L ^p-Boundedness for fractional oscillatory integral operator with rough kernel

Abstract

In this paper we give L^p-boundedness for the operator T_μ defined by

$$T_\mu f\left( x \right) = \int_R \cdot e^{iP\left( {x,y} \right)} \frac{{\Omega \left( {x - y} \right)}}{{\left| {x - y} \right|^{n - \mu } }}b\left( {\left| {x - y} \right|} \right)f\left( y \right)dy$$

where P(x,y) is a real nontrivial polynomial onR ⁿ×R ⁿ, Ω is homogeneous of degree zero, Ωε L^q(Sⁿ⁻¹),q>1/(1−μ) and b(r)εBV(R ₊). The result can be regarded as an improvement of F. Ricci and E. M. Stein’s result for fractional oscillatory integral operator with smoothness kernel[3].