A Paley-Wiener-Schwartz Theorem for smooth valuations on convex functions

TL;DR

This paper characterizes smooth valuations on convex functions using Fourier-Laplace transforms, provides integral representations, and classifies invariant subspaces, advancing the understanding of valuations in convex analysis.

Contribution

It introduces a Paley-Wiener-Schwartz type theorem for smooth valuations on convex functions, linking valuations to Fourier-Laplace transforms and classifying invariant subspaces.

Findings

Characterization of valuations via Fourier-Laplace transform.

Integral representations of smooth valuation vectors.

Complete classification of invariant subspaces.

Abstract

Continuous dually epi-translation invariant valuations on convex functions are characterized in terms of the Fourier-Laplace transform of the associated Goodey-Weil distributions. This description is used to obtain integral representations of the smooth vectors of the natural representation of the group of translations on the space of these valuations. As an application, a complete classification of all closed and affine invariant subspaces is established, yielding density results for valuations defined in terms of mixed Monge-Amp\`ere operators.