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 1 0 Chapter 2 . Th e relatio n betwee n potentia l theor y an d geometr y fo r planar domain s 1 3 1. Smoot h domain s 1 4 2. No n smoot h domain s 1 5 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 1 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 10 0
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