THE FRACTIONAL TYPE MARCINKIEWICZ INTEGRALS AND COMMUTATORS ON WEIGHTED HARDY SPACES

This paper is devoted to studying the behaviors of the fractional type Marcinkiewicz integralsμΩ,βand the commutatorsμΩ,βb generated byμΩ,βwith b b∈Lloc(Rn)on weighted Hardy spaces.Under the assumption of that the homogeneous kernelΩsatisfies certain regularities,the authors obtain the boundedness ofμΩ,βfrom the weighted Hardy spaces Hωpp(Rn)to the weighted Lebesgue spaces Lωqq(Rn)for n/(n+β)≤<p≤1 with 1/q=1/p-β/n,as well as the same(Hωpp,Lωqq)-boudedness ofμΩ,βb when b belongs to BMOωp,p(Rn),which is a non-trivial subspace of BMO(Rn).

Commuting Space of Primary and Secondary Schools in the Context of Child Friendly Cities:A Case Study of Bajiao Street of Shijingshan District

With the continuous promotion of the construction of child friendly cities,the school commuting space is an important component of the construction of child friendly roads.Based on the background of child friendly cities,the commuting space of 11 primary and secondary schools in Bajiao Street is analyzed through literature analysis and field research methods.Firstly,the relevant literature on school commuting space is sorted out,and the characteristics of school commuting space are summarized,including transportation,landscape,culture,leisure,and security.Secondly,the characteristics of commuting space of primary and secondary schools in Bajiao Street are analyzed from three aspects:in front of the school gate,path space,and node space.This paper aims to provide reference and guidance for the future construction of children’s walking school commuting and promote the construction of a child friendly city.

Analysis of Commuting Modal Shift in Consideration of Social Interaction of Consciousness for Environment

It is the matter for achievement of the low carbon transport system that the excessive use of private vehicles can be controlled appropriately.Not only improvement of service level of modes except private vehicle,but also consciousness for environmental problem of individual trip maker is important for eco-commuting promotion.On the other hand,consciousness for environment would be changed by influence of other person.Accordingly,it is aimed in the study that the structure of decision-making process for modal shift to the eco-commuting mode in the local city is described considering environmental consciousness and social interaction.For the purpose,the consciousness for the environment problem and the travel behavior of the commuter at the suburban area in the local city are investigated by the questionnaire survey.The covariance structure about the eco-consciousness is analyzed with the database of the questionnaire survey by structural equation modeling.As the result,it can be confirmed with the structural equation model that the individual environmental consciousness is strongly related with the intention of self-sacrifice and is influenced with the local interaction of the individual connections.On the other hand,the intention of modal shift for the commuting mode is analyzed with the database of the questionnaire survey.It can be found out that the environmental consciousness is not statistically significant for commuting mode choice with the present poor level of service of public transport.However,the intention of self-sacrifice for the prevention of the global warming is statistically confirmed as the factor of modal shift with the operation of eco-commuting bus service with the RP/SP integrated estimation method.As the result,the multi-agent simulation system with social interaction model for eco consciousness is developed to measure the effect of the eco-commuting promotion.For the purpose,the carbon dioxide emission is estimated based on traffic demand and road network condition in the traffic environment model.On the other hand,the relation between agents is defined based on the small world network.The proposed multi-agent simulation is applied to measure the effect of the eco-commuting promotion such as improvement of level of service on the public transport or education of eco-consciousness.The effect of the promotion plan can be observed with the proposed multi-agent system.Finally,it can be concluded that the proposed multi-agent simulation with social interaction for eco-consciousness is useful for planning of eco-commuting promotion.

L^p Boundedness of Commutators of Calderón-Zygmund Singular Integral Operators

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COMMUTATORS OF SINGULAR INTEGRAL OPERATORS WITH NON-SMOOTH KERNELS

Let T be a singular integral operator bounded on Lp(Rn) for some p, 1 < p < ∞. The authors give a sufficient condition on the kernel of T so that when b ∈BMO, the commutator [b,T](f) = T(bf) - bT(f) is bounded on the space Lp for all p, 1 < p < ∞. The condition of this paper is weaker than the usual pointwise Hormander condition.

Multilinear singular integrals and commutators in variable exponent Lebesgue spaces

Boundedness of multilinear singular integrals and their commutators from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces are obtained. The vector-valued case is also considered.

WEIGHTED ESTIMATES FOR COMMUTATORS OF MULTILINEAR CALDERóN-ZYGMUND OPERATORS WITH NON-DOUBLING MEASURES

Let μ be a nonnegative Radon measure on Rd, which only satisfies the polynomial growth condition. Under this assumption, the authors obtain some weighted weaktype estimates for the commutators generated by the multilinear CalderSn-Zygmund op- erators and RBMO（μ） functions.

INTERIOR ESTIMATES IN MORREY SPACES FOR SOLUTIONS OF ELLIPTIC EQUATIONS AND WEIGHTED BOUNDEDNESS FOR COMMUTATORS OF SINGULAR INTEGRAL OPERATORS

It is proved that, for the nondivergence elliptic equations Σi,jn=1aijuxixj=f, if f belongs to the generalized Morrey spaces Lp, (w), then uxixj ∈ Lp, (w), where u is the W2,p-solution of the equations. In order to obtain this, the author first establish the weighted boundedness for the commutators of some singular integral operators on Lp, (w).

