On the Maximal Directional Hilbert Transform in Three Dimensions

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On the Maximal Directional Hilbert Transform

For any dimension n ≥ 2, we consider the maximal directional Hilbert transform HU on R associated with a direction set U ⊆ Sn−1: HUf(x) := 1 π sup v∈U ∣∣∣p.v.∫ f(x− tv) dt t ∣∣∣. The main result in this article asserts that for any exponent p ∈ (1,∞), there exists a positive constant Cp,n such that for any finite direction set U ⊆ Sn−1, ||HU ||p→p ≥ Cp,n √ log #U, where #U denotes the cardinali...

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ژورنال

عنوان ژورنال: International Mathematics Research Notices

سال: 2018

ISSN: 1073-7928,1687-0247

DOI: 10.1093/imrn/rny138