It is also possible, formally at least, to adapt the Hölder inequality argument to reach the same conclusion. For instance if one has a bounded bilinear operator for then one can then define adjoint bilinear operators and obeying the relationsĪnd with exactly the same operator norm as. The first argument also extends to some extent to multilinear operators. Then for any reasonable function on, we haveīy (1) and Hölder dividing out by we obtain, and a similar argument also recovers the reverse inequality. For sake of exposition let us make the simplifying assumption that (and hence also ) maps non-negative functions to non-negative functions, and ignore issues of convergence or division by zero in the formal calculations below. There is a slightly different way to proceed using Hölder’s inequality. Of (and similarly for ), one can show that has the same operator norm as. If one has a bounded linear operator for some measure spaces and exponents, then one can define an adjoint linear operator involving the dual exponents, obeying (formally at least) the duality relation for suitable test functions on respectively. To motivate matters let us review the classical theory of adjoints for linear operators. In this paper, we observe that the family of multilinear inequalities known as the Brascamp-Lieb inequalities (or Holder-Brascamp-Lieb inequalities) admit an adjoint formulation, and explore the theory of these adjoint inequalities and some of their consequences. Jon Bennett and I have just uploaded to the arXiv our paper “ Adjoint Brascamp-Lieb inequalities“.
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