Oscillatory hyper Hilbert transforms along general curves

doi:10.1007/s11464-016-0574-3

Front. Math. China

2017, Vol. 12

Issue (2) : 281-299 https://doi.org/10.1007/s11464-016-0574-3

RESEARCH ARTICLE

Oscillatory hyper Hilbert transforms along general curves

Jiecheng CHEN,Belay Mitiku DAMTEW,Xiangrong ZHU(

)

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

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Abstract

We consider the oscillatory hyper Hilbert transform $H γ, α, β f (x) = ∫ 0 ∞ f (x − Γ (t)) e i t − β t − (1 + α) d t$ , where Γ(t) = (t, γ(t)) in $? 2$ is a general curve. When γ is convex, we give a simple condition on γ such that H_γ,_α,_β is bounded on L² when $β ≥ 3 α, β > 0$ . As a corollary, under this condition, we obtain the L^p-boundedness of H_γ,_α,_β when $2 β / (2 β − 3 α) < p < 2 β (3 α)$ . When Γ is a general nonconvex curve, we give some more complicated conditions on γ such that H_γ,_α,_β is bounded on L². As an application, we construct a class of strictly convex curves along which H_γ,_α,_βis bounded on L² only if β>2α>0.

Keywords Hilbert transform oscillatory integral oscillatory hyper Hilbert transform

Corresponding Author(s): Xiangrong ZHU

Issue Date: 27 December 2016

Cite this article:

Jiecheng CHEN,Belay Mitiku DAMTEW,Xiangrong ZHU. Oscillatory hyper Hilbert transforms along general curves[J]. Front. Math. China, 2017, 12(2): 281-299.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-016-0574-3
https://academic.hep.com.cn/fmc/EN/Y2017/V12/I2/281

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