Preface This memoir is devoted to a question in planar Harmonic Analysis, a subject which is a circle of problems all related to the Besicovitch set. This anomalous set has zero Lebesgue measure, yet contains a line segment of unit length in each direction of the plane. It is a known, since the 1970’s, that such sets must necessarily have full Hausdorff dimension. The existence of these sets, and the full Hausdorff dimension, are intimately related to other, independently interesting issues [26]. An important tool to study these questions is the so-called Kakeya Maximal Function, in which one computes the maximal average of a function over rectangles of a fixed eccentricity and arbitrary orientation. Most famously, Charles Fefferman showed [10] that the Besicovitch set is the obstacle to the boundedness of the disc multiplier in the plane. But as well, this set is intimately related to finer questions of Bochner-Riesz summability of Fourier series in higher dimensions and space-time regularity of solutions of the wave equa- tion. This memoir concerns one of the finer questions which center around the Besi- covitch set in the plane. (There are not so many of these questions, but our purpose here is not to catalog them!) It concerns a certain degenerate Radon transform. Given a vector field v on R2, one considers a Hilbert transform computed in the one dimensional line segment determined by v, namely the Hilbert transform of a function on the plane computed on the line segment {x + tv(x) | |t| ≤ 1}. The Besicovitch set itself says that choice of v cannot be just measurable, for you can choose the vector field to always point into the set. Finer constructions show that one cannot take it to be H¨ older continuous of any index strictly less than one. Is the sharp condition of H¨ older continuity of index one enough? This is the question of E. M. Stein, motivated by an earlier question of A. Zygmund, who asked the same for the question of differentiation of integrals. The answer is not known under any condition of just smoothness of the vector field. Indeed, as is known, and we explain, a positive answer would necessarily imply Carleson’s famous theorem on the convergence of Fourier series, [6]. This memoir is concerned with reversing this implication: Given the striking recent successes related to Carleson’s Theorem, what can one say about Stein’s Conjecture? In this direction, we introduce a new object into the study, a Lipschitz Kakeya Maximal Function, which is a variant of the more familiar Kakeya Maximal Function, which links the vector field v to the ‘Besicovitch sets’ associated to the vector field. One averages a function over rectangles of arbitrary orientation and—in contrast to the classical setting—arbitrary eccentricity. But, the rectangle must suitably localize the directions in which the vector field points. This Maximal Function admits a favorable estimate on L2, and this is one of the main results of the Memoir. vii

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