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).

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

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

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

[4] J. Bourgain, H. Brezis, P. Mironescu, Limiting embedding theorems for $W^{s,p}$ when $s\uparrow 1$ and applications, J. Anal. Math. 87 (2002), 77 -101 | MR | Zbl

[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 | Numdam | MR | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl