On the Maximal Directional Hilbert Transform
نویسندگان
چکیده
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 cardinality of U . As a consequence, the maximal directional Hilbert transform associated with an infinite set of directions cannot be bounded on L(R) for any n ≥ 2 and any p ∈ (1,∞). This completes a result of Karagulyan [9], who proved a similar statement for n = 2 and p = 2.
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