Oscillatory integrals on unit square along surfaces

Abstract

Let Q² = [0, 1]² be the unit square in two-dimensional Euclidean space {\mathrm{ℝ}}^{\mathrm{2}}. We study the L^p boundedness of the oscillatory integral operator T_α,β defined on the set ({\mathrm{ℝ}}^{\mathrm{2}+n}) of Schwartz test functions by

T_{\alpha ,\beta }f(u,v,x)={\displaystyle {\int }_{Q^{\mathrm{2}}}}{\displaystyle \frac{f(u-t,v-s,x-\gamma (t,s))}{t^{\mathrm{1}+{\alpha }_{\mathrm{1}}}{s}^{\mathrm{1}+{\alpha }_{\mathrm{2}}}}}{\mathrm{e}}^{\mathrm{i}{t}^{-{\beta }_{\mathrm{1}}}{s}^{-{\beta }_{\mathrm{2}}}}\mathrm{d}t\mathrm{d}s,

where x\in {\mathrm{ℝ}}^n, (u,v)\in {\mathrm{ℝ}}^{\mathrm{2}}, (t,s,\gamma (t,s))=(t,s,t^{p_{\mathrm{1}}}{s}^{q_{\mathrm{1}}},t^{p_{\mathrm{2}}}{s}^{q_{\mathrm{2}}},\cdots ,t^{p_n}{s}^{q_n}) is a surface on {\mathrm{ℝ}}^{n+\mathrm{2}}, and β₁>α₁, β₂>α₂. Our results extend some known results on {\mathrm{ℝ}}^{\mathrm{3}}.

Keywords

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

References

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