Science in China Series A: Mathematics

, Volume 51, Issue 10, pp 1895–1903

Certain oscillatory integrals on unit square and their applications

Article

DOI: 10.1007/s11425-008-0076-1

Cite this article as:
Fan, D. & Wu, H. Sci. China Ser. A-Math. (2008) 51: 1895. doi:10.1007/s11425-008-0076-1

Abstract

Let Q2 = [0, 1]2 be the unit square in two dimension Euclidean space ℝ2. We study the Lp boundedness properties of the oscillatory integral operators Tα,β defined on the set S(ℝ3) of Schwartz test functions f by
$$ \mathcal{T}_{\alpha ,\beta } f(x,y,z) = \int_{Q^2 } {f(x - t,y - s,z - t^k s^j )e^{ - it^{ - \beta _1 } s^{ - \beta 2} } t^{ - 1 - \alpha _1 } s^{ - 1 - \alpha _2 } dtds} , $$
where β1 > α1 ⩾ 0, β2 > α2 ⩾ 0 and (k, j) ∈ ℝ2. As applications, we obtain some Lp boundedness results of rough singular integral operators on the product spaces.

Keywords

oscillatory integral singular integral rough kernel unit square product space 

MSC(2000)

42B10 42B15 42B20 

Copyright information

© Science in China Press and Springer-Verlag GmbH 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Wisconsin-MilwaukeeMilwaukeeUSA
  2. 2.School of Mathematical SciencesXiamen UniversityXiamenChina

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