Degree theory and BMO. II: Compact manifolds with boundaries. (Appendix with Petru Mironescu).

*(English)*Zbl 0868.58017In 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.

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)

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\textit{H. Brézis} and \textit{L. Nirenberg}, Sel. Math., New Ser. 2, No. 3, 309--368 (1996; Zbl 0868.58017)

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##### References:

[1] | M. Avellaneda and F. H. Lin [1]. Compactness methods in the theory of homogenization.Comm. Pure Appl. Math. 40 (1987), 803–847. · Zbl 0632.35018 |

[2] | H. Brezis and L. Nirenberg [1] (=[BNI]). Degree theory and BMO; Part I: Compact Manifolds without Boundaries.Selecta Mathematica, New Series 1 (1995), 197–263. · Zbl 0852.58010 |

[3] | R. G. Douglas [1]. Banach Algebra Techniques in Operator Theory.Acad. Press (1972). · Zbl 0247.47001 |

[4] | R. G. Douglas [2]. Banach Algebra Techniques in the Theory of Toeplitz Operators.C.B.M.S., Vol. 15, Amer. Math. Soc. (1973). · Zbl 0252.47025 |

[5] | C. Fefferman [1]. Characterizations of bounded mean oscillation.Bull. Amer. Math. Soc. 77 (1971), 587–588. · Zbl 0229.46051 |

[6] | C. Fefferman and E. Stein [1].H p spaces of several variables.Acta Math. 129 (1972), 137–193. · Zbl 0257.46078 |

[7] | I. Gohberg and N. Krupnik [1].One Dimensional Linear Singular Integral Equations. Vol. II, Birkhäuser (1992). · Zbl 0781.47038 |

[8] | L. Greco, T. Iwaniec, C. Sbordone and B. Stroffolini [1]. Degree formulas for maps with nonintegrable Jacobian (to appear). · Zbl 0854.58005 |

[9] | P. Jones [1]. Extension Theorems for BMO.Indiana Univ. Math. J. 29 (1980), 41–66. · Zbl 0432.42017 |

[10] | S. Lang [1].Real and Functional Analysis. Springer (third edition, 1993). · Zbl 0831.46001 |

[11] | D. Sarason [1]. Functions of vanishing mean oscillation.Trans. AMS 207 (1975), 391–405. · Zbl 0319.42006 |

[12] | D. Sarason [2]. Toeplitz operators with semi-almost periodic symbols.Duke Math. J. 44 (1977), 357–364. · Zbl 0356.47018 |

[13] | D. Sarason [3]. Toeplitz operators with piecewise quasicontinuous symbols.Indiana Univ. Math. J. 26 (1977), 817–838. · Zbl 0359.47016 |

[14] | D. Sarason [4]. Function Theory on the Unit Circle.Lecture notes, Virginia Polytech. Inst. (1978). · Zbl 0398.30027 |

[15] | E. Stein [1]. Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals.Princeton Univ. Press (1993). · Zbl 0821.42001 |

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