An update on differentiation theorems

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 p>1; unfortunately our scaling analysis worked only for p\geq 2, therefore our unrestricted maximal estimates and differentiation theorems were only valid in that range. However, just a few days after we posted the paper, Andreas Seeger sent us a “bootstrapping” scaling argument that works for p between 1 and 2. With Andreas’s kind permission, this is now included in the new version. The updated maximal theorem is as follows.

Theorem 1. There is a decreasing sequence of sets S_k \subseteq [1,2] with the following properties:

  • each S_k is a disjoint union of finitely many intervals,
  • |S_k| \searrow 0 as k \rightarrow \infty,
  • the densities \phi_k=\mathbf 1_{S_k}/|S_k| converge to a weak limit \mu,
  • the maximal operators

    {\mathcal M} f(x):=\sup_{t>0, k\geq 1} \frac{1}{|S_k|} \int_{S_k} |f(x+ty)|dy

    and

    {\mathfrak M} f(x) =  \sup_{t > 0} \int \left| f(x + ty) \right| d\mu(y)

    are bounded on L^p({\mathbb R}) for p >1.

Our differentiation theorem has been adjusted accordingly.

Theorem 2. Let S_k and \mu be given by Theorem 1. Then the family {\cal S} =\{ rS_k:\ r>0, n=1,2,\dots \} differentiates L^p( {\mathbb R}) for all p>1, in the sense that for every f \in L^p we have

\lim_{r\to 0} \sup_{n} \frac{ 1 }{ r|S_n| } \int_{ x+rS_n } f(y)dy = f(x) for a.e. x\in {\mathbb R}.

Furthermore,

\lim_{r\to 0} \int f(x+ry) d \mu (y)  =f(x) for a.e. x\in {\mathbb R}.

What about p=1? I had the good luck of meeting David Preiss in Barcelona right after Malabika and I had finished the first version of the preprint. I explained our work; we also spent some time speculating on whether such results could be true in L^1. Next day, David sent me a short proof that our Theorem 2 cannot hold with p=1 for any singular measure \mu supported away from 0. (The same goes for sequences of sets S_k as above, by a slight modification of his argument.) We are grateful to David for letting us include his proof in the new version of our paper.

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 S_k so that the limiting set S would also be a Salem set. The argument is very similar to the construction in our earlier paper on arithmetic progressions, so we only sketch it very briefly.

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.

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