Pdf version here: link.
A question by Ben Krause, whom I met here at the Hausdorff Institute, made me think back of one of the earliest posts of this blog. The question is essentially how to make sense of the fact that the (perhaps iterated) convolution of a (singular) measure with itself is in general smoother than the measure you started with, in a variety of settings. It’s interesting to me because in this phase of my PhD experience I’m constantly trying to build up a good intuition and learn how to use heuristics effectively.
So, let’s take a measure on with compact support (assume inside the unit ball wlog). We ask what we can say about , or higher iterates and more often1 about . In particular, we’re interested in the case where is singular, i.e. its support has zero Lebesgue measure.
Before starting though, I would like to give a little motivation as to why such convolutions are interesting. Consider the model case where you have an operator defined by where are some singular measures and . One asks whether the operator is bounded on , and the natural tool to use is Cotlar-Stein lemma, or almost-orthogonality (from which this blog takes its name). Then we need to verify that
and same for . But what is ? It’s simply
i.e. another convolution operator. Estimates on the convolution2 are likely to help estimate the norm of then. But if this measure is not smooth enough, one can go forward, and since
one sees that estimates on are likely to help, and so on, until a sufficient number of iterations gives a sufficiently smooth measure. This isn’t quite the iteration of a measure with itself, but in many cases one has an operator which then splits into the above sum by a spatial or frequency cutoff at dyadic scales. Then it becomes a matter of rescaling and the case can be reduced to that of and further reduced to that of by exploiting an iterate of the above norm inequality, namely that
Another possibility is to write and consider working with instead, to obtain results in term of . I will say more in the end, here I just wanted to show that they arise as natural objects.