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

Jonas Knoerr

TL;DR

The paper develops a Paley–Wiener–Schwartz framework for smooth valuations on Convex functions by translating valuation data into Goodey–Weil distributions and studying their Fourier–Laplace transforms. It reveals a precise module-theoretic description: the relevant transforms lie in a submodule generated by quadratic products of k-minors, linking to Monge–Ampère operators via the differential cycle and MAVal_k(ℝ^n). This yields a PW-type characterization of smooth valuations with support in a compact set and provides a complete classification of affine-invariant closed subspaces, showing they have finite codimension and are densely generated by Monge–Ampère–based valuations. The results unify representation-theoretic, geometric, and Fourier-analytic perspectives, yielding density results and a deep structural understanding of valuations on convex functions and their Monge–Ampère operators. The work thus advances the analytic description and classification of dually epi-translation invariant valuations and their affine-invariant subspaces with potential applications to convex-analytic and geometric-analytic problems.

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-Ampère operators.

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

TL;DR

The paper develops a Paley–Wiener–Schwartz framework for smooth valuations on Convex functions by translating valuation data into Goodey–Weil distributions and studying their Fourier–Laplace transforms. It reveals a precise module-theoretic description: the relevant transforms lie in a submodule generated by quadratic products of k-minors, linking to Monge–Ampère operators via the differential cycle and MAVal_k(ℝ^n). This yields a PW-type characterization of smooth valuations with support in a compact set and provides a complete classification of affine-invariant closed subspaces, showing they have finite codimension and are densely generated by Monge–Ampère–based valuations. The results unify representation-theoretic, geometric, and Fourier-analytic perspectives, yielding density results and a deep structural understanding of valuations on convex functions and their Monge–Ampère operators. The work thus advances the analytic description and classification of dually epi-translation invariant valuations and their affine-invariant subspaces with potential applications to convex-analytic and geometric-analytic problems.

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-Ampère operators.

Paper Structure

This paper contains 22 sections, 63 theorems, 239 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Representation theoretic/analytic perspective
  4. Geometric perspective
  5. Fourier analytic perspective
  6. Plan of the article
  7. Preliminaries
  8. Convex functions
  9. Polynomials and $k$-minors on $\mathop{\mathrm{\mathrm{Mat}}}\nolimits_{n,k}(\mathbb{C})$
  10. Some modules of entire functions generated by polynomials
  11. Dually epi-translation invariant valuations
  12. Support and topology
  13. Monge--Ampère operators and the differential cycle
  14. Basic properties of smooth valuations
  15. Structure of the Fourier--Laplace transform of Goodey--Weil distributions
  16. ...and 7 more sections

Key Result

Theorem 1.1

Linear combinations of mixed volumes span a dense subspace of $\mathop{\mathrm{Val}}\nolimits(\mathbb{R}^n)$.

Theorems & Definitions (110)

  • Theorem 1.1: Alesker AleskerDescriptiontranslationinvariant2001
  • Theorem 1.2: Alesker AleskerDescriptiontranslationinvariant2001
  • Theorem 1
  • Definition 1.3
  • Theorem 1.4
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 2.1
  • Proposition 2.2: Knoerrsupportduallyepi2021 Proposition 2.4
  • ...and 100 more