\），\（{\ Omega} \ subset \ mathbb {R} ^ {N} \）与严格正雅可比实际上是连续的：它也为分数Sobolev空间真\（W ^ {S，\压裂{N} {S}}（{\欧米茄}）\）为任何\（s \ geq \ frac {n} {n + 1} \），其中从分布意义上理解雅可比行列上的符号条件。在此过程中，我们还获得了W ¹的度数估计的分数Sobolev空间\（W ^ {s，\ frac {n} {s}} \\}的扩展^，其中n映射具有正或非负Jacobian，例如作为保持感觉的属性。