The Commutator of the Kato Square Root for Second Order Elliptic Operators on R~n

Let L=-div(A?) be a second order divergence form elliptic operator,and A be an accretive,n×n matrix with bounded measurable complex coefficients in R~n.We obtain the L~p bounds for the commutator generated by the Kato square root L~(1/2) and a Lipschitz function,which recovers a previous result of Calderón,by a different method.In this work,we develop a new theory for the commutators associated to elliptic operators with Lipschitz function.The theory of the commutator with Lipschitz function is distinguished from the analogous elliptic operator theory.     

Weighted Norm Inequalities with General Weights for the Commutator of Calderón

In this paper, by a sharp function estimate and an idea of Lerner, the authors establish someweighted estimates for the m-multilinear integral operator which is bounded from L1(Rn)×···×L1 (Rn)to L1/m,∞ (Rn),, and the associated kernel K(x; y1, . . . , ym)) enjoys a regularity on the variable x. As anapplication, weighted estimates with general weights are given for the commutator of Calderón.     

Estimates of commutators for multilinear operators on Herz-type spaces

In this paper, the boundedness of mulitilinear commutator [b,T] on Herz-type space is considered, where T is a standard Calderón-Zygmund singular operator and b ∈ (BMO(Rn))m.     

Commutators of Calderón-Zygmund operators related to admissible functions on spaces of homogeneous type and applications to Schrödinger operators

Let X be an RD-space. In this paper, the authors establish the boundedness of the commutator Tbf = bTf-T(bf) on Lp , p∈(1,∞), where T is a Calderón-Zygmund operator related to the admissible function ρ and b∈BMOθ(X)BMO(X). Moreover, they prove that Tb is bounded from the Hardy space H1ρ(X) into the weak Lebesgue space L1weak(X). This can be used to deal with the Schrdinger operators and Schrdinger type operators on the Euclidean space Rn and the sub-Laplace Schrdinger operators on the stratified Lie group G.     

奇异积分算子交换子在变指数空间上的特征

The main purpose of this paper is to characterize the Lipschitz space by the boundedness of commutators on Lebesgue spaces and Triebel-Lizorkin spaces with variable exponent.Based on this main purpose, we first characterize the Triebel-Lizorkin spaces with variable exponent by two families of operators. Immediately after, applying the characterizations of TriebelLizorkin space with variable exponent, we obtain that b ∈■β if and only if the commutator of Calderón-Zygmund singular integral operator is bounded, respectively, from■ to■,from■ to■ with■. Moreover, we prove that the commutator of Riesz potential operator also has corresponding results.     

ON MULTILINEAR COMMUTATORS OF Θ-TYPE CALDERóN-ZYGMUND OPERATORS

In this paper, we discuss the multilinear commutator of θ-type Calderón- 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.     

Weighted estimates for iterated commutators of multilinear Calderón-Zygmund operators on non-homogeneous metric measure spaces

Let(X, d, μ) be a non-homogeneous metric measure space satisfying the so-called upper doubling and geometrically doubling conditions, which includes the space of homogeneous type and the Euclidean space with the non-doubling measure as special cases. Let T be a multilinear Calderón-Zygmund operator and ■:=(b1,..., bm) be a finite family of ■(μ) functions. In this paper, some weak-type multiple weighted estimates for the iterated commutator ■Tgenerated by T and ■ are obtained.     

SOME ENDPOINT ESTIMATES FOR COMMUTATORS OF θ-TYPE CALDERóN-ZYGMUND OPERATORS

In this note, the authors prove that the commutator Tb generated by θ-type Calderón-Zygmund operator T and a Lipschitz function b is bounded from Lp(Rn) into Lip(β- np)(Rn) and also maps from L βn (Rn) into BMO(Rn).     

Boundedness of Singular Integral Operators on Herz-Morrey Spaces with Variable Exponent

Let ? ∈ L~s(S~(n-1))(s1) be a homogeneous function of degree zero and b be a BMO function or Lipschitz function. In this paper, the authors obtain some boundedness of the Calderón-Zygmund singular integral operator T_? and its commutator [b, T_?] on Herz-Morrey spaces with variable exponent.     

