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

Mironescu, Petru; Molnar, Ioana

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 5, pp. 965-1013

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 ${\left|\varphi \right|}_{W^{s,p}}\lesssim F\left({|e^{ı\varphi }|}_{W^{s,p}}\right)$.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 $0<s<1$ 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 $sp\ge 1$, 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).

Keywords: Unimodular maps, Lifting, Sobolev spaces

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