Consider the oscillatory hyper-Hilbert transform Hn,α,βf（x）=∫0^1 f（x-Г（t））e^it-βt^-1-α dt along the curve P（t） = （tp1, tP2,..., tpn）, where β 〉 α ≥ 0 and 0 〈 p1 〈 p2 〈 ... 〈 Pn. We prove that ...Consider the oscillatory hyper-Hilbert transform Hn,α,βf（x）=∫0^1 f（x-Г（t））e^it-βt^-1-α dt along the curve P（t） = （tp1, tP2,..., tpn）, where β 〉 α ≥ 0 and 0 〈 p1 〈 p2 〈 ... 〈 Pn. We prove that H n,α,β is bounded on L2 if and only if β ≥ （n ＋ 1）α. Our work extends and improves some known results.展开更多
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...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.展开更多
We consider the boundedness of the n-dimension oscillatory hyper- Hilbert transform along homogeneous curves on the α-modulation spaces, including the inhomogeneous Besov spaces and the classical modulation spaces. T...We consider the boundedness of the n-dimension oscillatory hyper- Hilbert transform along homogeneous curves on the α-modulation spaces, including the inhomogeneous Besov spaces and the classical modulation spaces. The main theorems significantly improve some known results.展开更多
We prove the boundedness from Lp（T2） to itself, 1 〈 p 〈∞, of highly oscillatory singular integrals Sf（x, y） presenting singularities of the kind of the double Hilbert transform on a non-rectangular domain of in...We prove the boundedness from Lp（T2） to itself, 1 〈 p 〈∞, of highly oscillatory singular integrals Sf（x, y） presenting singularities of the kind of the double Hilbert transform on a non-rectangular domain of integration, roughly speaking, defined by ｜y′｜ 〉 ｜x′｜, and presenting phases λ（Ax ＋ By） with 0≤ A, B ≤ 1 and λ≥ 0. The norms of these oscillatory singular integrals are proved to be independent of all parameters A1 B and A involved. Our method extends to a more general family of phases. These results are relevant to problems of almost everywhere convergence of double Fourier and Walsh series.展开更多
文摘Consider the oscillatory hyper-Hilbert transform Hn,α,βf（x）=∫0^1 f（x-Г（t））e^it-βt^-1-α dt along the curve P（t） = （tp1, tP2,..., tpn）, where β 〉 α ≥ 0 and 0 〈 p1 〈 p2 〈 ... 〈 Pn. We prove that H n,α,β is bounded on L2 if and only if β ≥ （n ＋ 1）α. Our work extends and improves some known results.
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.
文摘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 The authors are thankful to the referees for their careful reading and useful comments. This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11501516, 11471288） and the Natural Science Foundation of Zhejiang Province （No. LQ15A010003）.
文摘We consider the boundedness of the n-dimension oscillatory hyper- Hilbert transform along homogeneous curves on the α-modulation spaces, including the inhomogeneous Besov spaces and the classical modulation spaces. The main theorems significantly improve some known results.
文摘We prove the boundedness from Lp（T2） to itself, 1 〈 p 〈∞, of highly oscillatory singular integrals Sf（x, y） presenting singularities of the kind of the double Hilbert transform on a non-rectangular domain of integration, roughly speaking, defined by ｜y′｜ 〉 ｜x′｜, and presenting phases λ（Ax ＋ By） with 0≤ A, B ≤ 1 and λ≥ 0. The norms of these oscillatory singular integrals are proved to be independent of all parameters A1 B and A involved. Our method extends to a more general family of phases. These results are relevant to problems of almost everywhere convergence of double Fourier and Walsh series.
