Acta Mathematica Sinica, English Series

, Volume 26, Issue 4, pp 653–658

Sharp L2 boundedness of the oscillatory hyper-Hilbert transform along curves

Article

DOI: 10.1007/s10114-010-7396-0

Cite this article as:
Chen, J.C., Fan, D.S. & Zhu, X.R. Acta. Math. Sin.-English Ser. (2010) 26: 653. doi:10.1007/s10114-010-7396-0

Abstract

Consider the oscillatory hyper-Hilbert transform
$$ H_{n,\alpha ,\beta } f(x) = \int_0^1 {f(x - \Gamma (t))e^{it^{ - \beta } } t^{ - 1 - \alpha } dt} $$
along the curve Γ(t) = (tp1, tp2, ..., tpn), where β > α ≥ 0 and 0 < p1 < p2 < ... < pn. We prove that Hn,α,β is bounded on L2 if and only if β ≥ (n + 1)α. Our work extends and improves some known results.

Keywords

oscillatory hyper-Hilbert transform 

MR(2000) Subject Classification

42B25 

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer Berlin Heidelberg 2010

Authors and Affiliations

  • Jie Cheng Chen
    • 1
  • Da Shan Fan
    • 2
  • Xiang Rong Zhu
    • 3
  1. 1.Department of MathematicsZhejiang UniversityHangzhouP. R. China
  2. 2.Department of MathematicsUniversity of Wisconsin-MilwaukeeMilwaukeeUSA
  3. 3.Mathematics SectionICTPTriesteItaly