Degree theory of BMO. I: Compact manifolds without boundaries
type: Item
instance of: scholarly article
title: Degree theory of BMO. I: Compact manifolds without boundaries
zbMATH Open document ID: 0852.58010
DOI: 10.1007/BF01671566
author: Louis Nirenberg
author: Haïm Brezis
published in: Selecta Mathematica. New Series
publication date: 1996-12-09T00:00:00Z
review text: The 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 ar…
Mathematics Subject Classification ID: 58C35
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 58D15
zbMATH DE Number: 810907
zbMATH Keywords: Sobolev space
zbMATH Keywords: VMO vanishing mean oscillation
zbMATH Keywords: BMO bounded mean oscillation
zbMATH Keywords: degree theory
zbMATH Keywords: BMO topology
zbMATH Keywords: John-Nirenberg inequality
reviewed by: Witold Mozgawa
MaRDI profile type: Publication
cites work: Q4073917
cites work: The approximation problem for Sobolev maps between two manifolds
cites work: Approximations in trace spaces defined between manifolds
cites work: Density of smooth functions between two manifolds in Sobolev spaces
cites work: A boundary value problem related to the Ginzburg-Landau model
cites work: Large solutions for harmonic maps in two dimensions
cites work: A degree theory for almost continuous functions
cites work: Q4003026
cites work: Q5660202
cites work: Sobolev maps with integer degree and applications to Skyrme’s problem
cites work: Remarks on the degree theory
cites work: Fixed Points for Certain Noncontinuous Transformations
cites work: On functions of bounded mean oscillation
cites work: Q4774620
cites work: Functions of bounded mean oscillation and quasiconformal mappings
cites work: Functions of Vanishing Mean Oscillation
cites work: Boundary regularity and the Dirichlet problem for harmonic maps
cites work: Fixed point theorems for connectivity maps
cites work: Bounded Toeplitz Operators on H 1 and Applications of the Duality Between H 1 and the Functions of Bounded Mean Oscillation
cites work: Q3142876
full work available at URL: https://doi.org/10.1007/bf01671566
OpenAlex ID: W4238175110
description: scientific article; zbMATH DE number 810907