### Abstract

H. Brezis and L. Nirenberg proved that if (gk) ⊂ C^{0} (S^{N},S^{N}) and g ∈ C^{0} (S^{N},S^{N}) (N ≥ 1) are such that g_{k} → g in BMO(S^{N}), then deg g_{k} →deg g. On the other hand, if g ∈ C^{1} (S^{N},S^{N}), then Kronecker's formula asserts that deg. Consequently, converges to provided g_{k} → g in BMO(S^{N}). In the same spirit, we consider the quantity, for all ψ ∈ C^{1} (S^{N},R) and study the convergence of J(g_{k},ψ). In particular, we prove that J(g_{k},ψ) converges to J(g,ψ) for any ψ ∈ C^{1} (S^{N},R) if g_{k} converges to g in C^{0,α}(S^{N}) for some α >. Surprisingly, this result is "optimal" when N > 1. In the case N = 1 we prove that if g_{k} → g almost everywhere and lim sup_{k→∞}|g_{k} - g|BMO is sufficiently small, then J(g_{k},ψ) → J(g,ψ) for any ψ ∈ C^{1} (S^{1}, R). We also establish bounds for J(g,ψ) which are motivated by the works of J. Bourgain, H. Brezis, and H.-M. Nguyen and H.-M. Nguyen. We pay special attention to the case N = 1.

Original language | English (US) |
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Pages (from-to) | 1141-1183 |

Number of pages | 43 |

Journal | Annals of Mathematics |

Volume | 173 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1 2011 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

^{N}into S

^{N}in fractional Sobolev and Ḧolder spaces.

*Annals of Mathematics*,

*173*(2), 1141-1183. https://doi.org/10.4007/annals.2011.173.2.15