A sharp nonlinear Hausdorff–Young inequality for small potentials

Kovač, Vjekoslav; Oliveira e Silva, Diogo; Rupčić, Jelena

doi:10.1090/proc/14268

A sharp nonlinear Hausdorff–Young inequality for small potentials
HTML articles powered by AMS MathViewer

by Vjekoslav Kovač, Diogo Oliveira e Silva and Jelena Rupčić PDF

Proc. Amer. Math. Soc. 147 (2019), 239-253 Request permission

Abstract:

The nonlinear Hausdorff–Young inequality follows from the work of Christ and Kiselev. Later Muscalu, Tao, and Thiele asked if the constants can be chosen independently of the exponent. We show that the nonlinear Hausdorff–Young quotient admits an even better upper bound than the linear one, provided that the function is sufficiently small in the $\operatorname {L}^1$-norm. The proof combines perturbative techniques with the sharpened version of the linear Hausdorff–Young inequality due to Christ.

References

Mark J. Ablowitz, David J. Kaup, Alan C. Newell, and Harvey Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Studies in Appl. Math. 53 (1974), no. 4, 249–315. MR 450815, DOI 10.1002/sapm1974534249
K. I. Babenko, An inequality in the theory of Fourier integrals, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 531–542 (Russian). MR 0138939
William Beckner, Inequalities in Fourier analysis, Ann. of Math. (2) 102 (1975), no. 1, 159–182. MR 385456, DOI 10.2307/1970980
M. Christ, A sharpened Hausdorff–Young inequality, preprint (2014), available at arXiv:1406.1210.
Michael Christ and Alexander Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), no. 2, 409–425. MR 1809116, DOI 10.1006/jfan.2000.3687
Michael Christ and Alexander Kiselev, WKB asymptotic behavior of almost all generalized eigenfunctions for one-dimensional Schrödinger operators with slowly decaying potentials, J. Funct. Anal. 179 (2001), no. 2, 426–447. MR 1809117, DOI 10.1006/jfan.2000.3688
Vjekoslav Kovač, Uniform constants in Hausdorff-Young inequalities for the Cantor group model of the scattering transform, Proc. Amer. Math. Soc. 140 (2012), no. 3, 915–926. MR 2869075, DOI 10.1090/S0002-9939-2011-11078-5
Elliott H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math. 102 (1990), no. 1, 179–208. MR 1069246, DOI 10.1007/BF01233426
Camil Muscalu and Wilhelm Schlag, Classical and multilinear harmonic analysis. Vol. II, Cambridge Studies in Advanced Mathematics, vol. 138, Cambridge University Press, Cambridge, 2013. MR 3052499
Camil Muscalu, Terence Tao, and Christoph Thiele, A Carleson theorem for a Cantor group model of the scattering transform, Nonlinearity 16 (2003), no. 1, 219–246. MR 1950785, DOI 10.1088/0951-7715/16/1/314
Camil Muscalu, Terence Tao, and Christoph Thiele, A counterexample to a multilinear endpoint question of Christ and Kiselev, Math. Res. Lett. 10 (2003), no. 2-3, 237–246. MR 1981900, DOI 10.4310/MRL.2003.v10.n2.a10
T. Tao, An introduction to the nonlinear Fourier transform (2002), unpublished note, available at http://www.math.ucla.edu/~tao/preprints/Expository/nonlinear_ft.dvi .
T. Tao and C. Thiele, Nonlinear Fourier Analysis, IAS/Park City Graduate Summer School, unpublished lecture notes (2003), available at arXiv:1201.5129.
V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Ž. Èksper. Teoret. Fiz. 61 (1971), no. 1, 118–134 (Russian, with English summary); English transl., Soviet Physics JETP 34 (1972), no. 1, 62–69. MR 0406174