Science in China Series A: Mathematics

, Volume 51, Issue 10, pp 1895–1903 | Cite as

Certain oscillatory integrals on unit square and their applications

Article

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 

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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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