| The boundedness of integral operators and their commutators has always been one of the main topics of harmonic analysis.This thesis mainly studies the boundedness of the Calderon integral operator with variable kernel,the maximal multilinear Calderon-Zygmund singular integrals,the pa.rametrized Littlewood-Paley operators,the parametrized Marcinkiewicz integrals,and their commutators generated by BMO functions on several kinds of function spaces.The bounded properties of these operators and their commutators are widely used in harmonic analysis,partial differential equations and probability theory.The main contents of this thesis are as follows:The first chapter is the introduction,which mainly introduces the research background and the research status at home and abroad,and briefly introduces some basic knowledge and main research contents required in this paper.In Chapter 2,let b ∈ Lip(Rn),by using the properties of Littlewood-Paley functions,Fourier transform and the spherical harmonic representation,we establish the L2(Rn)boundedness for the Calderón commutator[b,T1]with Ω(x,z’)∈L∞(Rn)x Lq(Sn-1)(q>2(n-1)/n)satisfying certain cancellation conditions in the sense that the exponent q>2(n-1)/n is optimal.Our result weakens the requirements for smoothness of kernel functions in a previous result of Calderon.In Chapter 3,let p(·),pi(-)∈P(Rn)∩LH(Rn),λi 0,i=1,...,m,λ=λ1+…·λm and 1/p(·)+=1/p1(·)+…+1/pm(·),we show that the maximal multilinear Calderon-Zygmund operator T*is bounded from Bp1(·),λ1(Rn)×…×Bpm(·),λm(Rn)to Bp(·),λ(Rn).Moreover,similar bounded properties are obtained for the commutators of multilinear Calderon-Zygmund operators as well as for the corresponding fractional integrals.In Chapter 4,We prove the boundedness of parametrized area integral μS(?)and Littlewood-Paley’s gλ*-funtion μλ*,(?)on grand variable Herz spaces Kq(·)α(·),p),θ(Rn),where the two main indices a and q are variable exponents.Furthermore,the boundedness of their higher-order commutators[bm,μS(?)]and[bm,μλ*,(?)]with BMO functions are also established on these spaces.The results are still new even when m=1 and α(·)≡α is constant.In Chapter 5,We study the boundedness of the parametrized Marcinkiewicz integrals μΩρ on the Herz spaces Kp(·),q(·)α(·)(Rn)with three variable exponents.Similar result holds for its higher-order commutators[bm,μΩρ]with BMO symbols.In particular,the result is also new when m=1.The main innovations of this thesis are as follows:1.We weaken the condition Ω∈L∞(Rn)× L∞(Sn-1)to Ω∈L∞(Rn)× Lq(Sn-1)(q>2(n-1)/n)and study the L2 boundedness of commutator of Calderón integral operator with variable kernel.In this case,the original rotation method for dealing with the smooth kernel case is no longer applicable.We use the properties of Littlewood-Paley function,Fourier transform and spherical harmonic function expansion to effectively replace the rotation method,and show that the exponent q>2(n-1)/n is optimal by constructing a counterexample.2.We usually use the boundedness of commutators on Lp(·)(Rn)spaces to study the boundedness of commutators generated by integral operators and BMO functions on variable exponent function spaces.However,when we study the boundedness of commutators of the multilinear Calderón-Zygmund operator with CBMO functions on products of central Morrey spaces with variable exponent,since the behavior of CBMO may be quite different from that of BMO space,then the commutator Tb*is not necessarily of type(Lpi(·)×…×Lpm(·),Lp(·)).We solve this problem by using function decomposition,the theory of variable exponent function spaces and the generalized central BMO norms.3.We use the generalized BMO norms and the expansion of the binomial theorem and obtain the boundedness of the higher-order commutators of paramet-rized Littlewood-Paley operators and paramet rized Marcinkiewicz integrals on Herz spaces with variable exponents,which is different from the usual method of dividing the higher-order commutators into two parts. |
