Banach's Indicatrix Reloaded
Classical Analysis and ODEs
2025-06-05 v2 Analysis of PDEs
Functional Analysis
Abstract
Banach famously related the smoothness of a function to the size of its level sets. More precisely, he showed that a continuous function is of bounded variation exactly when its "indicatrix" is integrable. In a similar vein, we connect the smoothness of the function -- measured now by its integral modulus of continuity -- to the structure of its superlevel sets. Our approach ultimately reduces to a continuum incidence problem for quantifying the regularity of open sets. The pay off is a refinement of Banach's original theorem and an answer to a question of Garsia--Sawyer.
Cite
@article{arxiv.2410.10668,
title = {Banach's Indicatrix Reloaded},
author = {Ikemefuna Agbanusi},
journal= {arXiv preprint arXiv:2410.10668},
year = {2025}
}
Comments
4 Figures, Submitted. Some expository changes. In particular, the computations in Examples 1 and 4 include clarifying details