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

Abstract

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 $|\phi {|}_{W^{s,p}}\lesssim F(|e^{ı\phi }{|}_{W^{s,p}})$.

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 $sp\ge 1$, 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).