Phases of unimodular complex valued maps: optimal estimates, the factorization method, and the sum-intersection property of Sobolev spaces
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 5, p. 965-1013
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We address and answer the question of optimal lifting estimates for unimodular complex valued maps: given s>0 and 1p<, find the best possible estimate of the form |ϕ| W s,p F(|e ıϕ | W s,p ).The most delicate case is sp<1. In this case, we extend the results obtained in [3,4] for p=2 (using L 2 Fourier analysis and optimal constants in the Sobolev embeddings) by developing non-L 2 estimates and an approach based on symmetrization. Following an idea of Bourgain (presented in [3]), our proof also relies on averaged estimates for martingales. As a byproduct of our arguments, we obtain a characterization of fractional Sobolev spaces with 0<s<1 involving averaged martingale estimates.Also when sp<1, we propose a new phase construction method, based on oscillations detection, and discuss existence of a bounded phase.When sp1, we extend to higher dimensions a result on optimal estimates of Merlet [20], based on one-dimensional arguments. This extension requires new ingredients (factorization techniques, duality methods).

DOI : https://doi.org/10.1016/j.anihpc.2014.04.005
Keywords: Unimodular maps, Lifting, Sobolev spaces
@article{AIHPC_2015__32_5_965_0,
     author = {Mironescu, Petru and Molnar, Ioana},
     title = {Phases of unimodular complex valued maps: optimal estimates, the factorization method, and the sum-intersection property of Sobolev spaces},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {32},
     number = {5},
     year = {2015},
     pages = {965-1013},
     doi = {10.1016/j.anihpc.2014.04.005},
     zbl = {1339.46037},
     mrnumber = {3400439},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2015__32_5_965_0}
}
Mironescu, Petru; Molnar, Ioana. Phases of unimodular complex valued maps: optimal estimates, the factorization method, and the sum-intersection property of Sobolev spaces. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 5, pp. 965-1013. doi : 10.1016/j.anihpc.2014.04.005. http://www.numdam.org/item/AIHPC_2015__32_5_965_0/

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