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Preprint Number 944
944. Michael Christ, James Demmel, Nicholas Knight, Thomas Scanlon, Katherine
Yelick On Holder-Brascamp-Lieb inequalities for torsion-free discrete Abelian
groups E-mail: Submission date: 14 October 2015 Abstract: Hölder-Brascamp-Lieb inequalities provide upper bounds for a class
of multilinear expressions, in terms of Lp norms of the functions
involved. They have been extensively studied for functions defined on
Euclidean spaces. Bennett-Carbery-Christ-Tao have initiated the study of
these inequalities for discrete Abelian groups and, in terms of suitable
data, have characterized the set of all tuples of exponents for which
such an inequality holds for specified data, as the convex polyhedron
defined by a particular finite set of affine inequalities.
In this paper we advance the theory of such inequalities for
torsion-free discrete Abelian groups in three respects. The optimal
constant in any such inequality is shown to equal 1 whenever it is
finite. An algorithm that computes the admissible polyhedron of
exponents is developed. It is shown that nonetheless, existence of an
algorithm that computes the full list of inequalities in the
Bennett-Carbery-Christ-Tao description of the admissible polyhedron for
all data, is equivalent to an affirmative solution of Hilbert's Tenth
Problem over the rationals. That problem remains open. Mathematics Subject Classification: 26D15, 11U05 Keywords and phrases: |

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