Lecture 5: One-Parameter Carleson Measures

We now turn to our study of Carleson measures in the one-parameter setting. We work on $R$ and the upper half plane since some of the computations are slightly easier and the resulting geometric pictures are very easy to draw and understand.

A non-negative measure $\mu$ is a Carleson measure if for any interval $I\subset R$ we have
$$\mu(T(I))\leq C\vert I\vert$$
where $T(I)$ is the tent'' over $I$. Namely the collection of points $(x,y)$ with $xin I$ and $0\leq y\leq 1-\vert I\vert$.

In this lecture we will show that this geometric class of measures actually connects back to some nice function theory that we have already touched upon. Carleson measures are so ubiquitous in analysis that it is good to have some exposure to them.