L2 BOUNDS FOR A MAXIMAL DIRECTIONAL HILBERT TRANSFORM

摘要

Given any finite direction set Omega of cardinality N in Euclidean space, we consider the maximal directional Hilbert transform H-Omega associated to this direction set. Our main result provides an essentially sharp uniform bound, depending only on N, for the L-2 operator norm of H-Omega in dimensions 3 and higher. The main ingredients of the proof consist of polynomial partitioning tools from incidence geometry and an almost-orthogonality principle for H-Omega. The latter principle can also be used to analyze special direction sets Omega and derive sharp L-2 estimates for the corresponding operator H-Omega that are typically stronger than the uniform L-2 bound mentioned above. A number of such examples are discussed.

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