Some Estimates For Marcinkiewicz Type Operators And Its Commutators

It is well-known that the Marcinkiewicz integral operator is very importantin harmonic analysis. Marcinkiewicz [1] considered the expressionμ(f)(x) givenbyμ(f)(x)=(intrgral form n=[0,2Ï€](F(x+t)+F(x-t)-2F(x))/t³ dt^1/2,x∈[0, 2Ï€],where F(x)=integral from n=[0,x] f(t)dt. In 1958, Stein introduced the higher-dimensionalMarcinkiewicz integrals operatorμ_Ωdefined byμ_Ωf(x)=intrgral from n=0 to∞|integral from to |y|≤t f(x-y),Ω(y)/(|y|^n-1)dy|² dt/t³_<sup>1/2,for f∈L_loc¹(Rⁿ).Stein [2] showed that ifΩ∈Lip_α(S^n-1)for some 0＜α≤1, thenμ_Ωis a boundedoperator on L^p(Rⁿ) for 1＜p≤2, and a bounded mapping from L¹(Rⁿ) to weakL¹(Rⁿ). In 1990, Torchinsky and Wang Shilin[13] introduced the definition of thecommutator of the Marcinkiewicz integral. For b∈L_loc(Rⁿ), the commutator ofthe Marcinkiewicz integralμ_{Ω, b} is defined asμ_Ω,b(f)(x)=intrgral from n=0 to∞|F_Ω,b,t(x)|²dt/t³)^1/2,where F_{Ω, b, t}(x)=integral from n=|x-y|≤t(Ω(x-y))/(|x-y|^n-1)(b(x)-b(y))f(y)dy.If b(x)∈BMO(Rⁿ), they [13]showed that ifΩ∈Lip_α(S^n-1)(0＜α≤1),then for 1＜p＜∞, w∈A_p(the weight class of Muckenhoupt's)μ_Ω, b is boundedon L^p(w).In chapter 1, the boundedness on the weighted Herz spaces is established fora class of Marcinkiewicz integral commutators generated by BMO(Rⁿ) functionsand Marcinkiewicz integrals with rough kernelsΩ∈L^r(S^n-1), 1＜r≤∞, andintegral from n=S^n-1Ω(x)dσ(x)=0.In chapter 2, the boundedness on Herz spies is established for the high ordercommutators of Marcinkiewicz integral generated by BMO(Rⁿ)functions withΩ∈L(logL)^m+1(S^n-1).In chapter 3, we study the boundeness of the commutator of Marcinkiewiczintegral operator on BMO^p(Rⁿ)(BMO^p(Rⁿ)=BMO(Rⁿ)∩L^p(Rⁿ) (1＜p＜∞)).In chapter 4, we show that the Marcinkiewicz oscillatory integrals is boundedon L^p(Rⁿ)(2≤p＜∞). Furthermore, we also give the endpoint estimate forthe Marcinkiewicz oscillatory integrals on Hardy spaces.