X - Rectifiability and Quadratic Expressions

Summary

First we introduce three topics, two old and one new.

1. (i) The first topic is the classical Lusin area function. The Lusin area function gives another description of H^p functions and another almost everywhere necessary and sufficient condition for the existence of nontangential limits. The area function is discussed in Section 1. In Appendix M we prove the Jerison–Kenig theorem that the area function determines the H^p class of an analytic function on a chord-arc domain.

2. (ii) The second topic is the characterizations of subsets of rectifiable curves in terms of certain square sums. These theorems, from Jones [1990], are proved in Sections 2 and 3.

3. (iii) The third topic is the Schwarzian derivative, which measures how much an analytic function deviates from a Möbius transformation. Section 4 is a brief introduction to the Schwarzian derivative.

Then we turn to the chapter's main goal, an exposition of the two papers [1990] and [1994] by Bishop and Jones. In Section 5 the Schwarzian derivative is estimated by the Jones square sums and by a second related quantity. In Section 6 rectifiable quasicircles are characterized by a quadratic integral akin to the Lusin function but featuring the Schwarzian derivative. In Sections 7, 8, and 9 the same quadratic integral gives new criteria for the existence of angular derivatives and further characterizations of BMO domains. Section 10 brings together most of the ideas from the chapter to prove a local version of the F. and M. Riesz theorem, and in Section 11 this F. and M. Riesz theorem leads us to the most general form of the Hayman–Wu theorem.