The goal of this chapter is to solve the Dirichlet problem on an arbitrary plane domain Ω. There are three traditional ways to solve this problem:

1. (i) The Wiener method is to approximate Ω from inside by subdomains Ωn of the type studied in Chapter II and to show that the harmonic measures ω(z, E, Ωn) converge weak-star to a limit measure on ∂Ω. With Wiener's method one must prove that the limit measure ω(z, E, Ω) does not depend on the approximating sequence Ωn.

2. (ii) The Perron method associates to any bounded function f on ∂Ω a harmonic function Pf on Ω. The function Pf is the upper envelop of a family of subharmonic functions constrained by f on ∂Ω. Perron's method is elegant and general. With Perron's method the difficulty is linearity; one must prove that P−f = −Pf, at least for f continuous.

3. (iii) The Brownian motion approach, originally from Kakutani [1944a], identifies ω(z, E, Ω) with the probability that a randomly moving particle, starting at z, first hits ∂Ω in the set E. This method has considerable intuitive appeal, but it leaves many theorems hard to reach.

We follow Wiener and use the energy integral to prove that the limit ω(z, E, Ω) is unique. This leads to the notions of capacity, equilibrium distribution, and regular point and to the characterization of regular points by Wiener series.

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 Hp 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 Hp 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.