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

References

[1] Chen J., Fan D., Wang M., Zhu X. L ^p bounds for oscillatory hyper Hilbert transforms along curves. Proc Amer Math Soc, 2008, 136, 3145-3153

[2] Chen J., Fan D., Zhu X. Sharp L ² boundedness of the oscillatory hyper Hilbert transform along curves. Acta Math Sin (Engl Ser), 2010, 26 3 653-658

[3] Chandarana S. L ^p bounds for hypersingular integral operators along curves. Pacific J Math, 1996, 175 2 389-416

[4] Fan D., Wu H. Certain oscillatory integrals on unit square and their applications. Sci in China, Ser A, 2008, 51 10 1895-1903

[5] Fefferman C., Stein E. M. H ^p spaces of several variables. Acta Math, 1972, 129, 137-193

[6] Hirschman I. I. Multiplier transforms, I. Duke Math J, 1956, 26, 222-242

[7] Stein E. M., Wainger S. Problems in harmonic analysis related to curvatures. Bull Amer Math Soc, 1978, 84, 1239-1295

[8] Wainger S. Special Trigonometric Series in k Dimension. Mem Amer Math Soc, 1965, Providence: AMS.

[9] Ye X. Boundedness of certain operators on function spaces. Ph D Thesis, 2006, Hangzhou, China: Zhejiang Univ.

[10] Zielinski M. Highly oscillatory singular integrals along curves. Ph D Thesis, 1985, WI, USA: Univ of Wisconsin-Madison.