论文标题
理性同喻组中分数Sobolev映射的定量估计值
Quantitative estimates for fractional Sobolev mappings in rational homotopy groups
论文作者
论文摘要
令$ \ mathcal {n} \ subset \ mathbb {r}^m $为平滑的简单连接的紧凑型歧管，而没有边界。 $π_{n}（\ Mathcal {n}）$的有理同副本子组由同型\ [{\ rm deg}：π_{n}（\ Mathcal {n}）\ to \ MathB {R}。 \ Mathcal {n} $，我们对其合理同质元素元素$ {\ rm deg}（[f]）\ in \ Mathbb {rm} $进行定量估计，就其分数Sobolev-norm或HölderNorm而言。也就是说，我们证明了所有$β\ in（β_0（{\ rm deg}），1] $，\ [ | {\ rm deg}（[f]）| \ leq c（{\ rm deg}）\，[f] _ {w^{w^{β，\ frac {n}β}（\ mathbb {s}^n）} | {\ rm deg}（[f]）| \ leq c（{\ rm deg}）\，[f] _ {c^β（\ Mathbb {s}^n）}^{\ frac {\ frac {n+l（{\ rm deg}）}β}。 \ \]这里$ c（{\ rm deg}）> 0 $，$ l（{\ rm deg}）\ in \ mathbb {n} $，$β_0（{\ rm deg}）\在（0,1）$ in（0,1）$可从$ {\ rm rm deg} $代表的合理同质组中计算。这扩展了Van Schaftingen的早期作品和Hopf学位的第二作者的工作，以诺维科夫（Novikov）的整体代表性为理性同型小组的积分代表，由Sullivan，Novikov，Hardt和Rivière开发。
Let $\mathcal{N} \subset \mathbb{R}^M$ be a smooth simply connected compact manifold without boundary. A rational homotopy subgroup of $π_{N}(\mathcal{N})$ is represented by a homomorphism \[{\rm deg}: π_{N}(\mathcal{N}) \to \mathbb{R}.\] For maps $f: \mathbb{S}^N \to \mathcal{N}$ we give a quantitative estimate of its rational homotopy group element ${\rm deg}([f]) \in \mathbb{R}$ in terms of its fractional Sobolev-norm or Hölder norm. That is, we show that for all $β\in (β_0({\rm deg}),1]$, \[ |{\rm deg}([f])|\leq C({\rm deg})\, [f]_{W^{β,\frac{N}β}(\mathbb{S}^N)}^{\frac{N+L({\rm deg})}β}, \] and \[ |{\rm deg}([f])|\leq C({\rm deg})\, [f]_{C^β(\mathbb{S}^N)}^{\frac{N+L({\rm deg})}β}. \] Here $C({\rm deg}) > 0$, $L({\rm deg}) \in \mathbb{N}$, $β_0({\rm deg}) \in (0,1)$ are computable from the rational homotopy group represented by ${\rm deg}$. This extends earlier work by Van Schaftingen and the second author on the Hopf degree to the Novikov's integral representation for rational homotopy groups as developed by Sullivan, Novikov, Hardt and Rivière.
