Two quantitative versions of the Nonlinear Carleson Conjecture

Sergey A. Denisov

doi:10.5802/crmath.806

Research article - Harmonic analysis

Two quantitative versions of the Nonlinear Carleson Conjecture

Sergey A. Denisov ¹

¹University of Wisconsin – Madison, Department of Mathematics, 480 Lincoln Dr., Madison, WI, 53706, USA

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1533-1541

Abstract (Original language)
Résumé

In this note, we state and compare two quantitative versions of the Nonlinear Carleson Conjecture (NCC). We provide motivations for our conjectures and show that they both imply the NCC. We also obtain some applications to the zero distribution of polynomials orthogonal on the unit circle and to their pointwise asymptotics.

Dans cette note, nous formulons et comparons deux versions quantitatives de la conjecture non linéaire de Carleson (NCC). Nous en exposons les motivations et montrons qu’elles impliquent toutes deux la NCC. Nous en déduisons également certaines applications à la distribution des zéros des polynômes orthogonaux sur le cercle unité, ainsi qu’à leurs asymptotiques ponctuelles.

Received: 2025-05-11
Revised: 2025-11-10
Accepted: 2025-11-10
Published online: 2025-12-12

DOI: 10.5802/crmath.806

Classification: 42C05, 30H35
Keywords: Maximal function, variational norm, product of matrices, nonlinear Carleson conjecture, orthogonal polynomials, Schur functions
Mots-clés : Fonction maximale, norme variationnelle, produit de matrices, conjecture non linéaire de Carleson, polynômes orthogonaux, fonctions de Schur

Author's affiliations:

Sergey A. Denisov ¹

¹ University of Wisconsin – Madison, Department of Mathematics, 480 Lincoln Dr., Madison, WI, 53706, USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Sergey A. Denisov},
     title = {Two quantitative versions of the {Nonlinear} {Carleson} {Conjecture}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1533--1541},
     year = {2025},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     doi = {10.5802/crmath.806},
     language = {en},
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Sergey A. Denisov. Two quantitative versions of the Nonlinear Carleson Conjecture. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1533-1541. doi: 10.5802/crmath.806

References
Cited by

[1] Michel Alexis; Gevorg Mnatsakanyan; Christoph Thiele Quantum signal processing and nonlinear Fourier analysis, Rev. Mat. Complut., Volume 37 (2024) no. 3, pp. 655-694 | DOI | MR | Zbl

[2] Roman Bessonov; Sergey Denisov Zero sets, entropy, and pointwise asymptotics of orthogonal polynomials, J. Funct. Anal., Volume 280 (2021) no. 12, 109002, 38 pages | DOI | MR | Zbl

[3] Michael Christ; Alexander Kiselev Maximal functions associated to filtrations, J. Funct. Anal., Volume 179 (2001) no. 2, pp. 409-425 | DOI | MR | Zbl

[4] Michael Christ; Alexander Kiselev WKB and spectral analysis of one-dimensional Schrödinger operators with slowly varying potentials, Commun. Math. Phys., Volume 218 (2001) no. 2, pp. 245-262 | DOI | MR | Zbl

[5] Michael Christ; Alexander Kiselev WKB asymptotic behavior of almost all generalized eigenfunctions for one-dimensional Schrödinger operators with slowly decaying potentials, J. Funct. Anal., Volume 179 (2001) no. 2, pp. 426-447 | DOI | MR | Zbl

[6] Sergey Denisov; Liban Mohamed Generalizations of Menchov–Rademacher theorem and existence of wave operators in Schrödinger evolution, Can. J. Math., Volume 73 (2021) no. 2, pp. 360-382 | DOI | MR | Zbl

[7] John B. Garnett Bounded analytic functions, Pure and Applied Mathematics, 96, Academic Press Inc., 1981 | MR | Zbl

[8] Miguel de Guzmán Real variable methods in Fourier analysis, North-Holland Mathematics Studies, North-Holland, 1981 no. 46 | MR | Zbl

[9] Ole G. Jørsboe; Leif Mejlbro The Carleson–Hunt theorem on Fourier series, Lecture Notes in Mathematics, 911, Springer, 1982 | MR | DOI | Zbl

[10] B. S. Kashin; A. A. Saakyan Orthogonal series (Ben Silver, ed.), Translations of Mathematical Monographs, 75, American Mathematical Society, 1989 | DOI | MR | Zbl

[11] Sergei Khrushchev Schur’s algorithm, orthogonal polynomials, and convergence of Wall’s continued fractions in $L^{2} (𝕋)$ , J. Approx. Theory, Volume 108 (2001) no. 2, pp. 161-248 | DOI | MR | Zbl

[12] Vjekoslav Kovač; Diogo Oliveira e Silva; Jelena Rupčić A sharp nonlinear Hausdorff–Young inequality for small potentials, Proc. Am. Math. Soc., Volume 147 (2019) no. 1, pp. 239-253 | MR | DOI | Zbl

[13] Ferenc Móricz; Károly Tandori An improved Menshov–Rademacher theorem, Proc. Am. Math. Soc., Volume 124 (1996) no. 3, pp. 877-885 | MR | DOI | Zbl

[14] Camil Muscalu; Terence Tao; Christoph Thiele A counterexample to a multilinear endpoint question of Christ and Kiselev, Math. Res. Lett., Volume 10 (2003) no. 2–3, pp. 237-246 | DOI | MR | Zbl

[15] Richard Oberlin; Andreas Seeger; Terence Tao; Christoph Thiele; James Wright A variation norm Carleson theorem, J. Eur. Math. Soc., Volume 14 (2012) no. 2, pp. 421-464 | DOI | MR | Zbl

[16] Barry Simon Orthogonal polynomials on the unit circle. Part 1, Colloquium Publications, 54, American Mathematical Society, 2005 | DOI | MR

[17] E. M. Stein On limits of seqences of operators, Ann. Math. (2), Volume 74 (1961), pp. 140-170 | DOI | MR | Zbl

[18] P. K. Suetin V. A. Steklov’s problem in the theory of orthogonal polynomials, Mathematical analysis, Vol. 15 (Russian) (Itogi Nauki i Tekhniki), Akademii Nauk SSSR, 1977, pp. 5-82 | MR | Zbl

[19] John Sylvester; Dale P. Winebrenner Linear and nonlinear inverse scattering, SIAM J. Appl. Math., Volume 59 (1999) no. 2, pp. 669-699 | DOI | MR | Zbl

[20] Terence Tao; Christoph Thiele; Ya-Ju Tsai The nonlinear Fourier transform https://www.math.uni-bonn.de/... (Unpublished lecture notes)

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