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

  • Petru Mironescu

    Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France
  • Ioana Molnar

    Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France

Abstract

We address and answer the question of optimal lifting estimates for unimodular complex valued maps: given and , find the best possible estimate of the form .

The most delicate case is . In this case, we extend the results obtained in [3,4] for (using Fourier analysis and optimal constants in the Sobolev embeddings) by developing non- 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 involving averaged martingale estimates.

Also when , we propose a new phase construction method, based on oscillations detection, and discuss existence of a bounded phase.

When , 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).

Cite this article

Petru Mironescu, Ioana Molnar, Phases of unimodular complex valued maps: optimal estimates, the factorization method, and the sum-intersection property of Sobolev spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 5, pp. 965–1013

DOI 10.1016/J.ANIHPC.2014.04.005