BOUNDEDNESS OF GENERALIZED HIGHER COMMUTATORS OF MARCINKIEWICZ INTEGRALS

Let b = （b1,…,bm） be a finite family of locally integrable functions. Then, we introduce generalized higher commutator of Marcinkiwicz integral as follows：（ μΩ^b^→（f）（x）=（∫ 0^∞｜FΩ^b^→,t（f）（x）｜^2t/dt）^1/2,where（FΩ^b^→,t（f）（x）=1/t∫｜x-y｜≤t Ω（x-y）/｜x-y｜^n-1 Лj=1^m（bj（x）-bj（y））f（y）dy.）When（bj∈Aβj,1≤j≤m,0〈βj〈1∑j=1^mβj=β〈n）and Ω is homogeneous of degreezero and satisfies the cancelation condition, we prove that μΩ^b^→is bounded from L^p（R^n）to L^8（R^n）,where1〈p〈βand 1/s=1/p-β/n,Moreover,if Ω also satisties some L^q -Dini condition,then μΩ^b^→ isbounded from L^p（R^n）to Fp^β,∞（R^n）and on certain Hardy spaces.The article extends some known results.

Commutators of Multilinear Singular Integrals with Lipschitz Functions

The boundedness of commutators of multilinear singular integrals with Lipschitz functions in product Lebesgue spaces is obtained.

Multilinear commutators of BMO functions and multilinear singular integrals with non-smooth kernels

This paper is concerned with certain multilinear commutators of BMO functions and multilinear singular integral operators with non-smooth kernels. By the sharp maximal functions estimates, the weighted norm inequalities for this kind of commutators are established.

Lipschitz Estimates for Commutators of N-dimensional Fractional Hardy Operators

In this paper, it is proved that the commutator Hβ,b which is generated by the n-dimensional fractional Hardy operator Hβ and b ∈λα （R^n） is bounded from L^p（R^n） to L^q（R^n）, where 0 〈 α 〈 1, 1 〈 p, q 〈 ∞ and 1/P - 1/q = （α＋β）/n. Furthermore, the boundedness of Hβ,b on the homogenous Herz space Kq^α,p（R^n） is obtained.

A∞ weight estimates for vector-valued commutators of multilinear fractional integral

In this paper, the authors get the Coifman type weighted estimates and weak weighted LlogL estimates for vector-valued generalized commutators of multilinear fractional integral with w ∈ A∞. Furthermore, both the boundedness of vector-valued multilinear frac- tional integral and the weak weighted LlogL estimates for vector-valued multilinear fractional integral are also obtained.

BOUNDEDNESS OF COMMUTATORS RELATED TO MARCINKIEWICZ INTEGRALS ON HARDY TYPE SPACES

In this paper, the authors study the boundedness of the operator [μΩ,b], the commutator generated by a function b ∈ Lipβ(Rn)(0 <β≤1) and the Marcinkiewicz integrals μΩ, on the classical Hardy spaces and the Herz-type Hardy spaces in the case Ω ∈ Lipα(Sn-1)(0 <α≤ 1).

THE BOUNDEDNESS OF MULTILINEAR COMMUTATORS ON WEIGHTED SPACES AND HERZ-TYPE SPACES

The boundedness of maximal multilinear commutator on certain weighted spaces is obtained. The boundedness of mulitilinear commutators of singular integrals with Calderon-Zygmund kernel on Herz-type spaces is also considered.

Parameterized Littlewood-Paley Operators and Their Commutators on Lebesgue Spaces with Variable Exponent

In this paper, by applying the technique of the sharp maximal function and the equivalent representation of the norm in the Lebesgue spaces with variable exponent, the boundedness of the parameterized Litflewood-Paley operators, including the parameterized Lusin area integrals and the parameterized Littlewood-Paley gλ^＊- functions, is established on the Lebesgue spaces with variable exponent. Furthermore, the boundedness of their commutators generated respectively by BMO functions and Lipschitz functions are also obtained.

WEIGHTED NORM INEQUALITIES FOR THE COMMUTATORS OF MULTILINEAR SINGULAR INTEGRAL OPERATORS

In this paper, the authors consider the weighted estimates for the commutators of multilinear Calderón-Zygmund operators.By introducing an operator which shifts the commutation, and establishing the weighted estimates for this new operator, the authors prove that, if p_1 ∈ （1,∞）, p_2,…,p_m ∈（1,∞], p ∈ （0,∞） with 1/p =Σ1≤k≤ m 1/pk, then for any weight w, the commutators of m-linear Galderón-Zygmund operator are bounded from L P1（R n,M_l（logL） σw）× p2（Rn,M~w）×...×Lpm（Rn,Mw） to Lp（Rn,w）with σ to be a constant depending only on p_1 and the order of commutator

ON MULTILINEAR COMMUTATORS OF Θ-TYPE CALDERóN-ZYGMUND OPERATORS

In this paper, we discuss the multilinear commutator of Θ-type Calder6n- Zygmund operators, and obtain that this kind of multilinear commutators is bounded from LP（Rn） to Lq（Rn）, from LP（Rn） to Triebel-Lizorkin spaces and on certain Hardy type spaces.

ENDPOINT ESTIMATE FOR MAXIMAL COMMUTATORS WITH NON-DOUBLING MEASUES

The authors establish the weak type endpoint estimate for the maximal commutators generated by Calderon-Zygmund singular integrals and Orlicz type functions with non-doubling measures.

Commutators of Weighted Lipschitz Functions and Multilinear Singular Integrals with Non-Smooth Kernels

This paper is devoted to studying the commutators of the multilinear singular integral operators with the non-smooth kernels and the weighted Lipschitz functions. Some mapping properties for two types of commutators on the weighted Lebesgue spaces, which extend and generalize some previous results, are obtained.