# Analysis and Mathematical Physics

## The adjoint Brascamp-Lieb inequality

The Brascamp-Lieb inequality is a fundamental inequality in analysis, generalizing more classical inequalities such as Holder's inequality, the Loomis-Whitney inequality, and Young's convolution inequality: it controls the size of a product of pullbacks of low-dimensional functions to a common high-dimensional space by the size of the individual functions. One can take an "adjoint" of this inequality, and control the size of a high-dimensional function by the size of various marginals. This adjoint is equivalent to an entropy inequality of Carlen and Cordero-Erasquin, and implies a Holder inequality for the Gowers uniformity norms of non-negative functions that had only previously been known in special cases. The best constant for this adjoint inequality however differs slightly from that of the original Brascamp-Lieb inequality. These results arise from joint work in progress with Jonathan Bennett.