Phases of unimodular complex valued maps: optimal estimates, the factorization method, and the sum-intersection property of Sobolev spaces

Mironescu, Petru; Molnar, Ioana

Mironescu, Petru ; Molnar, Ioana

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015), p. 965-1013 / Harvested from Numdam

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Résumé

We address and answer the question of optimal lifting estimates for unimodular complex valued maps: given $s > 0$ and $1 \leq p < \infty$ , find the best possible estimate of the form ${| ϕ |}_{W^{s, p}} ≲ F ({| e^{ı ϕ} |}_{W^{s, p}})$ .The most delicate case is $s p < 1$ . 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 $s p < 1$ , we propose a new phase construction method, based on oscillations detection, and discuss existence of a bounded phase.When $s p \geq 1$ , 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).

Publié le : 2015-01-01

MR 3400439 | Zbl 1339.46037

DOI : https://doi.org/10.1016/j.anihpc.2014.04.005

@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},
     volume = {32},
     year = {2015},
     pages = {965-1013},
     doi = {10.1016/j.anihpc.2014.04.005},
     mrnumber = {3400439},
     zbl = {1339.46037},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 32 (2015) pp. 965-1013. doi : 10.1016/j.anihpc.2014.04.005. http://gdmltest.u-ga.fr/item/AIHPC_2015__32_5_965_0/

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