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Author:Maasalo, Outi
Title:BMO-funktioiden karakterisaatioita
Characterisations of BMO
Publication type:Master's thesis
Publication year:2004
Pages:55      Language:   fin
Department/School:Teknillisen fysiikan ja matematiikan osasto
Main subject:Matematiikka   (Mat-1)
Supervisor:Nevanlinna, Olavi
Instructor:Kinnunen, Juha
OEVS:
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Location:P1 Ark TF80     | Archive
Keywords:BMO
metric space
doubling measure
John-Nirenberg lemma
Coifman-Rochberg theorem
Muckenhoupt weigths
reverse Hölder inequality
BMO
metrinen avaruus
tuplaava mitta
John-Nirenbergin lemma
Coifman-Rochbergin lause
Muckenhouptin painot
käänteinen Hölderin epäyhtälö
Abstract (eng): In this Master's Thesis we consider functions of Bounded Mean Oscillation in metric measure spaces.
We concentrate on two characterisation theorems when the underlying measure is doubling.
The John-Nirenberg lemma and the Coifman-Rochberg theorem are our main interests.
We intend to show that the essential features of the presented BMO-theory are independent of the linear structure of Rn.
We proceed mainly by using maximal functions with Vitali -type covering theorems.

This work does not contain new results.
It is often straightforward to generalize the familiar results in Rn to a metric space.
However the main results are an exception.
In the proof of the John-Nirenberg lemma we follow the presentation of Mateau, Mattila, Nicolau and Orobitg [14].
We prove the Coifman-Rochberg theorem by generalizing the work of Garcia-Cuerva and Rubio de Francia [6].
Our approach to the reverse Holder inequality is also different from the metric space tradition.
We present an improved Calderon-Sigmund theorem for balls and generalise the proof in Rn presented by Coifman and Fefferman [2].

The thesis is loosely based on Javier Duoandikoetxea's Fourier Analysis [4], Elias M.
Stein's Harmonic Analysis [18] and Juha Heinonen's Lectures on Analysis on Metric Spaces [8] as well as recent articles of BMO.
The reader is assumed to be familiar with the basic concepts of metric spaces and also of measure and integral theory.
Abstract (fin): Tässä työssä tarkastelemme keskimääräiseltä heilunnaltaan rajoitettuja eli Bounded Mean Oscillation -funktioita metrisissä mitta-avaruuksissa.
Keskitymme erityisesti BMO-funktioiden karakterisaatiolauseisiin käytettävän mitan ollessa tuplaava.
Työn päätulokset ovat John-Nirenbergin lemma ja Coifman-Rochbergin lause.
Keskeisenä ideana on osoittaa analyysin perusteemojen olevan riippumattomia avaruuden Rn lineaarisesta rakenteesta.
Usein tämä tehdään hyödyntämällä vektoriavaruuden ominaisuuksien sijaan maksimaalifunktiotekniikoita ja Vitali-tyyppisiä peitelauseita.

Kaikki tulokset ovat n-ulotteisten reaaliavaruuksien tapauksessa tunnettuja ja suurelta osin yleistys metriseen avaruuteen on vaivatonta.
Päätulosten osalta tilanne on monimutkaisempi.
John-Nirenbergin lemman todistus on Mateaun, Mattilan, Nicolaun ja Orobitgin mukainen [14].
Coifman-Rochbergin lauseen todistamme yleistämällä Garcia-Cuervan ja Rubio de Francian version metriseen avaruuteen [6].
Esitämme myös käänteiselle Holderin epäyhtälölle metrisen avaruuden perinteestä poikkeavan todistuksen yleistämällä Coifmanin ja Feffermanin Rn-todistuksen [2].
Tätä varten todistamme Calderón-Zygmundin hajotelmasta parannetun version palloille.

Pääasiallisina lähteinä on käytetty reaaliavaruuksien analyysia käsitteleviä teoksia kuten Javier Duoandikoetxean Fourier Analysis [4] ja Elias M.
Steinin Harmonic Analysis [18] sekä useita BMO-funktioita käsitteleviä ajankohtaisia artikkeleita.
Metrisen avaruuden analyysiin on haettu ajatuksia Juha Heinosen kirjasta Lectures on Analysis on Metric Spaces [8].
Oletamme lukijalta perustiedot metrisistä avaruuksista sekä mitta- ja integrointiteoriasta.
ED:2004-11-26
INSSI record number: 26525
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