New Calderón reproducing formulae with exponential decay on spaces of homogeneous type

Assume that(X,d,μ) is a space of homogeneous type in the sense of Coifman and Weiss(1971,1977). In this article, motivated by the breakthrough work of Auscher and Hyt(o|¨)nen(2013) on orthonormal bases of regular wavelets on spaces of homogeneous type, we introduce a new kind of approximations of the identity with exponential decay(for short, exp-ATI). Via such an exp-ATI, motivated by another creative idea of Han et al.(2018) to merge the aforementioned orthonormal bases of regular wavelets into the frame of the existed distributional theory on spaces of homogeneous type, we establish the homogeneous continuous/discrete Calderón reproducing formulae on(X, d,μ), as well as their inhomogeneous counterparts. The novelty of this article exists in that d is only assumed to be a quasi-metric and the underlying measure μ a doubling measure,not necessary to satisfy the reverse doubling condition. It is well known that Calderón reproducing formulae are the cornerstone to develop analysis and, especially, harmonic analysis on spaces of homogeneous type.     

带变量Calderón-Zygmund核的奇异积分算子在加权Morrey空间上的有界性（英文）

In this paper, we will study the boundedness of the singular integral operator with variable Calder′on-Zygmund kernel on the weighted Morrey spaces Lp,κ(ω) for q′≤ p ∞and 0 κ 1. Furthermore, the boundedness for the commutator with BMO functions is also obtained.     

BOUNDEDNESS AND COMPACTNESS FOR THE COMMUTATOR OF THE ω-TYPE CALDERóN-ZYGMUND OPERATOR ON LORENTZ SPACE

In this paper,the authors consider the ω-type Calderón-Zygmund operator T_ω and the commutator [b,T_ω] generated by a symbol function b on the Lorentz space L^p,r(X)over the homogeneous space(X,d,μ).The boundedness and the compactness of the commutator [b,T_ω] on Lorentz space L^p,r(X) are founded for any p ∈(1,∞) and r ∈ [1,∞).     

THE BOUNDEDNESS FOR COMMUTATORS OF ANISOTROPIC CALDERóN-ZYGMUND OPERATORS

Let T be an anisotropic Calderón-Zygmund operator and φ:R~n×[0,∞)→[0,∞) be an anisotropic Musielak-Orlicz function with φ(x,·) being an Orlicz function andφ(·,t) being a Muckenhoupt A_∞(A) weight.In this paper,our goal is to study two boundedness theorems for commutators of anisotropic Calderon-Zygmund operators.Precisely,when b∈BMO_w(R~n,A)(a proper subspace of anisotropic bounded mean oscillation space BMO(R~n,A)),the commutator [b,T] is bounded from anisotropic weighted Hardy space H_ω~1(R~n,A) to weighted Lebesgue space L_ω~1(R~n) and when b∈BMO(R~n)(bounded mean oscillation space),the commutator [b,T] is bounded on Musielak-Orlicz space L~φ(R~n),which are extensions of the isotropic setting.     

Stochastic Hamiltonian flows with singular coefficients Dedicated to the 60th Birthday of Professor Michael R¨ockner

In this paper, we study the following stochastic Hamiltonian system in R~(2d)(a second order stochastic differential equation):dX_t = b(X_t,X_t)dt + σ(X_t,X_t)dW_t,(X_0,X_0) =(x, v) ∈ R~(2d),where b(x, v) : R~(2d)→ R~d and σ(x, v) : R~(2d)→ R~d ? R~d are two Borel measurable functions. We show that if σ is bounded and uniformly non-degenerate, and b ∈ H_p~(2/3,0) and ?σ∈ L~p for some p 2(2 d + 1), where H_p~(α,β)is the Bessel potential space with differentiability indices α in x and β in v, then the above stochastic equation admits a unique strong solution so that(x, v) → Z_t(x, v) :=(Xt,Xt)(x, v) forms a stochastic homeomorphism flow,and(x, v) → Z_t(x, v) is weakly differentiable with ess.sup_(x,v)E(sup_(t∈[0,T])|?Z_t(x, v)|~q) ∞ for all q ≥ 1 and T≥ 0. Moreover, we also show the uniqueness of probability measure-valued solutions for kinetic Fokker-Planck equations with rough coefficients by showing the well-posedness of the associated martingale problem and using the superposition principle established by Figalli(2008) and Trevisan(2016).     

