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A field guide for Hilbert transforms with new estimates on an associated maximal directional operator Marshall, Caleb


We give an overview of Hilbert transforms, followed by new results concerning maximal directional Hilbert transforms. Historically, the Hilbert transform motivated the development of many tools in harmonic analysis, such as interpolation theorems and more general singular integrals. Over time, variants of the Hilbert transform were studied as prototypical examples of singular integrals and maximal directional operators. In our research, we are especially concerned with maximal directional Hilbert transforms. After rigorously constructing the Hilbert transform and directional Hilbert transforms, we proceed to define the maximal directional Hilbert transforms. We then prove general L² mapping estimates for maximal directional Hilbert transforms, followed by specific examples which sharpen these estimates. Finally, we prove sharp L²(R²) to L²(R²) estimates for a large class of maximal directional Hilbert transforms.

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