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The Oscillatory Hyper-Hilbert Transform Associated with Plane Curves

Published online by Cambridge University Press:  20 November 2018

Junfeng Li  and
Haixia Yu
Junfeng Li
Affiliation:
Laboratory of Math and Complex systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China, e-mail : lijunfeng@bnu.edu.cn, yuhaixia@mail.bnu.edu.cn
Haixia Yu
Affiliation:
Laboratory of Math and Complex systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China, e-mail : lijunfeng@bnu.edu.cn, yuhaixia@mail.bnu.edu.cn
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Abstract

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In this paper, the bounded properties of oscillatory hyper-Hilbert transformalong certain plane curves $\gamma \left( t \right)$,

$${{T}_{\alpha ,\beta }}f\left( x,\,y \right)\,=\,\int_{0}^{1}{f\left( x\,-\,t,\,y\,-\,\gamma \left( t \right) \right){{e}^{i{{t}^{-\beta }}}}\frac{\text{d}t}{{{t}^{1}}+\alpha }}$$

are studied. For general curves, these operators are bounded in ${{L}^{2}}\left( {{\mathbb{R}}^{2}} \right)$ if $\beta \,\ge \,3\alpha $. Their boundedness in ${{L}^{p}}\left( {{\mathbb{R}}^{2}} \right)$ is also obtained, whenever $\beta \,\ge \,3\alpha $ and $\frac{2\beta }{2\beta -3\alpha }\,<\,p\,<\,\frac{2\beta }{3\alpha }$.

Keywords

42B2042B35oscillatory hyper-Hilbert transformoscillatory integral
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 4 , 01 December 2018 , pp. 802 - 811
Copyright
Copyright © Canadian Mathematical Society 2018

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