Study On Bessel Operators And Other Related Operators

The main purpose of this thesis is to study the boundedness of some integral operators associated with the second order elliptic operators,Bessel operators and Schrodinger operators in some function spaces.These three kinds of operators are generalized forms of operators extracted from elliptic equation,Laplace equation and Schrodinger equation.The main innovations of this thesis are summarized as follows:1.The second order elliptic operator is more complex than the Laplacian operator,and the rotation method dealing with Calderon commutators is invalid for the commutators of second order elliptic operator.The method of Sobolev Calderon-Zygmund decomposition and off-diagonal estimation was used to replace the rotation method effectively,the weak(1,1)boundedness of the commutators for Sobolev function and square root associated with second order elliptic operator was re-estimated.Finally,the index of p=2 in the gradient estimate for the commutator of the square root of the second order elliptic operator and the Sobolev function was enlarged to p-(L)<p<p+(L).2.The phase function Semigroups of square root type square function operators can not be completely written into the differential form of heat semigroups.That is,the kernel functions of these operators have no specific expression of heat kernel.By using the method of functional calculus and the properties of the kernel function of the thermal semigroup of the Bessel operator,the upper bound estimate of the kernel of square root type square function operator was estimated.Thus,the boundedness proof on the various function spaces can be realized.3.A new type of BMO space related to generalized Schrodinger operator was defined,which is larger than BMO space related to classical Schrodinger operator,and the boundedness of littlewood-Paley g-function on this new space was verified.The contents of this thesis are as follows:In Chapter 2,by using the method of Sobolev Calderon-Zygmund decompo-sition and off diagonal estimate,the commutator[b,(?)]which is formed by Kato square root(?)and Sobolev function b satisfying ▽b ∈Ln(Rn)(n>2)was bounded from Sobolev space,L1p(Rn)to LP(Bn),with(p-(L)<p<p+(L)).In Chapter 3,the equivalence between the square roots of two classes of Bessel operators and their corresponding differential operators under LP norm.In addi-tion,by using holomorphic functional calculus,we obtain the boundedness of weak(1,1),H1 to L1.Finally,the boundedness of the square root type square function associated with the Bessel operator Sλ on BMO space also be proved.In Chapter 4,on the basis of the estimate for the square function kernel of the Bessel operator in Chapter 3,It is further verified that the commutator of square function associated with △λ,[b,g△λ],which is bounded(or compact)on LP(R+,x2λdx),if and only if b ∈ BMO(R+,x2λdx)(or b ∈ CMO(R+,x2λdx)).Thus,we get the commutator[b,g△λ]can describe BMO(or CMO)space.In Chapter 5,let(?)=△+μ be the generalized Schrodinger operator on Rn,n≥3,where μ≠0 is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions.A new BMO space associated to the generalized Schrodinger operator(?)was defined,called BMOθ,(?),which is bigger than the BMO spaces related to the classical Schrodinger operators A＝-△+V,where V is a potential satisfying a reverse Holder inequality.Besides,the boundedness of the Littlewood-Paley g-function associated to(?)in BMOθ，(?)also be proved.In Chapter 6,on the one hand,the Lp-boundedness of the commutators[b,▽(?)-1/2]of Riesz transforms associated to generalized Schrodinger operator ▽(?)-1/2 and b with b ∈ BMO(Rn)was studied.On the other hand,by using the compactness criteria of commutator related to Schrodinger operator,the LP-compactness of the commutators[b,(?)-1/2▽]also be proved.