Degree theory of BMO. I: Compact manifolds without boundaries.

*(English)*Zbl 0852.58010The authors consider the degree theory for mappings \(u\) from a compact smooth manifold \(X\) to a connected compact smooth manifold \(Y\) of the same dimension. The notion of degree can be extended to continuous maps from \(X\) to \(Y\) because if \(u,v \in C^1 (X,Y)\) are close in the \(C^0\) topology then they have the same degree. For a \(C^1\)-map there is an integral formula for the degree. The integral formulas suggest the possibility of extending degree theory to another class of maps which need not be continuous namely maps in appropriate Sobolev spaces. This was done by several authors and the list of references is given in the paper. Among them, L. Boutet de Monvel and O. Gabber introduced a degree for maps \(u \in H^{1/2} (S^1, S^1)\) and made an observation that this notion makes sense for maps in the class VMO (vanishing mean oscillation): the closure of the set of smooth maps in the BMO (bounded mean oscillation) topology. Namely, if \(u \in \text{VMO} (S^1, S^1)\) and \(\overline u_\varepsilon (\theta) = {1 \over 2 \varepsilon} \int^{\theta + \varepsilon}_{\theta - \varepsilon} u(s) ds\) then \(|\overline u_\varepsilon (\theta) |\to 1\) uniformly in \(\theta\), in spite of the fact that \(u\) need not be continuous. Then, for \(\varepsilon\) small,
\[
u_\varepsilon (\theta) = {\overline u_\varepsilon (\theta) \over \bigl |\overline u_\varepsilon (\theta) \bigr |}
\]
has a well defined degree which is independent of \(\varepsilon\). In the paper under review, the authors develop this concept for maps between \(n\)-dimensional manifolds \(X,Y\) and establish its basic properties. The degree is defined via approximation, in the BMO topology. The content of the paper is as follows:

In Section I.1 they recall the notion of BMO and VMO maps on Euclidean spaces and describe its extension to maps between manifolds. The next section takes up various examples of BMO and VMO maps. The degree for VMO maps is defined in Section I.3 and its standard properties are described in the next section. In Section I.5 the authors consider a natural question concerning maps from \(X\) to \(Y\) not necessarily of the same dimension. The last section deals with the question of the possibility of lifting a map \(u \in \text{BMO} (X, S^1)\) to \(\text{BMO} (X, \mathbb{R})\). The proofs of many technical statements are given in Appendix A. The proofs of results of Section I.6 are technical and use the John-Nirenberg inequality, various forms of which are presented in Appendix B. The authors announce that Part II of this paper will consider the degree theory for VMO maps on manifolds with boundary.

In Section I.1 they recall the notion of BMO and VMO maps on Euclidean spaces and describe its extension to maps between manifolds. The next section takes up various examples of BMO and VMO maps. The degree for VMO maps is defined in Section I.3 and its standard properties are described in the next section. In Section I.5 the authors consider a natural question concerning maps from \(X\) to \(Y\) not necessarily of the same dimension. The last section deals with the question of the possibility of lifting a map \(u \in \text{BMO} (X, S^1)\) to \(\text{BMO} (X, \mathbb{R})\). The proofs of many technical statements are given in Appendix A. The proofs of results of Section I.6 are technical and use the John-Nirenberg inequality, various forms of which are presented in Appendix B. The authors announce that Part II of this paper will consider the degree theory for VMO maps on manifolds with boundary.

Reviewer: W.Mozgawa (Lublin)

##### MSC:

58C35 | Integration on manifolds; measures on manifolds |

58C25 | Differentiable maps on manifolds |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

58D15 | Manifolds of mappings |

##### Keywords:

Sobolev space; VMO vanishing mean oscillation; BMO bounded mean oscillation; degree theory; BMO topology; John-Nirenberg inequality
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\textit{H. Brézis} and \textit{L. Nirenberg}, Sel. Math., New Ser. 1, No. 2, 197--263 (1995; Zbl 0852.58010)

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