# Function theory III, fall 2010

### What is going on?

Lectures have ended, thanks for participation !

Those who urgently need to get marks of the course during December, please come to meet me to discuss the essays you are writing.

### Exercises

SUGGESTIONS FOR SOLUTIONS:

### Lecture notes (scetches only)

Rami Luisto has written an account of a self-contained proof (presented in the last lecture) of Jordan's curve theorem (with two proofs) and of the lemma of Janizewski, based only on results of FTII. The text is found below.

In turn, the text by Tomas Soto shows how the Caratheodory theorem of boundary extension for conformal maps (from the disc to Jordan domains), and as a simple consequence the Schoenfliess theorem, are proven by assuming Jordan theorem and Janitzewski's lemma.

Jordan's curve theorem

Schoenfliess theorem

### Content

We will cover several basic elements of function theory that have been not covered by Function Theory I-II.

Basic topic covered include modulus of curve families, boundary extensions of conformal maps, basic distortion estimates of univalent functions, hyperbolic and quasihyperbolic metrics with applications to conformal maps, modular groups and functions, the Picard theorems, harmonic measure, elliptic functions, and (if time allows) the uniformization theorem will be discussed.

### Prerequisities

Function theory II. Measure and integral. For the bits of topology, real or functional

analysis that will possibly be needed, careful references are given.

### Scope

10 op.

### Type

Advanced studies.

### Literature

Suitable sections from various books will be mentioned during the lectures.

### Lecturer

### Lectures

Most of the weeks 36-42 and 44-50 Tu 10-12, We 9-12 C123 (during couple of weeks extra lectures on Fr 10-12 in C123).

Exercise classes are held only every second week, the answers should be returned in writing before the exercise class takes place.

If there are participants who do not know Finnish, the lectures will be held in English.

### Passing the course

The course can be passed by returning written exercises to the instructor before the instruction classes take place.

Also, most probably, one will be asked to write a small essay on a topic that will be chosen jointly with the lecturer.

It is also possible to pass the course by an oral examination (this sounds bad, I know, but the questions will be

easy and the examiner will help to answer them).

### Register

Did you forget to register? What to do.

### Exercise groups

Every second week, exact timing will be announced later on.

Group | Day | Time | Location | Instructor |
---|---|---|---|---|

1. | Fri | 10-12 | C123 | Istvan Prause |

WELLCOME!

### Logbook

8.9 Definition of modulus of curve families, modulus of simplest configurations (rectangle, annulus), basic properties

10.9 Properties of modulus (continued). Continuous extension to the boundary (beginning)

14.9 Finite or local connectivity along the boundary. Continuous extension of conformal maps onto the boundary.

15.9 Caratheodory-Osgood theorem (homeomorphic extensions). Properties of Jordan domains and conformal maps between them.

Prime ends via modulus (beginning)

21.9 Prime ends via modulus (completion)

22.9 Area theorem of conformal maps. Koebe mapping. Koebe 1/4-theorem, Koebe estimate. Automorphisms of the unit disc.

Conformal invariance of the hyperbolic metric. Shortest hyperbolic distance, Non-Euclidean geometry.

28.9 Hyperbolic metric for simply connected domains, analytic functions as contraction of the hyperboli metric, kvasihyperbolic metric,

Whitney tilings, rigidity of conformla maps between simply connected domains.

29.9 Growth estimates for univalent functions on the unit disc. Convergence properties of conformal maps:

the Caratheodory kernel theorem on convergence (beginning).

1.10 Caratheodory kernel theorem on convergence (completion).

5.10 The modular group. Construction of the fundamental domain for the (mod 2) subgroup.

6.10 Fundamental domains continued. Construction of the modular function.

12.10 Construction and properties of the modular function (continued). Lifts of functions through the covering space.

13.10 Picard's little and great theorems. Schottky theorem. Fundamental group of the full modular group. Functions with one period.

19.10 Discrete period modules. Jacobi's theorem. Period parallegrams. Basic properties of elliptic functions: order and equidistribution.

20.10 Relation between sums of poles and zeros. Construction of Weierstrass p-function, and Weierstrass zeta- and eta-functions.

Representation of elliptic functions in terms of p and p'. Characterization via zeros and poles.

8.11 The differential equation of p-function. Every elliptic function satisfies an algebraic differential equation.

Weierstrass characterization of meromorphic functions admitting an algebraic addition theorem (beginning).

9.11 Weierstrass characterization of meromorphic functions admitting an algebraic addition theorem (completion). Mapping properties of elliptic integrals. Connection to modular functions. Construction of elliptic functions in terms of the Jacobi theta-functions.

16.11. Harmonic measure for intervals on the unit circle. Lindelöf's generalized maximum principle. Lindelöf's theorem on boundary limits of analytic functions.

17.11. Harmonic measure for Borel subsets of the unit circle. Caharcterization of harmonic functions that are Poisson extensions of boundary Borel measures. Non-tangential convergence. Domination of the non-tangential maximal function by the Hardy-Littlewood maximal function on the boundary.

Fatou's theorem for boundary limits of Poisson integrals.

23.11 Harmonic h^p-spaces, representation in terms of the radian limit function, norm equivalence.

24.11 Analytic Hardy spaces H^p, completeness, characterization in terms of the Fourier coefficients, Blaschke products, zeroes of Hardy functions satisfy the Blaschke condition.

1.12 Factroization theorem of hardy functions. Uniqueness via boundary values. Boundedness of th radial maximal function on H^p for p>0. Brothers Riesz theorem

on absolute continuity. Conformal maps onto Jordan domains with rectifiable boundary: (another) Brothers Riesz theorem.

7.12 Harmonic measure on Jordan domains with rectifiable boundary. Discussion of further topics in function theory.

8.12 (last lecture) the Jordan curve theorem (proven by Rami Luisto). Discussion of the Schoenfliess theorem.