Lecture 8: Examples of Multi-Parameter Carleson Measures

We now turn to giving some examples of multi-parameter Carleson measures. The examples we have in mind are motivated by what happens in one parameter and in function theory. In one parameter (and one variable) we saw that the gradient of bounded functions times an expression related to the distance to the boundary is a Carleson measure. It turns out that examples in several variables and multi-parameters is of the same nature. Namely for the harmonic extension of a bounded function on the boundary of the bidisc, we will have that the gradient(s) times some expressions related to the distance to the boundary are in fact Carleson measures. The method of proof is necessarily more complicated since we have to work with all open sets (as seen in the last lecture).

This will be the last lecture of the semester. The projects for this years topic will be posted sometime in late January or early February.