一类超奇异Marcinkiewicz积分的加权有界性

Weighted Boundedness of Hyper Singular Marcinkiewicz Integral Operators

考虑粗糙核超奇异Marcinkiewicz积分算子为:α≥0,其中,核函数Ω∈Hq(Sn-1),q=(n-1/)(n-1+α),且Ω是零次齐次函数,同时满足[(n-1)(1/q-1)]次消失性;b(r)∈L∞(R+)为径向函数.建立了上述算子μΩb,α从加权齐次Sobolev空间Lαp(ω)到加权空间Lp(ω)的有界性,其中ω是适当的Ap权,1<p<∞.同时也证明了当2≤p<∞时,相应于gλ*函数和面积积分函数的Marcinkiewicz积分算子μΩ,λ,α *,b和μΩb,S,α的Lαp(ω)到Lp(ω)的有界性。

We discuss a class weighted norm inequality for the following Marcinkiewicz integral operator μΩ.α^b（f）=（∫0^∞｜∫｜x-y｜≤tΩ（x-y/｜x-y｜^n-1）b（｜x-y｜）f（y）dy｜^2dt/t^3＋2a）^1/2,a≥0 Where Ω∈H^q（S^n-1）,q=（n-1/）（n-1＋α） satisfying certain cancelation conditions, b（r）∈L^∞（R＋）, and ω∈Ap. We also give the weighted boundedness for a class of Marcinkiewicz integral operators with rough kernels μΩ.λ.α^·,b andμΩ.s.α^b, respectively.