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arXiv · 1712.02673

On the maximal directional Hilbert transform in three dimensions

Francesco Di Plinio·Ioannis Parissis
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Abstract

We establish the sharp growth rate, in terms of cardinality, of the $L^p$ norms of the maximal Hilbert transform $H_\Omega$ along finite subsets of a finite order lacunary set of directions $\Omega \subset \mathbb R^3$, answering a question of Parcet and Rogers in dimension $n=3$. Our result is the first sharp estimate for maximal directional singular integrals in dimensions greater than 2. The proof relies on a representation of the maximal directional Hilbert transform in terms of a model maximal operator associated to compositions of two-dimensional angular multipliers, as well as on the usage of weighted norm inequalities, and their extrapolation, in the directional setting.

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Francesco Di Plinio, Ioannis Parissis. 2017-12-07. On the maximal directional Hilbert transform in three dimensions. https://doi.org/10.1093/imrn%2Frny138

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