Abstract
The Hilbert transform along curves is defined by the principal value integral. The pointwise existence of the principal value Hilbert transform can be educed from the appropriate estimates for the corresponding maximal Hilbert transform. By using the estimates of Fourier transforms and Littlewood-Paley theory, we obtain L p -boundedness for the maximal Hilbert transform associated to curves (t,P(γ(t))), where 1<p<∞, P is a real polynomial and γ is convex on [0,∞). Then, we can conclude that the Hilbert transform along curves (t,P(γ(t))) exists in pointwise sense.MSC:42B20, 42B25.
Highlights
For n ≥, let : R → Rn be a curve in Rn with ( ) =
To transform H which is defined as a principal value integral we associate the Hilbert
The Lp-boundedness for the Hilbert transform H and the maximal function M above have been well studied by many scholars
Summary
The Lp-boundedness for the Hilbert transform H and the maximal function M above have been well studied by many scholars. Under the assumptions (i), (ii) and (iii) on the γ , the maximal Hilbert transform H∗ is a bounded operator in Lp(R ) for < p < ∞. The decay of associated multipliers is essential for the proof of Theorem. This set of techniques originated from the work [ ] and [ ]. Littlewood-Paley theory and interpolation theorem are effective tools to treat those problems Those ideas are due to the contribution of Córdoba, Nagel, Vance, Wainger, Rubio de Francia. In Section , we list some key properties concerning the polynomial and give some lemmas for the proof of the main result.
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