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) no. 5, pp. 965-1013.

We address and answer the question of optimal lifting estimates for unimodular complex valued maps: given s>0 and 1p<, find the best possible estimate of the form |ϕ| W s,p F(|e ıϕ | 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 sp1, 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).

DOI : https://doi.org/10.1016/j.anihpc.2014.04.005
Mots clés : Unimodular maps, Lifting, Sobolev spaces
@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},
     pages = {965--1013},
     publisher = {Elsevier},
     volume = {32},
     number = {5},
     year = {2015},
     doi = {10.1016/j.anihpc.2014.04.005},
     zbl = {1339.46037},
     mrnumber = {3400439},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2014.04.005/}
}
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) no. 5, pp. 965-1013. doi : 10.1016/j.anihpc.2014.04.005. http://archive.numdam.org/articles/10.1016/j.anihpc.2014.04.005/

[1] R.A. Adams, Sobolev Spaces, Pure and Applied Mathematics vol. 65 , Academic Press, New York–London (1975) | MR 450957 | Zbl 0186.19101

[2] G. Bourdaud, Ondelettes et espaces de Besov, Rev. Mat. Iberoam. 11 no. 3 (1995), 477 -512 | EuDML 39488 | MR 1363202 | Zbl 0912.42024

[3] J. Bourgain, H. Brezis, P. Mironescu, Lifting in Sobolev spaces, J. Anal. Math. 80 (2000), 37 -86 | MR 1771523 | Zbl 0967.46026

[4] J. Bourgain, H. Brezis, P. Mironescu, Limiting embedding theorems for W s,p when s1 and applications, J. Anal. Math. 87 (2002), 77 -101 | MR 1945278 | Zbl 1029.46030

[5] J. Bourgain, H. Brezis, P. Mironescu, H 1/2 maps with values into the circle: minimal connections, lifting, and the Ginzburg–Landau equation, Publ. Math. Inst. Hautes Études Sci. 99 (2004), 1 -115 | EuDML 104206 | Numdam | MR 2075883 | Zbl 1051.49030

[6] J. Bourgain, H. Brezis, P. Mironescu, Lifting, degree, and distributional Jacobian revisited, Commun. Pure Appl. Math. 58 no. 4 (2005), 529 -551 | MR 2119868 | Zbl 1077.46023

[7] H. Brezis, P. Mironescu, Sobolev maps with values into the circle. Analytical, geometrical and topological aspects, in preparation.

[8] H. Brezis, P. Mironescu, Gagliardo–Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ. 1 no. 4 (2001), 387 -404 | MR 1877265 | Zbl 1023.46031

[9] H. Brezis, L. Nirenberg, Degree theory and BMO, Part I: Compact manifolds without boundaries, Sel. Math. New Ser. 1 no. 2 (1995), 197 -263 | MR 1354598 | Zbl 0852.58010

[10] H. Brezis, L. Nirenberg, Degree theory and BMO, Part II: Compact manifolds with boundaries, Sel. Math. New Ser. 2 (1996), 309 -368 | MR 1422201 | Zbl 0868.58017

[11] J.-Y. Chemin, Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and Its Applications vol. 14 , The Clarendon Press Oxford University Press, New York (1998) | MR 1688875 | Zbl 0927.76002

[12] J. Dávila, R. Ignat, Lifting of BV functions with values in S 1 , C. R. Math. Acad. Sci. Paris 337 no. 3 (2003), 159 -164 | MR 2001127 | Zbl 1046.46026

[13] J.B. Garnett, P.W. Jones, BMO from dyadic BMO, Pac. J. Math. 99 no. 2 (1982), 351 -371 | MR 658065 | Zbl 0516.46021

[14] A.M. Garsia, E. Rodemich, Monotonicity of certain functionals under rearrangement, Ann. Inst. Fourier (Grenoble) 24 no. 2 (1974), 67 -116 | EuDML 74178 | Numdam | MR 414802 | Zbl 0274.26006

[15] A.M. Garsia, E. Rodemich, H. Rumsey, A real variable lemma and the continuity of paths of some Gaussian processes, Indiana Univ. Math. J. 20 (1970/1971), 565 -578 | MR 267632 | Zbl 0252.60020

[16] L. Grafakos, Classical Fourier Analysis, Graduate Texts in Mathematics vol. 249 , Springer, New York (2008) | MR 2445437 | Zbl 1220.42001

[17] E.H. Lieb, M. Loss, Analysis, Graduate Studies in Mathematics , American Mathematical Society (2001) | MR 1817225 | Zbl 0966.26002

[18] B. Matei, P. Mironescu, On the sum-intersection property of classical function spaces, in preparation.

[19] V. Maz'Ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] vol. 342 , Springer, Heidelberg (2011) | MR 2777530 | Zbl 1217.46002

[20] B. Merlet, Two remarks on liftings of maps with values into S 1 , C. R. Math. Acad. Sci. Paris 343 no. 7 (2006), 467 -472 | MR 2267188 | Zbl 1115.46027

[21] P. Mironescu, Sobolev spaces of circle-valued maps, in preparation.

[22] P. Mironescu, Decomposition of 𝕊 1 -valued maps in Sobolev spaces, C. R. Math. Acad. Sci. Paris 348 no. 13–14 (2010), 743 -746 | MR 2671153 | Zbl 1205.46017

[23] P. Mironescu, 𝕊 1 -valued Sobolev mappings, J. Math. Sci. (N.Y.) 170 no. 3 (2010), 340 -355 | MR 2752641 | Zbl 1307.46024

[24] P. Mironescu, E. Russ, Traces and restrictions in function spaces. Old and new, in preparation.

[25] P. Mironescu, E. Russ, Y. Sire, Lifting in Besov spaces, in preparation.

[26] H.-M. Nguyen, Inequalities related to liftings and applications, C. R. Math. Acad. Sci. Paris 346 (2008), 957 -962 | MR 2449635 | Zbl 1157.46016

[27] T. Runst, W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications vol. 3 , Walter de Gruyter & Co., Berlin (1996) | MR 1419319 | Zbl 0873.35001

[28] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series vol. 43 , Princeton University Press, Princeton, NJ (1993) | MR 1232192 | Zbl 0821.42001

[29] H. Triebel, Theory of Function Spaces, Monographs in Mathematics vol. 78 , Birkhäuser Verlag, Basel (1983) | MR 781540 | Zbl 0546.46027

[30] S.V. Uspenskiĭ, Imbedding theorems for classes with weights, Tr. Mat. Inst. Steklova 60 (1961), 282 -303 | MR 136980 | Zbl 0198.46106

[31] M. Yamazaki, A quasihomogeneous version of paradifferential operators. II. A symbol calculus, J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 33 no. 2 (1986), 311 -345 | MR 866396 | Zbl 0659.47045