Lecture 7: Multiparameter Carleson Measures

Now that we know what Carleson measures look like in one-parameter, and some examples of them, we turn our attention to what two parameter Carleson measures should look like. Immediately we see that we will have to understand what a good definition of them should be, and will have two competing definitions of Carleson measures. We will show that the simplest definition is unfortunately not the correct one. This will be accomplished by constructing a very clever counterexample originally due to Carleson. We will build a collection of rectangles in the unit square that have in some sense large area, but will have large amounts of overlap at the same time.

P.S. I likely won't post a lecture next week since it is Thanksgiving week. The last lecture of the semester will be posted on November 28th.