# Dissertation: 4.12 Hardy-Orlicz Spaces of Quasiconformal Mappings and Conformal Densities (Benedict)

**Start date:**
Dec 04, 2014 12:00 AM

**End date:**
Dec 04, 2014 03:00 PM

**Location:**
Seminaarinmäki,
S212

MSc **Sita Benedict** defends her doctoral dissertation in Mathematics. Opponent University Researcher Dr **Maria-José Martin-Gómez** (University of Eastern Finland) and custos Academy Professor **Pekka Koskela** (University of Jyväskylä). The event is in English.

### Abstract

The general area of study of this thesis is Geometric Function Theory, which studies analytic functions and their generalizations from a geometric perspective. In mathematics, a ‘space’ is simply a collection of objects that are defined to have some common property. A function space, then, is a space of functions. This thesis deals with various types of function spaces known as Hardy (or H^p) spaces.

In the early 20th century G.H. Hardy published results on properties of the ‘integral means’ of analytic functions defined on disks in the complex plane. The integral means are integral averages of the modulus (distance from the origin) of image points under an analytic function along a circle. Later, F. Riesz considered only analytic functions defined on the unit disk satisfying a certain type of bounded growth at the boundary of the unit disk, defined using the integral means described above. He named these spaces H^p spaces, after Hardy’s work. Since then a rich theory has developed, and also many new Hardy-type spaces have been defined and studied.

This thesis defines and studies properties of a new type of Hardy space, that of conformal densities on the unit ball in dimensions 2 or greater. A conformal density is the generalization of the function |f’|, where f is a conformal (analytic and one-to-one) function on the unit disk. In essence, a density is a function that can be used to define length using line integrals. As an example, we can use |f’| to define a different notion of distance in the image of the unit disk under the conformal mapping f, by considering only lengths of curves inside the image domain. Then, we define the new Hardy space by replacing the usual euclidean distance (modulus) between f(x) and 0 with the ‘internal distance’ between f(x) and f(0). After defining these spaces, this thesis addresses two very natural questions. The first asks whether or not these newly defined spaces differ from the spaces of conformal mappings belonging to classical H^p. In other words, since the internal distance is always at least as big as the euclidean modulus, the spaces defined using the internal distance are trivially contained in the classical spaces, and so we are asking if this inclusion is strict or not. In this thesis we obtain a complete answer to this question for all 0 < p. The other question deals with the more general development of the theory of these spaces. We prove, in particular, a number of analogues to theorems that characterize the conformal mappings belonging to the classical Hardy spaces. We prove these analogues in the more general setting of conformal densities.

The dissertation is published in the series Report/University of Jyväskylä, Department of Mathematics and Statistics, no. 143, 16 p., Jyväskylä 2014, ISSN: 1457-8905, ISBN: 978-951-39-5946-3. It is available at the University Library’s Publications Unit, tel. +358 40 805 3825, myynti@library.jyu.fi

**Further information:**

Sita Benedict, sita.c.benedict@jyu.fi, +358 44 328 1377

Communications officer Anitta Kananen, tiedotus@jyu.fi, +358 40 805 4142