Oscillatory integrals on unit square along surfaces

Abstract

Let Q ² = [0, 1]² be the unit square in two-dimensional Euclidean space ℝ². We study the L ^p boundedness of the oscillatory integral operator T _α,β defined on the set ℒ(ℝ²⁺ⁿ) of Schwartz test functions by $T_{\alpha ,\beta } f(u,v,x) = \int_{Q^2 } {\frac{{f(u - t,v - s,x - \gamma (t,s))}} {{t^{1 + \alpha _1 } s^{1 + \alpha _2 } }}} e^{it - \beta _{1_s } - \beta _2 } dtds,$ where x ∈ ℝ ⁿ, (u, v) ∈ ℝ², (t, s, γ(t, s)) = (t, s, $t^{p_1 } s^{q_1 } ,t^{p_2 } s^{q_2 } ,...,t^{p_n } s^{q_n } $) is a surface on ℝ ⁿ⁺², and β ₁ > α ₁, β ₂ > α ₂. Our results extend some known results on ℝ³.

Keywords

Oscillatory integral / singular integral / unit square / surface / product space

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