Science in China Series A: Mathematics

, Volume 51, Issue 10, pp 1895–1903

Certain oscillatory integrals on unit square and their applications

Authors

  • DaShan Fan
    • Department of MathematicsUniversity of Wisconsin-Milwaukee
    • School of Mathematical SciencesXiamen University
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 integralsingular integralrough kernelunit squareproduct space

MSC(2000)

42B1042B1542B20

Copyright information

© Science in China Press and Springer-Verlag GmbH 2008