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Lifting, degree, and distributional Jacobian revisited
Author(s) -
Bourgain Jean,
Brezis Haïm,
Mironescu Petru
Publication year - 2005
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20063
Subject(s) - mathematics , marie curie , degree (music) , library science , humanities , algebra over a field , physics , philosophy , computer science , pure mathematics , european union , acoustics , business , economic policy
Let g : I = (0, 1) → S1. If g ∈ VMO, we may write g = eiφ for some φ ∈ VMO; this φ is unique modulo 2π (see [13] and the earlier work [14]). There is no control of |φ|BMO in terms of |g|BMO, since we always have |g|BMO ≤ 2 and |φ|BMO can be arbitrarily large; recall, however, that, when |g|BMO is sufficiently small, there is a linear estimate |φ|BMO ≤ C |g|BMO (see [13, theorem 4], [14], and Remark 0.2 below). We are going to establish that a norm slightly stronger than |g|BMO does control |φ|BMO. Consider, for 1 < p < ∞, 0 < s < 1, the fractional Sobolev space Ws,p(I ), equipped with its standard seminorm