单位方体上沿曲面的振荡积分的Sobolev有界性

The boundedness of certain oscillatory integrals on unit square along surfaces on Sobolev spaces

研究了欧氏空间R^2中单位方体Q^2=[0,1]2上沿曲面(t,s,t^ks^j)的振荡奇异积分算子-Tkα,βf(x,y,z)=∫f(x-t,y-s,z-tsj)e-itβ1-sβ2t-1-α-α1 s-1 2 dtdsQ2从Sobolev空间Lp r(R3)到Lp(R3)中的有界性,其中β_1>α_1≥0,β_2>α_2≥0,(k,j)∈R^2.最后,得到了乘积空间上粗糙核奇异积分算子的Sobolev有界性.

Let Q2= [0,1]2be the unit square in two dimension Euclidean space R^2. It was studied the boundedness properties from Sobolev spaces Lpr( R3) to Lp( R3) of the oscillatory integral operator τα,βdefined on the set S( R^3) of Schwartz test funtions f by-Tα,βf( x,y,z) =∫f( x- t,y- s,z- tksj) e- itβ1- sβ2t-1- α α1 s-1-2 dtds,Q2where( t,s,tksj) is a surface on R3,β1> α1≥0,β2> α2≥0 and( k,j) ∈R^2. As applications,some Sobolev boundedness results of rough singular integral operators on the product spaces were obtained.