Banach's Indicatrix Reloaded
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912412164161536 |
|---|---|
| author | Agbanusi, Ikemefuna |
| author_facet | Agbanusi, Ikemefuna |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10668 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Banach's Indicatrix Reloaded Agbanusi, Ikemefuna Classical Analysis and ODEs Analysis of PDEs Functional Analysis (Primary) 26A45, 26A15, (Secondary) 28A75, 28A80 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. |
| title | Banach's Indicatrix Reloaded |
| topic | Classical Analysis and ODEs Analysis of PDEs Functional Analysis (Primary) 26A45, 26A15, (Secondary) 28A75, 28A80 |
| url | https://arxiv.org/abs/2410.10668 |