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 $1\le p<\infty$, find the best possible estimate of the form ${|\varphi |}_{{W}^{s,p}}\lesssim F\left({|{e}^{ı\varphi }|}_{{W}^{s,p}}\right)$.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 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 $sp\ge 1$, 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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