Certain oscillatory integrals on unit square and their applications

Abstract

Let Q ² = [0, 1]² be the unit square in two dimension Euclidean space ℝ². We study the L ^p boundedness properties of the oscillatory integral operators T _α,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 L ^p boundedness results of rough singular integral operators on the product spaces.