Contents
Introduction i x
Chapter 1. Motivatio n an d statemen t o f th e mai n result s 1
1. Characterizatio n (l)
a
: Approximatio n wit h plane s 2
2. Characterizatio n (2)
a
: Introducin g BM O an d VM O 3
3. Multiplicativ e vs . additiv e formulation : Introducin g th e doublin g
condition 3
4. Characterizatio n (l)
a
an d flatnes s 4
5. Doublin g an d asymptoticall y optimall y doublin g measure s 7
6. Regularit y o f a domain an d doublin g characte r o f its harmoni c measur e 8
7. Regularit y o f a domai n an d smoothnes s o f its Poisso n kerne l 10
Chapter 2 . Th e relatio n betwee n potentia l theor y an d geometr y fo r
planar domain s 13
1. Smoot h domain s 14
2. No n smoot h domain s 15
3. Preliminarie s t o th e proof s o f Theorems 2. 7 and 2. 8 2 0
4. Proo f o f Theore m 2. 7 2 5
5. Proo f o f Theorem 2. 8 2 9
6. Note s 3 7
Chapter 3 . Preliminar y result s i n potentia l theor y 3 9
1. Potentia l theor y i n NT A domain s 3 9
2. A brief revie w o f the rea l variable theor y o f weights 4 6
3. Th e space s BM O an d VM O 4 8
4. Potentia l theor y i n C 1 domain s 5 2
5. Note s 5 3
Chapter 4 . Reifenber g flat an d chor d ar c domain s 5 5
1. Geometr y o f Reifenberg fla t domain s 5 5
2. Smal l constan t chor d ar c domain s 6 1
3. Note s 7
Chapter 5 . Furthe r result s o n Reifenber g fla t an d chor d ar c domain s 7 3
1. Improve d boundar y regularit y fo r 6— Reifenberg fla t domain s 7 4
2. Approximatio n an d Rellic h identit y 7 7
3. Note s 8 0
Chapter 6 . Fro m th e geometr y o f a domai n t o it s potentia l theor y 8 1
1. Potentia l theor y fo r Reifenber g domain s wit h vanishin g constan t 8 1
2. Potentia l theor y fo r vanishin g chor d ar c domain s 100
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