Chapter 1. Overview of principal results ........................ 1
Chapter 2. Besicovitch set and Carleson's Theorem ............... 7
Besicovitch set ................................................. 7
The Kakeya maximal function ..................................... 8
Carleson's Theorem .............................................. 8
The weak L2 estimate in Theorem 1.15 is sharp ................... 9
Chapter 3. The Lipschitz Kakeya maximal function ............... 11
The weak L2 estimate ........................................... 11
An obstacle to an Lp estimate, for 1 < p < 2 ................... 20
Bourgain's geometric condition ................................. 21
Vector fields that are a function of one variable .............. 24
Chapter 4. The L2 estimate ..................................... 27
Definitions and principal Lemmas ............................... 27
Truncation and an alternate model sum .......................... 31
Proofs of Lemmata .............................................. 34
Chapter 5. Almost orthogonality between annuli ................. 51
Application of the Fourier localization Lemma .................. 51
The Fourier localization estimate .............................. 64
References ..................................................... 71
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