On the Semicontinuity of Functionals on Function Spaces

2025/09/22 by Fernanda M. Baêta, Baêta, Fernanda M., Monika Ludwig +1 · 3 citations
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results

paper · pdf · doi:10.48550/arxiv.2509.17426

openalex publication_date 2025/09/22 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28

Abstract

Results on the upper and lower semicontinuity of functionals defined on spaces of convex and more general functions are established. In particular, the following result is obtained. Let ϕ(v; ⋅) be the density of the absolutely continuous part of a Radon measure Φ(v; ⋅) associated to a function v\colon X→ ℝ defined on the topological measure space (X,λ). For concave ζ\colon [0, ∞)→[0,∞) with lim_{t→ 0} ζ(t)=0 and lim_t→∞ζ(t)/t= 0, it is shown that the functional v ↦ ∫_X ζ(ϕ(v;x))dλ(x) depends upper semicontinuously on v. Examples include functional affine surface areas for convex functions.

Citations

• Weighted floating functions and weighted functional affine surface areas
• Continuous valuations on the space of Lipschitz functions on the sphere
• Floating functions
• Functional versions of L_p-affine surface area and entropy inequalities
• Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality

Cited by

• A Characterization of Functional Affine Surface Areas
• Upper semicontinuous valuations on convex functions of one variable
• Asymptotic Weighted Approximation of Convex Functions

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