That main formula again doesn’t show up.

]]>à propos: I haven’t actually checked whether we can we use the Poisson semi-group instead here. In the Bourgain/Carberry strong result, the Poisson semi-group is used instead and seems to be more appropriate. I also haven’t checked (maybe a naive comment, but an obvious one) whether an improvement on the weak (1,1) bound can be achieved for the metric by comparing to a suitable semi-group.

]]>You are absolutely right. I linked the wrong paper so feel free to edit my comment with the correct link (the collectanea paper is freely available here as far as I can tell).

*[OK, fixed. -T]*

Thanks for the Soria-Menarguez reference (though it appears that their Collectanea paper is more relevant here than the one you linked, which is mostly concerned with the 1D case). Yes, the localisation principle formalises the gap between powers of (where everything is understood) and powers of (which is essentially the full case) by saying that we can reduce to just the scales between, say, 1 and n. At this point, it seems that one cannot get any further just from the triangle inequality, and must now use something special about the underlying metric structure.

It was also pointed out to us that we neglected to mention some closely related work of Stromberg, Rochberg-Taibleson, and Cowling-Meda-Setti on the free group and on hyperbolic space (we had already intended to cite these papers, but somehow things got lost in the process). The next revision of the paper will address these issues.

]]>– It is my impression that in order to improve the known bounds in the Euclidean case (or in the -case), one has to “address all scales at the same time”. The Stein-Strömberg proof (via the covering lemma) as well as the Lindenstrauss lemma approach, use the localization principle you explain here and get the final bound by counting how many distinct scales there are. Another, more naive, way to put it is that everything happens for radii that are between powers of and powers of as you mention in your notes for the Lindenstrauss Lemma. Any sparser set of radii gives constants and any denser set than powers of is already the unrestricted maximal function (up to absolute constants). This has already been observed by ~~Soria and Menarguez~~ (corrected link here) to do them justice. Between these two endpoints, there are essentially scales. Philosophically speaking, this approach cannot ever give us anything better that . Contrast that to the Euclidean case; the comparison with the heat kernel does not use this localization, but rather the Euclidean symmetry for all scales at the same time (and it is enough to do that for large enough).

– Aiming somewhere in the middle, it would be interesting to understand if the bound for (say) lacunary radii , is best possible. We get this bound, again, as a consequence of the localization principle. But I don’t know of a direct approach to this, meaning an approach that’s specific to this scale.

]]>*[Corrected, thanks – T.]*