BCAM COURSE ON HARMONIC ANALYSIS: BRASCAMP–LIEB INEQUALITIES

Speaker: Marco Fraccaroli (BCAM)

Dates: 24-28 February, 2025. Monday to Thursday 15:00 - 17:00, Friday 10:00-12:00. 

The classical Brascamp–Lieb inequalities describe the interaction between geometric means and the integration of functions composed with linear maps. This general approach provides a unified framework to the study of classical inequalities such as Hölder, Loomis–Whitney, and Young ones, originating from the work of Brascamp and Lieb. In their seminal paper Bennett, Christ, Carbery, and Tao proved that the Brascamp–Lieb inequalities hold true if and only if certain simple linear algebraic conditions involving the exponents appearing in the geometric means and the kernels/images of the linear maps are satisfied. Many different research directions stemmed from this starting point. For example, we can replace some of the functions with singular kernels which barely defy the required integrability conditions but come with good additional properties, e.g. cancellation. A second variation on the theme is obtained replacing the linear maps with more general submersions which locally can be approximated by the formers.

References

• [Bar98] F. Barthe. On a reverse form of the Brascamp-Lieb inequality. Invent. Math., 134(2):335–361, 1998.
• [BBB+20] J. Bennett, N. Bez, S. Buschenhenke, M. G. Cowling, and T. C. Flock. On the nonlinear Brascamp-Lieb inequality. Duke Math. J., 169(17):3291–3338, 2020.
• [BBG13] J. Bennett, N. Bez, and S. Gutierrez. Global nonlinear Brascamp-Lieb inequalities.  J. Geom. Anal., 23(4):1806–1817, 2013.
• [BH09] F. Barthe and N. Huet. On Gaussian Brunn-Minkowski inequalities. Studia Math., 191(3):283–304, 2009.
• [BT23] J. Bennett and T. Tao. Adjoint brascamp-lieb inequalities. arXiv preprint arXiv:2306.16558, 2023.
• [CHV23] A. Carbery, T. S. Hanninen, and S. I. Valdimarsson. Multilinear duality and factorisation for Brascamp-Lieb- type inequalities. J. Eur. Math. Soc. (JEMS), 25(6):2057–2125, 2023.
• [DST22] P. Durcik, L. Slavíkova, and C. iele. Local bounds for singular Brascamp-Lieb forms with cubical structure. Math. Z., 302(4):2375–2405, 2022.
• [DT20] P. Durcik and C. Thiele. Singular Brascamp-Lieb inequalities with cubical structure. Bull. Lond. Math. Soc., 52(2):283–298, 2020.
• [DT21] P. Durcik and C. iele. Singular Brascamp-Lieb: a survey. In Geometric aspects of harmonic analysis, volume 45 of Springer INdAM Ser., pages 321–349. Springer, Cham, [2021] ©2021.
• [Dun21] J. Duncan. An algebraic Brascamp-Lieb inequality. J. Geom. Anal., 31(10):10136–10163, 2021.