Oscillatory hyper Hilbert transforms along general curves
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- 作者：Jiecheng Chen ; Belay Mitiku Damtew ; Xiangrong Zhu
- 关键词：Hilbert transform ; oscillatory integral ; oscillatory hyper Hilbert transform
- 刊名：Frontiers of Mathematics in China
- 出版年：2017
- 出版时间：April 2017
- 年：2017
- 卷：12
- 期：2
- 页码：281-299
- 全文大小：
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics, general;
- 出版者：Higher Education Press
- ISSN：1673-3576
- 卷排序：12
文摘
We consider the oscillatory hyper Hilbert transform Hγ,α,βf(x) = ∫0∞f(x - Γ(t))eit-βt-(1+α)dt; where Γ(t) = (t, γ(t)) in ℝ2 is a general curve. When γ is convex, we give a simple condition on γ such that Hγ,α,β is bounded on L2 when β > 3α, β > 0: As a corollary, under this condition, we obtain the Lp-boundedness of Hγ,α,β when 2β/(2β - 3α) < p < 2β/(3α). When Γ is a general nonconvex curve, we give some more complicated conditions on γ such that Hγ,α,β is bounded on L2: As an application, we construct a class of strictly convex curves along which Hγ,α,β is bounded on L2 only if β > 2α > 0.