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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ L^p$ bounds for oscillatory hyper-Hilbert transform along curves


Authors: Jiecheng Chen, Dashan Fan, Meng Wang and Xiangrong Zhu
Journal: Proc. Amer. Math. Soc. 136 (2008), 3145-3153
MSC (2000): Primary 42B25
Published electronically: May 2, 2008
MathSciNet review: 2407077
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Abstract: We study the oscillatory hyper-Hilbert transform

$\displaystyle H_{n,\alpha,\beta}f(x)=\int^1_0 f(x-\Gamma(t))e^{it^{-\beta}}t^{-1-\alpha}dt$ (1)

along the curve $ \Gamma(t)=(t^{p_1},t^{p_2},\cdots,t^{p_n})$, where $ p_1,p_2,\cdots,p_n,\alpha,\beta$ are some real positive numbers. We prove that if $ \beta>(n+1)\alpha$, then $ H_{n,\alpha,\beta}$ is bounded on $ L^p$ whenever $ p \in(\frac{2\beta}{2\beta-(n+1)\alpha},\frac{2\beta}{(n+1)\alpha})$. Furthermore, we also prove that $ H_{n,\alpha,\beta}$ is bounded on $ L^2$ when $ \beta=(n+1)\alpha$. Our work improves and extends some known results by Chandarana in 1996 and in a preprint. As an application, we obtain an $ L^p$ boundedness result for some strongly parabolic singular integrals with rough kernels.


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Additional Information

Jiecheng Chen
Affiliation: Department of Mathematics, Zhejiang, University, Hangzhou, Zhejiang, People’s Republic of China
Email: jcchen@mail.hz.zj.cn

Dashan Fan
Affiliation: Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201
Email: harmonic_analysis@yahoo.com

Meng Wang
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, People’s Republic of China
Email: mathdreamcn@zju.edu.cn

Xiangrong Zhu
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, People’s Republic of China
Email: zxr@zju.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09325-8
PII: S 0002-9939(08)09325-8
Received by editor(s): July 8, 2005
Received by editor(s) in revised form: May 23, 2007
Published electronically: May 2, 2008
Additional Notes: This work was supported by NSFC (10371043, 10571156, 10601046) and PDSFC (20060400336)
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2008 American Mathematical Society