ENDPOINT ESTIMATES FOR THE COMMUTATOR OF PSEUDO-DIFFERENTIAL OPERATORS

It is well known that the commutator Tb of the Calder′on-Zygmund singular integral operator is bounded on Lp(Rn) for 1 p +∞if and only if b∈BMO [1]. On the other hand, the commutator Tb is bounded from H1(Rn) into L1(Rn) only if the function b is a constant [2]. In this article, we will discuss the boundedness of commutator of certain pseudo-differential operators on Hardy spaces H1. Let Tσbe the operators that its symbol is S01,δwith 0≤δ 1, if b∈LMO∞, then, the commutator [b, Tσ] is bounded from H1(Rn) into L1(Rn) and from L∞(Rn) into BMO(Rn); If [b, Tσ] is bounded from H1(Rn) into L1(Rn) or L∞(Rn) into BMO(Rn), then, b∈LMO_(loc).     

Multilinear Commutators of θ-Type Calderón-Zygmund Operators on Non-Homogeneous Metric Measure Spaces

In this paper, the boundedness in Lebesgue spaces of commutators and multilinear commutators generated by θ-type Calderón-Zygmund operators with RB MO(μ) functions on non-homogeneous metric measure spaces is obtained.     

Weighted norm inequalities for the multilinear Calderón-Zygmund operators

In this paper, by proving some suitable weighted endpoint estimates, and then by multilinear interpolation and a new multilinear extrapolation lemma, the author establishes some weighted estimates for the multilinear Calderón-Zygmund operator. Also, the author gives a weighted estimate for the corresponding commutator.     

BOUNDEDNESS OF THE CALDERóN COMMUTATOR WITH A ROUGH KERNEL ON TRIEBEL-LIZORKIN SPACES

In this paper,we consider the boundedness on Triebel-Lizorkin spaces for the d-dimensional Calderón commutator defined by■,where Ω is homogeneous of degree zero,integrable on S^d-1 and has a vanishing moment of order one,and a is a function on R^d such that ?a∈L^∞(R^d),We prove that if 1 2q(S^d-1) with q=max{1/q,1/q’},then T_Ω,a is bounded on Triebel-Lizorkin spaces F_p^0,q(R     

新的齐型BMO, Lipschitz空间上的奇异积分算子

Let(X, d, μ) be a space of homogeneous type, BMO_A(X) and Lip_A(β,X) be the space of BMO type,lipschitz type associated with an approximation to the identity {A_t}_t0 and introduced by Duong,Yan and Tang, respectively. Assuming that T is a bounded linear operator on L~2(X), we find the sufficient condition on the kernel of T so that T is bounded from BMO(X) to BMO_A(X) and from Lip(β, X) to Lip_A(β, X). As an application, the boundedness of Calderón-Zygmund operators with nonsmooth kernels on BMO(R~n) and Lip(β, R~n) are also obtained.     

Weighted estimates for commutators of singular integral operators with non-doubling measures

Letμbe a nonnegative Radon measure on R~d which only satisfiesμ(B(x,r))≤C_0r~n for all x∈R~d,r>0,and some fixed constants C_0>0 and n∈(0,d].In this paper,some weighted weak type estimates with A_(p,(log L)~σ)~ρ(μ) weights are established for the commutators generated by Calder■n-Zygmund singular integral operators with RBMO(μ) functions.