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Degree theory and BMO. II: Compact manifolds with boundaries. (Appendix with Petru Mironescu). (English) Zbl 0868.58017
In an earlier paper H. Brezis and L. Nirenberg [Sel. Math., New Ser. 1, No. 2, 197-263 (1995; Zbl 0852.58010)] studied the degree theory for VMO (vanishing mean oscillation) maps between compact \(n\)-dimensional oriented manifolds without boundaries.
In this paper they study a class of maps \(u\) from a bounded domain \(\Omega \subset \mathbb{R}^n\) into \(\mathbb{R}^n\). A real function \(f\in L^1_{\text{loc}}(\Omega)\) is said to be in BMO\((\Omega)\) (bounded mean oscillation) if \[ |f|_{\text{BMO}(\Omega)}:= \sup_B -\hskip -12.5pt \int_B |f--\hskip -12.5pt \int_B f|< \infty,\tag{\(\ast\)} \] where sup is taken over all balls with closure in \(\Omega\). Now VMO is the closure of \(C^0(\overline \Omega)\) in the BMO norm. In addition to the bounded domains they also consider domains \(\Omega\) in a smooth open \(n\)-dimensional Riemannian manifold \(X_0\). BMO\((\Omega)\) is defined as in \((\ast)\); the sup is now taken over geodesic balls \(B_\varepsilon(x)\) with \(x<r_0\), the injectivity radius of \(\overline\Omega\). Furthermore, the space BMO\((\Omega)\) is independent of the Riemannian metric on \(X_0\). VMO is defined as above. They then consider VMO maps of \(\Omega\) into an \(n\)-dimensional smooth open manifold \(Y\) (which is smoothly embedded in some \(\mathbb{R}^N\)). If \(X_0\) and \(Y\) are oriented, and \(p\in Y\) is such that, in a suitable sense, \(p\notin u(\partial\Omega)\) then they define by approximation \(\text{deg}(u,\Omega,p)\).
The content of the paper is as follows: In §II.1 BMO and VMO are introduced together with associated properties. §II.2 takes up the definition of degree, various properties of degree are established, the invariance of degree under continuous deformations in the BMO topology provided some additional assumptions is proved. In §II.3 the behaviour of functions in VMO\((\Omega)\) on \(\partial\Omega\) is considered. Section §II.4 extends to a certain class of maps a standard result for continuous maps \(u:\overline{\Omega} \to\mathbb{R}^n\) with \(u\mid_{\partial\Omega}=\varphi\), and with \(\varphi \neq p\) on \(\partial\Omega\) for some point \(p\in \mathbb{R}^n\) that \[ \text{deg}(u,\Omega,p)=\text{deg}\left(\frac{\varphi-p}{|\varphi-p|},\;\partial\Omega, S^{n-1}\right). \] Appendix 1 contains the proofs of results of §II.1. In Appendix 2 written with P. Mironescu the authors consider Toeplitz operators on \(S^1\). Appendix 3 deals with properties of the harmonic extension of BMO and VMO maps.
Reviewer: W.Mozgawa (Lublin)

MSC:
58C35 Integration on manifolds; measures on manifolds
57N65 Algebraic topology of manifolds
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