TY - JOUR
T1 - Euclidean Forward–Reverse Brascamp–Lieb Inequalities
T2 - Finiteness, Structure, and Extremals
AU - Courtade, Thomas A.
AU - Liu, Jingbo
N1 - Funding Information:
This work was supported by NSF Grants CCF-1704967, CCF-0939370 and CCF-1750430.
Funding Information:
The first author is grateful for discussions with Franck Barthe, Michael Christ, Varun Jog, Joseph Lehec, and Pawe? Wolff. In particular, he thanks Franck Barthe for drawing our attention to references [5 , 6], and Pawe? Wolff for explaining the connections to the forward?reverse Brascamp?Lieb inequality.
Publisher Copyright:
© 2020, Mathematica Josephina, Inc.
PY - 2020
Y1 - 2020
N2 - A new proof is given for the fact that centered Gaussian functions saturate the Euclidean forward–reverse Brascamp–Lieb inequalities, extending the Brascamp–Lieb and Barthe theorems. A duality principle for best constants is also developed, which generalizes the fact that the best constants in the Brascamp–Lieb and Barthe inequalities are equal. Finally, as the title hints, the main results concerning finiteness, structure, and Gaussian-extremizability for the Brascamp–Lieb inequality due to Bennett, Carbery, Christ, and Tao are generalized to the setting of the forward–reverse Brascamp–Lieb inequality.
AB - A new proof is given for the fact that centered Gaussian functions saturate the Euclidean forward–reverse Brascamp–Lieb inequalities, extending the Brascamp–Lieb and Barthe theorems. A duality principle for best constants is also developed, which generalizes the fact that the best constants in the Brascamp–Lieb and Barthe inequalities are equal. Finally, as the title hints, the main results concerning finiteness, structure, and Gaussian-extremizability for the Brascamp–Lieb inequality due to Bennett, Carbery, Christ, and Tao are generalized to the setting of the forward–reverse Brascamp–Lieb inequality.
KW - Barthe inequalities
KW - Brascamp–Lieb inequalities
KW - Functional inequalities
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U2 - 10.1007/s12220-020-00398-y
DO - 10.1007/s12220-020-00398-y
M3 - Article
AN - SCOPUS:85083107206
VL - 31
SP - 3300
EP - 3350
JO - Journal of Geometric Analysis
JF - Journal of Geometric Analysis
SN - 1050-6926
IS - 4
ER -