Affine rigidity of functions with additive oscillation

TL;DR

This paper establishes that functions with a specific oscillation measure property are necessarily affine, revealing a rigidity phenomenon linking mean oscillation to affine functions.

Contribution

It proves that functions with mean oscillation extending to a locally finite measure are affine, introducing a new rigidity result for functions with additive oscillation.

Findings

Functions with mean oscillation as a locally finite measure are affine.

Functions satisfying the integro-differential identity are affine.

The result links oscillation measures to affine structure.

Abstract

We prove that a locally integrable function $f:(a,b) \to \mathbb R$ must be affine if its mean oscillation, considered as a function of intervals, can be extended to a locally finite Borel measure. In particular, we show that any function $f$ satisfying the integro-differential identity $|Df|(I)=4\text{osc}(f,I)$ for all intervals $I \subset {(a,b)}$ must be affine.