Endpoint Sobolev regularity of centered disk averages
The following describes the scope of the Lean formalization related to the following accompanying paper(s):
Scope
The centered disk maximal operator takes the supremum of averages of $|f|$ over disks centered at each point. The formalization proves that every real $f\in W^{1,1}(\mathbb R^2)$ has a maximal function that is finite almost everywhere, belongs to $W^{1,1}_{\mathrm{loc}}$, and has a globally integrable weak gradient satisfying $|\nabla Mf|_1\le C|\nabla f|_1$ for one absolute constant $C$.
The earlier signed finite-band estimate for smooth compactly supported functions is also retained. The general endpoint statement covers the Sobolev setting of the paper; a BV extension is outside these statements.
Comparator links
| Result | Comparator statement |
|---|---|
| Signed finite-band disk-maximal gradient bound | SignedFiniteBand.lean |
| Endpoint gradient bound for the centered disk maximal operator | DiskMaximal.lean |