Boundedness for commutators of multipliers on Hardy type spaces

In this paper, we study the commutators generalized by multipliers and a BMO function. Under some assumptions, we establish its boundedness properties from certain atomic Hardy space Hb^p（R^n） into the Lebesgue space L^p with p 〈 1.     

On the maximal operator associated with certain rotational invariant measures

The aim of this work is to investigate the integrability properties of the maximal operator Mu,associated with a non-doubling measure μ defined on Rn. We start by establishing for a wide class of radial and increasing measures μ that Mu is bounded on all the spaces Lu^p（R^n）,P〉1.Also,we show that there is a radial and increasing measure p for which Mμ does not map Lμ^p（R^n） into weak Lμ^p（R^n）,1≤p〈∞.     

COMMUTATORS OF MULTIPLIERS ON HARDY SPACES

Let T be the multiplier operator associated to a multiplier m, and [b, T] be the commutator generated by T and a BMO function b. In this paper, the authors have proved that [b,T] is bounded from the Hardy space H^1（R^n） into the weak L^1 （R^n） space and from certain atomic Hardy space Hb^1 （R^n） into the Lebesgue space L^1 （R^n）, when the multiplier m satisfies the conditions of Hoermander type.     

Solving a Class of Brouwer Fixed-point Problems via a Modified Aggregate Constraint Homotopy Method

In this paper, we provide an aggregate function homotopy interior point method to solve a class of Brouwer fixed-point problems. Compared with the homotopy method （proposed by Yu and Lin, Appl. Math. Comput., 74（1996）, 65）, the main adavantages of this method are as foUows： on the one hand, it can solve the Brouwer fixed-point problems in a broader class of nonconvex subsets Ω in R^n （in this paper, we let Ω={x∈ R^n ： gi（x） ≤0, i= 1,... , m}）; on the other hand, it can also deal with the subsets Ω with larger amount of constraints more effectively.     

WEAK TYPE（H^1 ,L^1） ESTIMATE FOR COMMUTATOR OF MARCINKIEWICZ INTEGRAL

Let μΩ,b be the commutator generalized by the n-dimensional Marcinkiewicz integral μΩ and a function b∈ BMO（R^n）. It is proved that μΩ,bis bounded from the Hardy space H^1 （R^n） into the weak L^1（R^n） space.     

REGULARITY OF SOLUTIONS TO NONLINEAR TIME FRACTIONAL DIFFERENTIAL EQUATION

We find an upper viscosity solution and give a proof of the existence-uniqueness in the space C^∞（t∈（0,∞）;H2^s＋2（R^n））∩C^0（t∈[0,∞）;H^s（R^n））,s∈R,to the nonlinear time fractional equation of distributed order with spatial Laplace operator subject to the Cauchy conditions ∫0^2p（β）D＊^βu（x,t）dβ=△xu（x,t）＋f（t,u（t,x））,t≥0,x∈R^n,u（0,x）=φ（x）,ut（0,x）=ψ（x）,（0.1） where △xis the spatial Laplace operator,D＊^β is the operator of fractional differentiation in the Caputo sense and the force term F satisfies the Assumption 1 on the regularity and growth. For the weight function we take a positive-linear combination of delta distributions concentrated at points of interval （0, 2）, i.e., p（β） =m∑k=1bkδ（β-βk）,0〈βk〈2,bk〉0,k=1,2,…,m.The regularity of the solution is established in the framework of the space C^∞（t∈（0,∞）;C^∞（R^n））∩C^0（t∈[0,∞）;C^∞（R^n））when the initial data belong to the Sobolev space H2^8（R^n）,s∈R.     

Parametrized Area Integrals on Hardy Spaces and Weak Hardy Spaces

In this paper, the authors prove that if Ω satisfies a class of the integral Dini condition, then the parametrized area integral μΩ,S^ρ is a bounded operator from the Hardy space H1 （R^n） to L1 （R^n） and from the weak Hardy space H^1,∞ （R^n） to L^1,∞ （R^n）, respectively. As corollaries of the above results, it is shown that μΩ,S^ρ is also an operator of weak type These conclusions are substantial improvement and （1, 1） and of type （p,p） for 1 〈 p 〈 2, respectively extension of some known results.     

Decompositions of the Hardy space Hz^1（Ω）

We give a decomposition of the Hardy space Hz^1（Ω） into ＂div-curl＂ quantities for Lipschitz domains in R^n. We also prove a decomposition of Hz^1（Ω） into Jacobians det Du, u ∈ W0^1,2 （Ω,R^2） for Ω in R^2. This partially answers a well-known open problem.     

An Endpoint Estimate for the Commutator of Singular Integrals

It is well known that the commutator Tb of the singular integral operator T with a BMO function b is bounded on L^P（R^n）, 1 〈 p 〈 ∞. In this paper, we consider the endpoint estimates for a kind of commutator of singular integrals. A BMO-type estimate for Tb is obtained under the assumption b ∈ LMO.     

