Planar Harmonic Analysis

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This grant will be used to research problems in the harmonic analysis of the plane. In particular, we will investigate the boundedness of certain directional maximal operators, both those corresponding to sets ofdirections and those related to Zygmund's conjecture on differentiation in the direction of a Lipschitz vector field. We will investigate some other geometrical problems in the plane which are related to spaces of functions with bounded mean oscillations.One of the most basic results in the history of mathematics is the fundamental theorem of calculus, which states that one can recover a function by differentiating its integral. The beauty of this result has led to a search for generalizations, giving rise to so-called differentiation theory. The purpose of this project is to study a problem posed by Zygmund some 40 years ago which tests how well differentiation works with respect to not-very-smooth choices of directions.