January  2021, 20(1): 369-388. doi: 10.3934/cpaa.2020271

On optimal autocorrelation inequalities on the real line

1. 

Department of Mathematics, University of California, Los Angeles, Portola Plaza 520, Los Angeles, California, 90095, USA

2. 

Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, RJ, 22460-320, Brazil

*Corresponding author

Received  August 2020 Revised  September 2020 Published  January 2021 Early access  November 2020

We study autocorrelation inequalities, in the spirit of Barnard and Steinerberger's work [1]. In particular, we obtain improvements on the sharp constants in some of the inequalities previously considered by these authors, and also prove existence of extremizers to these inequalities in certain specific settings. Our methods consist of relating the inequalities in question to other classical sharp inequalities in Fourier analysis, such as the sharp Hausdorff–Young inequality, and employing functional analysis as well as measure theory tools in connection to a suitable dual version of the problem to identify and impose conditions on extremizers.

Citation: José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure and Applied Analysis, 2021, 20 (1) : 369-388. doi: 10.3934/cpaa.2020271
References:
[1]

R.C. Barnard and S. Steinerberger, Three convolution inequalities on the real line with connection to additive combinatorics, J. Number Theory, 207 (2020), 42-55.  doi: 10.1016/j.jnt.2019.07.001.

[2]

W. Beckner, Inequalities in Fourier analysis, Ann. Math., 102 (1975), 159-182.  doi: 10.2307/1970980.

[3]

J. BourgainL. Clozel and J.P. Kahane, Principe d'Heisenberg et fonctions positives, Annales de l'institut Fourier, 60 (2010), 1215-1232. 

[4]

J. CillerueloI. Ruzsa and C. Trujillo, Upper and lower bounds for finite $Bh[g]$ sequences, J. Number Theory, 97 (2002), 26-34.  doi: 10.1006/jnth.2001.2767.

[5]

J. Cilleruelo, I. Ruzsa and C. Vinuesa, Generalized Sidon sets, Adv. Math., 225 (2010), 2786 –2807. doi: 10.1016/j.aim.2010.05.010.

[6]

A. Cloninger and S. Steinerberger, On suprema of autoconvolutions with an application to Sidon sets, P. Am. Math. Soc., 145 (2017), 3191-3200.  doi: 10.1090/proc/13690.

[7]

H. Cohn and F. Gonçalves, An optimal uncertainty principle in twelve dimensions via modular forms, Inventionnes Mathematicae, 217 (2019), 799-831.  doi: 10.1007/s00222-019-00875-4.

[8]

S. Fish, D. King and S. J. Miller, Extensions of Autocorrelation Inequalities with Applications to Additive Combinatorics, arXiv: 2001.02326.

[9] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, 2016. 
[10]

F. Gonçalves, D. Oliveira e Silva and J. P. G. Ramos, New sign uncertainty principles, Preprint, 2020.

[11]

F. Gonçalves, D. Oliveira e Silva and J. P. G. Ramos, On regularity and mass concentration phenomena for the sign uncertainty principle, Preprint, 2020.

[12]

F. GonçalvesD. Oliveira e Silva and S. Steinerberger, Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots, J. Math. Anal. Appl., 451 (2017), 678-711.  doi: 10.1016/j.jmaa.2017.02.030.

[13]

B. Green, The number of squares and $Bh[g]$ sets, Acta Arith., 100 (2001), 365-390.  doi: 10.4064/aa100-4-6.

[14]

G. Martin and K. O'Bryant, Constructions of generalized Sidon sets, J. Comb. Theory A, 113 (2006), 591-607.  doi: 10.1016/j.jcta.2005.04.011.

[15]

G. Martin and K. O'Bryant, The symmetric subset problem in continuous Ramsey theory, Exp. Math., 16 (2007), 145-166. 

[16]

M. Matolcsi and C. Vinuesa, Improved bounds on the supremum of autoconvolutions, J. Math. Anal. Appl., 372 (2010), 439-447.  doi: 10.1016/j.jmaa.2010.07.030.

[17]

G. Yu, An upper bound for $B2[g]$ sets, J. Number Theory, 122 (2007), 211-220.  doi: 10.1016/j.jnt.2006.04.008.

show all references

References:
[1]

R.C. Barnard and S. Steinerberger, Three convolution inequalities on the real line with connection to additive combinatorics, J. Number Theory, 207 (2020), 42-55.  doi: 10.1016/j.jnt.2019.07.001.

[2]

W. Beckner, Inequalities in Fourier analysis, Ann. Math., 102 (1975), 159-182.  doi: 10.2307/1970980.

[3]

J. BourgainL. Clozel and J.P. Kahane, Principe d'Heisenberg et fonctions positives, Annales de l'institut Fourier, 60 (2010), 1215-1232. 

[4]

J. CillerueloI. Ruzsa and C. Trujillo, Upper and lower bounds for finite $Bh[g]$ sequences, J. Number Theory, 97 (2002), 26-34.  doi: 10.1006/jnth.2001.2767.

[5]

J. Cilleruelo, I. Ruzsa and C. Vinuesa, Generalized Sidon sets, Adv. Math., 225 (2010), 2786 –2807. doi: 10.1016/j.aim.2010.05.010.

[6]

A. Cloninger and S. Steinerberger, On suprema of autoconvolutions with an application to Sidon sets, P. Am. Math. Soc., 145 (2017), 3191-3200.  doi: 10.1090/proc/13690.

[7]

H. Cohn and F. Gonçalves, An optimal uncertainty principle in twelve dimensions via modular forms, Inventionnes Mathematicae, 217 (2019), 799-831.  doi: 10.1007/s00222-019-00875-4.

[8]

S. Fish, D. King and S. J. Miller, Extensions of Autocorrelation Inequalities with Applications to Additive Combinatorics, arXiv: 2001.02326.

[9] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, 2016. 
[10]

F. Gonçalves, D. Oliveira e Silva and J. P. G. Ramos, New sign uncertainty principles, Preprint, 2020.

[11]

F. Gonçalves, D. Oliveira e Silva and J. P. G. Ramos, On regularity and mass concentration phenomena for the sign uncertainty principle, Preprint, 2020.

[12]

F. GonçalvesD. Oliveira e Silva and S. Steinerberger, Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots, J. Math. Anal. Appl., 451 (2017), 678-711.  doi: 10.1016/j.jmaa.2017.02.030.

[13]

B. Green, The number of squares and $Bh[g]$ sets, Acta Arith., 100 (2001), 365-390.  doi: 10.4064/aa100-4-6.

[14]

G. Martin and K. O'Bryant, Constructions of generalized Sidon sets, J. Comb. Theory A, 113 (2006), 591-607.  doi: 10.1016/j.jcta.2005.04.011.

[15]

G. Martin and K. O'Bryant, The symmetric subset problem in continuous Ramsey theory, Exp. Math., 16 (2007), 145-166. 

[16]

M. Matolcsi and C. Vinuesa, Improved bounds on the supremum of autoconvolutions, J. Math. Anal. Appl., 372 (2010), 439-447.  doi: 10.1016/j.jmaa.2010.07.030.

[17]

G. Yu, An upper bound for $B2[g]$ sets, J. Number Theory, 122 (2007), 211-220.  doi: 10.1016/j.jnt.2006.04.008.

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