A variation on Hölder–Brascamp–Lieb inequalities
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Abstract:
The class of Hölder–Brascamp–Lieb inequalities is a collection of multilinear inequalities generalizing a convolution inequality of Young and the Loomis–Whitney inequalities. The full range of exponents was classified in a paper of Bennett, Carbery, Christ, and Tao [Math. Res. Lett. 17 (2010), no. 4, 647–666]. In a setting similar to that of Ivanisvili and Volberg [J. Lond. Mat. Sci (2) (2015), no. 3, 657–674], we introduce a notion of size for these inequalities which generalizes $L^p$ norms. Under this new setup, we then determine necessary and sufficient conditions for a generalized Hölder–Brascamp–Lieb-type inequality to hold and establish sufficient conditions for maximizers to exist when the underlying linear maps match those of the convolution inequality of Young.References
- Franck Barthe, On a reverse form of the Brascamp-Lieb inequality, Invent. Math. 134 (1998), no. 2, 335–361. MR 1650312, DOI 10.1007/s002220050267
- William Beckner, Inequalities in Fourier analysis, Ann. of Math. (2) 102 (1975), no. 1, 159–182. MR 385456, DOI 10.2307/1970980
- Jonathan Bennett, Anthony Carbery, Michael Christ, and Terence Tao, The Brascamp-Lieb inequalities: finiteness, structure and extremals, Geom. Funct. Anal. 17 (2008), no. 5, 1343–1415. MR 2377493, DOI 10.1007/s00039-007-0619-6
- Jonathan Bennett, Anthony Carbery, Michael Christ, and Terence Tao, Finite bounds for Hölder-Brascamp-Lieb multilinear inequalities, Math. Res. Lett. 17 (2010), no. 4, 647–666. MR 2661170, DOI 10.4310/MRL.2010.v17.n4.a6
- Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
- Herm Jan Brascamp and Elliott H. Lieb, Best constants in Young’s inequality, its converse, and its generalization to more than three functions, Advances in Math. 20 (1976), no. 2, 151–173. MR 412366, DOI 10.1016/0001-8708(76)90184-5
- Almut Burchard and Hichem Hajaiej, Rearrangement inequalities for functionals with monotone integrands, J. Funct. Anal. 233 (2006), no. 2, 561–582. MR 2214588, DOI 10.1016/j.jfa.2005.08.010
- E. A. Carlen, E. H. Lieb, and M. Loss, A sharp analog of Young’s inequality on $S^N$ and related entropy inequalities, J. Geom. Anal. 14 (2004), no. 3, 487–520. MR 2077162, DOI 10.1007/BF02922101
- Michael Christ, Near-extremizers of Young’s inequality for Euclidean groups, Rev. Mat. Iberoam. 35 (2019), no. 7, 1925–1972. MR 4029788, DOI 10.4171/rmi/1055
- J. A. Crowe, J. A. Zweibel, and P. C. Rosenbloom, Rearrangements of functions, J. Funct. Anal. 66 (1986), no. 3, 432–438. MR 839110, DOI 10.1016/0022-1236(86)90067-4
- James Demmel and Alex Rusciano, Parallelepipeds obtaining HBL lower bounds, arXiv:1611.05944, Nov 2016.
- Thomas S. Fergusonm Linear programming, https://www.math.ucla.edu/~tom/lp.pdf
- P. Ivanisvili and A. Volberg, Hessian of Bellman functions and uniqueness of the Brascamp-Lieb inequality, J. Lond. Math. Soc. (2) 92 (2015), no. 3, 657–674. MR 3431655, DOI 10.1112/jlms/jdv040
- Stanisław Saks, Theory of the integral, Second revised edition, Dover Publications, Inc., New York, 1964. English translation by L. C. Young; With two additional notes by Stefan Banach. MR 0167578
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- Stefán Ingi Valdimarsson, The Brascamp-Lieb polyhedron, Canad. J. Math. 62 (2010), no. 4, 870–888. MR 2674705, DOI 10.4153/CJM-2010-045-2
Additional Information
- Kevin O’Neill
- Affiliation: Department of Mathematics, University of California, Berkeley, Berkeley, California 94720-3840
- Address at time of publication: Department of Mathematics, University of California, Davis, One Shields Ave, Davis, California 95616
- MR Author ID: 1013580
- Email: oneill@math.berkeley.edu; oneill@math.ucdavis.edu
- Received by editor(s): August 27, 2018
- Received by editor(s) in revised form: November 27, 2018, August 14, 2019, and October 28, 2019
- Published electronically: May 26, 2020
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 5467-5489
- MSC (2010): Primary 42B99; Secondary 26D15, 42B35
- DOI: https://doi.org/10.1090/tran/8070
- MathSciNet review: 4127883