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 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
The hyper Hilbert transform Tnf（x） =∫-1^1 f（x - Γ（t））e^-i｜t｜-β｜t｜^-1-αdt along an appropriate curve Γ（t） on R^n is investigated,where β 〉 α 〉 0.An L^p boundedness theorem of T4 is obtained,which i...The hyper Hilbert transform Tnf（x） =∫-1^1 f（x - Γ（t））e^-i｜t｜-β｜t｜^-1-αdt along an appropriate curve Γ（t） on R^n is investigated,where β 〉 α 〉 0.An L^p boundedness theorem of T4 is obtained,which is an extension of some earlier results of n = 2 and n = 3.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
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 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.
基金Supported by the National Natural Science Foundation of China (1057115610701064)+1 种基金ZJNSF (RC97017)the Zijin Project of Zhejiang University
文摘The hyper Hilbert transform Tnf（x） =∫-1^1 f（x - Γ（t））e^-i｜t｜-β｜t｜^-1-αdt along an appropriate curve Γ（t） on R^n is investigated,where β 〉 α 〉 0.An L^p boundedness theorem of T4 is obtained,which is an extension of some earlier results of n = 2 and n = 3.
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.
