带拟微分算子高阶交换子的加权L2有界性

摘要

The boundedness of higher order commutators generated by Pseudo-differential operators Tf(x) and BMO functions is studied on weighted L2 spaces,which generalized the Chanillo' s conclusions in 1997,and better result was obtained.When w ∈ A2,T ∈ Lmρ,δ,0 ≤δ ＜ρ ＜ 1/2 and m ＜ 0,let b ∈ BMO,under the assumption conclusion has been established for t-1 order,it is investigatedf∈ L2 (we2bcosθ) for arbitrary θ ∈ [0,2πrr] according to the linear property of pseudo-differential operator and using the stein-weiss restricted interpolation theorem.Finally,using the Minkowski inequality and Plancherel theorem,it is proven that the conclusion was also correct for t order.From which,the weighted L2 boundedness for higher order commutators with variable symbol of pseudo-differential operator [b,T] mf(x) =∫Rna(x,z) f(z) e2πix·ξ (b (x)-b (z)) mdz is obtained.%研究一类带变象征的拟微分算子Tf(x)的高阶交换子的L2有界性,推广了Chanillo的结论,并得到更优的结果.当ω∈A2,T∈Lmρ,δ,0≤δ＜ρ＜1/2且m＜0时,若b∈BMO,假设结论对t-1阶成立,根据拟微分算子的线性性质,运用Stein-Weiss限制性插值定理,得到对于任意的θ∈[0,2π],有f∈L2(we2bcosθ).利用Minkowski不等式和Plancherel定理,证明结论对t阶也成立,由此得到带变象征拟微分算子的高阶交换子[b,T]mf(x)=∫Rna(x,z)f(z)e2πix·ξ(b(x)-b(z))mdz的加权L2有界性质.