The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1259–1301 | Cite as

A Constrained Optimization Problem for the Fourier Transform: Quantitative Analysis

  • Dominique MaldagueEmail author


Among functions f majorized by indicator functions \(1_E\), which functions have maximal ratio \(\Vert {\widehat{f}}\Vert _q/|E|^{1/p}\)? We establish a quantitative answer to this question for exponents q sufficiently close to even integers, building on previous work proving the existence of such maximizers.


Harmonic analysis Extremization Fourier transform Quantitative analysis 

Mathematics Subject Classification

42A16 42B10 


  1. 1.
    Beckner, W.: Inequalities in Fourier analysis. Ann. Math. (2) 102(1), 159–182 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bianchi, G., Egnell, H.: A note on the Sobolev inequality. J. Funct. Anal. 100(1), 1824 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boas, R .P.: Entire Functions. Academic Press, New York (1954)zbMATHGoogle Scholar
  4. 4.
    Burchard, A.: Cases of equality in the Riesz rearrangement inequality. Ann. Math. (2) 143(3), 499–527 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, S., Frank, R.L., Weth, T.: Remainder terms in the fractional Sobolev inequality. Indiana Univ. Math. J. 62(4), 1381–1397 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Christ, M.: A sharpened Riesz-Sobolev inequality, arXiv:1706.02007, (2017)
  7. 7.
    Christ, M.: Near-extremizers of Young’s inequality for \(R^d\) , math.CA arXiv:1112.4875, to appear, Rev. Mat. Iberoamericana
  8. 8.
    Christ, M.: A sharpened Hausdorff-Young inequality, arXiv:1406.1210, (2014)
  9. 9.
    Christ, M.: On an extremization problem concerning Fourier coefficients, arXiv:1506.00153, (2015)
  10. 10.
    Cianchi, A., Fusco, N., Maggi, F., Pratelli, A.: The sharp Sobolev inequality in quantitative form. J. Eur. Math. Soc. (JEMS) 11(5), 1105–1139 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Figalli, A., Maggi, F., Pratelli, A.: A mass transportation approach to quantitative isoperimetric inequalities. Ann. Inst. H. Poincaré Anal. Non linéare 26(6), 2511–2519 (2009)CrossRefzbMATHGoogle Scholar
  12. 12.
    Fusco, N., Maggi, F., Pratelli, A.: The sharp quantitative isoperimetric inequality. Ann. Math. (2) 168(3), 941–980 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Frank, R.L., Lieb, E.H.: Sharp constants in several inequalities on the Heisenberg group. Ann. Math. 176(2), 349–381 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Lieb, E.: Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. Math. (2) 118(2), 349–374 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lieb, E.: Gaussian kernels have only Gaussian maximizers. Invent. Math. 102, 179–208 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Maldague, D.: A constrained optimization problem for the Fourier transform: existence of extremizers, arXiv:1802.01743, (2017)
  17. 17.
    Talenti, G.: Best constants in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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