On the Maximal Directional Hilbert Transform in Three Dimensions
نویسندگان
چکیده
منابع مشابه
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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For a Schwartz function f on the plane and a non-zero v ∈ R2 define the Hilbert transform of f in the direction v to be Hvf(x) = p.v. ∫ R f(x− vy) dy y Let ζ be a Schwartz function with frequency support in the annulus 1 ≤ |ξ| ≤ 2, and ζf = ζ ∗ f . We prove that the maximal operator sup|v|=1|Hvζf | maps L2 into weak L2, and Lp into Lp for p > 2. The L2 estimate is sharp. The method of proof is ...
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ژورنال
عنوان ژورنال: International Mathematics Research Notices
سال: 2018
ISSN: 1073-7928,1687-0247
DOI: 10.1093/imrn/rny138