积域上沿曲线的振荡积分算子在调幅函数空间上的有界性

Boundedness of Oscillatory Integral Operators Along Curves in Product Domain on Modulation Space

设Q^(2)=[0,1]×[0,1]表示一个2维的单位方体.本文主要证明了算子Υ_(α,β)f(x,y)=∫_(Q2)f(x−γ_(1)(t),y−γ_(2)(s))e^(−2πit−β_(1s)−β_(2))t^(−α1−1s−α2−1)dtds在调幅函数空间M_(s)^(p,q)(Rn×Rm)上的有界性,其中(x,y)∈Rn×Rm,γ1(t)=(tp1,tp2,⋯,tpn,γ2(s)=(s^(q1),s^(q2),⋯,s^(qm)分别是R^(n)和R^(m)上的曲线,以及β_(1)>α1>0,β_(2)>α2>0.

Let Q^(2)=[0,1]×[0,1]be the unit square in two dimensions.In this paper,we mainly study the boundedness of the oscillatory integral operatorsΥ_(α,β)on modulation space,which are defined byΥ_(α,β)f(x,y)=∫_(Q2)f(x−γ_(1)(t),y−γ_(2)(s))e^(−2πit−β_(1s)−β_(2))t^(−α_(1)−1_(s)−α_(2−1))dtds where(x,y)∈Rn×Rm,γ_(1)(t)=(tp1,tp2,⋯,tpn,γ2(s)=(s^(q1),s^(q2),⋯,s^(qm)are curves on R^(n)and R^(m),β_(1)>α_(1)>0,β_(2)>α2>0.