Kakeya–Brascamp–Lieb inequalities


We prove a sharp common generalization of endpoint multilinear Kakeya and local discrete Brascamp–Lieb inequalities.

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  1. 1.

    Bennett, J., Bez, N., Gutiérrez, S.: Transversal multilinear Radon-like transforms: local and global estimates. Rev. Mat. Iberoam. 29(3), 765–788 (2013). https://doi.org/10.4171/RMI/739

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Bennett, J., Carbery, A., Tao, T.: On the multilinear restriction and Kakeya conjectures. Acta Math. 196(2), 261–302 (2006). https://doi.org/10.1007/s11511-006-0006-4. arXiv:math/0509262

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Bourgain, J., Demeter, C.: The proof of the \(l^2\) decoupling conjecture. Ann. Math. (2) 182(1), 351–389 (2015). https://doi.org/10.4007/annals.2015.182.1.9. arXiv:1403.5335 [math.CA]

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Bourgain, J., Demeter, C.: A study guide for the \(l^2\) decoupling theorem. Chin. Ann. Math. Ser. B 38(1), 173–200 (2017). https://doi.org/10.1007/s11401-016-1066-1. arXiv:1604.06032 [math.CA]

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Bourgain, J., Demeter, C., Guth, L.: Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three. Ann. Math. (2) 184(2), 633–682 (2016). https://doi.org/10.4007/annals.2016.184.2.7. arXiv:1512.01565 [math.NT]

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Bennett, J., et al.: Stability of the Brascamp–Lieb constant and applications. Am. J. Math. 140(2), 543–569 (2018). https://doi.org/10.1353/ajm.2018.0013. arXiv:1508.07502 [math.CA]

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Bennett, J., et al.: The Brascamp–Lieb inequalities: finiteness, structure and extremals. Geom. Funct. Anal. 17(5), 1343–1415 (2008). https://doi.org/10.1007/s00039-007-0619-6. arXiv:math/0505065

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Bennett, J., et al.: Finite bounds for Hölder–Brascamp–Lieb multilinear inequalities. Math. Res. Lett. 17(4), 647–666 (2010). https://doi.org/10.4310/MRL.2010.v17.n4.a6. arXiv:math/0505691

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Bennett, J., et al.: Behaviour of the Brascamp–Lieb constant. Bull. Lond. Math. Soc. 49(3), 512–518 (2017). https://doi.org/10.1112/blms.12049. arXiv:1605.08603 [math.CA]

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Bourgain, J., Guth, L.: Bounds on oscillatory integral operators based on multilinear estimates. Geom. Funct. Anal. 21(6), 1239–1295 (2011). https://doi.org/10.1007/s00039-011-0140-9. arXiv:1012.3760 [math.CA]

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Brascamp, H.J., Lieb, E.H.: Best constants in Young’s inequality, its converse, and its generalization to more than three functions. Adv. Math. 20(2), 151–173 (1976). https://doi.org/10.1016/0001-8708(76)90184-5

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Carbery, A., Hänninen, T. S., Valdimarsson, S.I.: Multilinear duality and factorisation for Brascamp–Lieb-type inequalities with applications. Preprint (2018). arXiv:1809.02449 [math.FA]

  13. 13.

    Carbery, A., Valdimarsson, S.I.: The endpoint multilinear Kakeya theorem via the Borsuk–Ulam theorem. J. Funct. Anal. 264(7), 1643–1663 (2013). https://doi.org/10.1016/j.jfa.2013.01.012. arXiv:1205.6371 [math.CA]

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Finner, H.: A generalization of Hölder’s inequality and some probability inequalities. Ann. Probab. 20(4), 1893–1901 (1992)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Fulton, W.: Intersection Theory, 2nd edn. Springer, Berlin (1998). ISBN: 978-0-387-98549-7. https://doi.org/10.1007/978-1-4612-1700-8

  16. 16.

    Garg, A., et al.: Algorithmic and optimization aspects of Brascamp–Lieb inequalities, via operator scaling. Geom. Funct. Anal. 28(1), 100–145 (2018). https://doi.org/10.1007/s00039-018-0434-2. arXiv:1607.06711 [cs.CC]

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Guth, L.: The endpoint case of the Bennett–Carbery–Tao multilinear Kakeya conjecture. Acta Math. 205(2), 263–286 (2010). https://doi.org/10.1007/s11511-010-0055-6. arXiv:0811.2251 [math.CA]

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Guth, L.: A short proof of the multilinear Kakeya inequality. Math. Proc. Cambr. Philos. Soc. 158(1), 147–153 (2015). https://doi.org/10.1017/S0305004114000589. arXiv:1409.4683 [math.AP]

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Guo, S., Zorin-Kranich, P.: Decoupling for moment manifolds associated to Arkhipov–Chubarikov–Karatsuba systems. Adv. Math. (2019). https://doi.org/10.1016/j.aim.2019.106889

  20. 20.

    John, F.: Extremum problems with inequalities as subsidiary conditions. In: Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, pp. 187–204. Interscience Publishers, Inc., New York (1948)

  21. 21.

    Lieb, E.H.: Gaussian kernels have only Gaussian maximizers. Invent. Math. 102(1), 179–208 (1990). https://doi.org/10.1007/BF01233426

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Loomis, L.H., Whitney, H.: An inequality related to the isoperimetric inequality. Bull. Am. Math. Soc 55, 961–962 (1949). https://doi.org/10.1090/S0002-9904-1949-09320-5

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Maldague, D.: Weak Hölder–Brascamp–Lieb inequalities. Preprint (2019). arXiv:1904.06450 [math.CA]

  24. 24.

    Matouek, J.: Using the Borsuk–Ulam theorem. Universitext. Lectures on topological methods in combinatorics and geometry, Written in cooperation with Anders Björner and Günter M. Ziegler. Springer, Berlin (2003). ISBN: 3-540-00362-2. https://doi.org/10.1007/978-3-540-76649-0

  25. 25.

    Rogers, C.A., Shephard, G.C.: Convex bodies associated with a given convex body. J. Lond. Math. Soc. 33, 270–281 (1958). https://doi.org/10.1112/jlms/s1-33.3.270

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Schep, A.R.: Factorization of positive multilinear maps. Illinois J. Math. 28(4), 579–591 (1984)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Zhang, R.: The endpoint perturbed Brascamp–Lieb inequalities with examples. Anal. PDE 11(3), 555–581 (2018). https://doi.org/10.2140/apde.2018.11.555. arXiv:1510.09132 [math.CA]

    MathSciNet  Article  MATH  Google Scholar 

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The extension of Theorem 1.3 beyond the scale-invariant case (1.9) is motivated by ongoing joint work with Shaoming Guo and Ruixiang Zhang. This work was partially supported by the Hausdorff Center for Mathematics (DFG EXC 2047). I thank the anonymous referees for numerous corrections and helpful suggestions pertaining to exposition.

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Correspondence to Pavel Zorin-Kranich.

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Zorin-Kranich, P. Kakeya–Brascamp–Lieb inequalities. Collect. Math. 71, 471–492 (2020). https://doi.org/10.1007/s13348-019-00273-2

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Mathematics Subject Classification

  • Primary 26D15
  • Secondary 42B99
  • 52C07