Title

Prostori harmonijskih funkcija i harmonijska kvazikonformna preslikavanja

Creator

Shkheam, Abejela

Copyright date

2013

Object Links

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License description

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Language

English

Cobiss-ID

Theses Type

Doktorska disertacija

description

Datum odbrane: 09.10.2013.

Other responsibilities

mentor

Arsenović, Miloš, 1962-

član komisije

Božin, Vladimir

član komisije

Mateljević, Miodrag, 1949-

član komisije

Manojlović, Vesna

član komisije

Mihić, Olivera, 1974-

University

Univerzitet u Beogradu

Faculty

Matematički fakultet

Alternative title

Spaces of harmonic functions and harmonic quasiconformal mappings

Publisher

[A. Shkheam]

Format

III, 64 lista

description

Mathematics - Complex analysis / Matematika - Kompleksna analiza

Abstract (en)

This thesis has been written under the supervision of my mentor, Prof. dr. Miloš
Arsenović at the University of Belgrade academic, and my co-mentor dr. Vladimir
Božin in year 2013. The thesis consists of three chapters. In the first chapter we start
from defnition of harmonic functions (by mean value property) and give some of their
properties. This leads to a brief discussion of homogeneous harmonic polynomials, and
we also introduce subharmonic functions and subharmonic behaviour, which we need
later. In the second chapter we present a simple derivation of the explicit formula for the
harmonic Bergman reproducing kernel on the ball in euclidean space and give a proof that
the harmonic Bergman projection is Lp bounded, for 1 < p < 1, we furthermore discuss
duality results. We then extend some of our previous discussion to the weighted Bergman
spaces. In the last chapter, we investigate the Bergman space for harmonic functions bp,
0 < p < 1 on RnnZn. In the planar case we prove that bp 6= f0g for all 0 < p < 1.
Finally we prove the main result of this thesis bq c bp for n=(k + 1) < q < p < n=k,
(k = 1; 2; :::). This chapter is based mainly on the published paper [44]. M. Arsenović,
D. Kečkić,[5] gave analogous results for analytic functions in the planar case. In the
plane the logarithmic function log jxj, plays a central role because it makes a diference
between analytic and harmonic case, but in the space the function /x/2-n; n > 2 hints at
the contrast between harmonic function in the plane and in higher dimensions.

Authors Key words

Bergman space, harmonic functions, subharmonic functions, analytic functions

Authors Key words

Bergmanovi prostori, harmonijske funkcije, subharmonijske funkcije, analitičke funkcije

Classification

517.57+517.54(043.3)

Type

Tekst

Abstract (en)

This thesis has been written under the supervision of my mentor, Prof. dr. Miloš
Arsenović at the University of Belgrade academic, and my co-mentor dr. Vladimir
Božin in year 2013. The thesis consists of three chapters. In the first chapter we start
from defnition of harmonic functions (by mean value property) and give some of their
properties. This leads to a brief discussion of homogeneous harmonic polynomials, and
we also introduce subharmonic functions and subharmonic behaviour, which we need
later. In the second chapter we present a simple derivation of the explicit formula for the
harmonic Bergman reproducing kernel on the ball in euclidean space and give a proof that
the harmonic Bergman projection is Lp bounded, for 1 < p < 1, we furthermore discuss
duality results. We then extend some of our previous discussion to the weighted Bergman
spaces. In the last chapter, we investigate the Bergman space for harmonic functions bp,
0 < p < 1 on RnnZn. In the planar case we prove that bp 6= f0g for all 0 < p < 1.
Finally we prove the main result of this thesis bq c bp for n=(k + 1) < q < p < n=k,
(k = 1; 2; :::). This chapter is based mainly on the published paper [44]. M. Arsenović,
D. Kečkić,[5] gave analogous results for analytic functions in the planar case. In the
plane the logarithmic function log jxj, plays a central role because it makes a diference
between analytic and harmonic case, but in the space the function /x/2-n; n > 2 hints at
the contrast between harmonic function in the plane and in higher dimensions.

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