摘要
We consider the oscillatory hyper Hilbert transform Hγ,α,βf（x） = ∫0^∞ f（x - Г（t））eit-βt-（1＋α）dt, where Г（t） = （t, γ（t）） in R^2 is a general curve. When γ is convex, we give a simple condition on γ such that Hγ,α,βis 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 F is a general nonconvex curve, we give some more complicated conditions on γ such that Hγ,α,βis 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.
We consider the oscillatory hyper Hilbert transform Hγ,α,βf（x） = ∫0^∞ f（x - Г（t））eit-βt-（1＋α）dt, where Г（t） = （t, γ（t）） in R^2 is a general curve. When γ is convex, we give a simple condition on γ such that Hγ,α,βis 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 F is a general nonconvex curve, we give some more complicated conditions on γ such that Hγ,α,βis 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.
基金
Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant Nos. 11671363, 11471288, 11371136）, the Natural Science Foundation of Zhejiang Province （No. LY14A010015）, and the China Scholarship Council.
