## Optimisers for the Brascamp-Lieb inequality.(English)Zbl 1159.26007

The Brascamp-Lieb inequality is a broad generalization of Hölder’ s or Young’s inequality dealing with integration of separable measures. It has the form $$\int_H \prod_i f_i^{p_i}(B_i x)dx \leq C \prod_i (\int_{H_i} f_i)^{p_i}$$ where $$H,H_i$$ are finite-dimensional Hilbert spaces, $$B_i$$ are linear maps, $$p_i$$ are nonnegative numbers, $$f_i$$ are nonnegative functions and $$C$$ is a constant. The author deals with the problem of characterizing the $$B_i$$ and $$p_i$$ such that the corresponding constant $$C$$ is finite and attains its smallest value.

### MSC:

 26D15 Inequalities for sums, series and integrals
Full Text:

### References:

 [1] K. Ball, Volumes of sections of cubes and related problems, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 251–260. [2] F. Barthe, On a reverse form of the Brascamp-Lieb inequality, Inventiones Mathimaticae 134 (1998), 335–361. · Zbl 0901.26010 [3] W. Beckner, Inequalities in Fourier analysis, Annals of Mathematics. Second Series 102 (1975), 159–182. · Zbl 0338.42017 [4] J. Bennett, A. Carbery, M. Christ, and T. Tao, Finite bounds for Hölder-Brascamp-Lieb multilinear inequalities, Mathematical Research Letters, to appear. · Zbl 1247.26029 [5] J. Bennett, A. Carbery, M. Christ, and T. Tao, The Brascamp-Lieb inequalities: finiteness, structure, and extremals, Geometric and Functional Analysis 17 (2008), 1343–1415. · Zbl 1132.26006 [6] H. J. Brascamp and E. H. Lieb, Best constants in Young’s inequality, its converse, and its generalization to more than three functions, Advances in Mathematics 20 (1976), 151–173. · Zbl 0339.26020 [7] E. A. Carlen, E. H. Lieb and M. Loss, A sharp analog of Young’s inequality on S N and related entropy inequalities, The Journal of Geometric Analysis 14 (2004), 487–520. · Zbl 1056.43002 [8] E. H. Lieb, Gaussian kernels have only Gaussian maximizers, Inventiones Mathematicae 102 (1990), 179–208. · Zbl 0726.42005 [9] S. I. Valdimarsson, The Brascamp-Lieb polyhedron, 2007, preprint. · Zbl 1196.44003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.