Malabika Pramanik and I have just uploaded to the arXiv the revised version of our paper on differentiation theorems. The new version is also available from my web page.

Here’s what happened. In the first version, we proved our restricted maximal estimates (with the dilation parameter restricted to a single scale) for all

Theorem 1.There is a decreasing sequence of setswith the following properties:

- each
is a disjoint union of finitely many intervals, as , - the densities
converge to a weak limit , - the maximal operators

and

are bounded on

for .

Our differentiation theorem has been adjusted accordingly.

Theorem 2.Letand be given by Theorem 1. Then the family differentiates for all , in the sense that for every we have

for a.e. Furthermore,

for a.e.

What about

We have also polished up the exposition, fixed up the typos and minor errors, etc. One other thing we have added (to the arXiv preprint – we are not including this in the version we are submitting for publication) is a short section on how to modify our construction of

I’ll be on vacation throughout the rest of July. I’ll continue to show up here on this blog – I might actually write here more often – and I’ll finish up a couple of minor commitments, but you should not expect any more serious mathematics from me in the next few weeks.