# Certain oscillatory integrals on unit square and their applications

## Authors

Article

- First Online:

- Received:
- Accepted:

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 where β

*Q*^{2}= [0, 1]^{2}be the unit square in two dimension Euclidean space ℝ^{2}. We study the*L*^{p}boundedness properties of the oscillatory integral operators*T*_{α,β}defined on the set*S*(ℝ^{3}) of Schwartz test functions*f*by$$
\mathcal{T}_{\alpha ,\beta } f(x,y,z) = \int_{Q^2 } {f(x - t,y - s,z - t^k s^j )e^{ - it^{ - \beta _1 } s^{ - \beta 2} } t^{ - 1 - \alpha _1 } s^{ - 1 - \alpha _2 } dtds} ,
$$

_{1}> α_{1}⩾ 0, β_{2}> α_{2}⩾ 0 and (*k*,*j*) ∈ ℝ^{2}. As applications, we obtain some*L*^{p}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