Nonlinear Parabolic Equations with Singularities in Colombeau Vector Spaces

We consider nonlinear parabolic equations with nonlinear non-Lipschitz＇s term and singular initial data like Dirac measure, its derivatives and powers. We prove existence-uniqueness theorems in Colombeau vector space yC^1,W^2,2（[0,T）,R^n）,n ≤ 3. Due to high singularity in a case of parabolic equation with nonlinear conservative term we employ the regularized derivative for the conservative term, in order to obtain the global existence-uniqueness result in Colombeau vector space yC^1,L^2（[0,T）,R^n）,n≤ 3.     

Toeplitz operators related to strongly singular Calderón-Zygmund operators

In this paper, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε (ℝⁿ) is discussed from L ^p(ℝⁿ) to L ^q(ℝⁿ), , and from L ^p(ℝⁿ) to Triebel-Lizorkin space . We also obtain the boundedness of generalized Toeplitz operator Θ _α0 ^b from L ^p(ℝⁿ) to L ^q(ℝⁿ), . All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L ^p(ℝⁿ), 1 < p < ∞.     

Boundedness of Some Marcinkiewicz Integral Operators Related to Higher Order Commutators on Hardy Spaces

In this paper, the authors study the boundedness properties of μΩ↑m,b generated by the function b ∈Lipβ（R^n）（0 〈β≤ 1/m） and the Marcinkiewicz integrals operator μΩ. The boundednesses are established on the Hardy type spaces Hb^m^p,n（R^n） and the Herz Hardy type spaces Hbm Kq^α,p（R^b）.     

Herz-type Triebel-Lizorkin Spaces, Ⅰ

Let s ∈R,0〈β≤∞, 0〈 q, p〈 ∞ and-n/q〈α. In this paper the authors introduce the Herz-type Triebel-Lizorkin spaces,Kq^α,pFβ^s（R^n）andKq^α,pFβ^s（R^n）which are the generalizations of the well-known Herz-type spaces and the inhomogeneous Triebel-Lizorkin spaces, Some properties on these Herz-type Triebel Lizorkin spaces are also given.     

Characterization for commutators of n -dimensional fractional Hardy operators

In this paper, it was proved that the commutator generated by an n-dimensional fractional Hardy operator and a locally integrable function b is bounded from L ^p1 (ℝ ⁿ ) to L ^p2 (ℝ ⁿ ) if and only if b is a CṀO(ℝ ⁿ ) function, where 1/p ₁ − 1/p ₂ = β/n, 1 < p ₁ < ∞, 0 ⩽ β < n. Furthermore, the characterization of on the homogenous Herz space (ℝ ⁿ ) was obtained. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10571014, 10371080) and the Doctoral Programme Foundation of Institute of Higher Education of China (Grant No. 20040027001)     

THE BOUNDEDNESS OF MULTILINEAR COMMUTATORS FROM L^P （R^n） TO TRIEBEL-LIZORKIN SPACES

Let (→b)=(b1,…,bm),bi∈Λβi(Rn),1≤i≤m,0＜βi＜β,0＜β＜1,[(→b),T]f(x)=∫Rn,(b1(x)-b1(y))…(bm(x)-bm(y)))K(x-y)f(y)dy where K is a Calder(o)n-Zygmund kernel.In this paper,we show that[(→b),T] is bounded from Lp (Rn) to Fβ,∞p(Rn),as well as[(→b,Iα)] from Lp(Rn) to Fβ,∞p(Rn),where 1/q=1/p-α/n.     

Hardy spaces estimates for a class of multilinear homogeneous operators

It is proved that a class of multilinear singular integral operators with homogeneous kernels are bounded operators from the product spaces to the Hardy spacesH ^r , (ℝ ⁿ ) and the weak Hardy spaceH ^r,∞ (ℝ ⁿ . As an application of this result, the L ^p ,(ℝ ⁿ ) boundedness of a class of commutator for the singular integral with homogeneous kernels is obtained. Project supparted in part by the National Natural Science Foundation of Chind (Grant No. 19131080) of China and Doctoral Programme Foundation of Institution of Higher Education (Grant No. 98002703) of China.     

Darboux integrability of real polynomial vector fields on regular algebraic hypersurfaces

In this paper we extend the Darboux theory of integrability in ℝ ⁿ to the regular algebraic hypersurfaces. Then we apply the extended theory first to the 3-dimensional generalized cylinders ×ℝ^3−r of ℝ⁴ forr=0, 1, 2, 3; and after to then-dimensional sphere of ℝ ⁿ⁺¹.     

Certain oscillatory integrals on unit square and their applications

Let Q2 = [0, 1]2 be the unit square in two dimension Euclidean space R2. We study the Lp boundedness properties of the oscillatory integral operators Tα,β defined on the set S(R3) of Schwartz test functions f by Tα,βf(x,y,z) = Q2 f(x - t,y - s,z - tksj)e-it-β1s-β2t-1-α1s-1-α2dtds, where β1 > α1 0, β2 > α2 0 and (k, j) ∈ R2. As applications, we obtain some Lp boundedness results of rough singular integral operators on the product spaces.     

Polynomial solutions of quasi-homogeneous partial differential equations

By means of a method of analytic number theory the following theorem is proved. Letp be a quasi-homogeneous linear partial differential operator with degreem,m > 0, w.r.t a dilation given by ( a₁, …, a_n). Assume that either a₁, …, a_n are positive rational numbers or for some Then the dimension of the space of polynomial solutions of the equationp[u] = 0 on ℝⁿ must be infinite