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

We address and answer the question of optimal lifting estimates for unimodular complex valued maps: given s > 0 and 1 < p < co, find the best possible estimate of the form broken vertical bar phi broken vertical bar W-s,W-p less than or similar to F (broken vertical bar e(I phi)broken vertical bar W-s,W-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 >= 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). (C) 2014 Elsevier Masson SAS. All rights reserved.