Calderon-Zygmund Singular Integral Operators Research Articles

On Some Properties of Dyadic Operators

In this paper, the objects of our investigation are some dyadic operators, including dyadic shifts, multilinear paraproducts and multilinear Haar multipliers. We mainly focus on the continuity and compactness of these operators. First, we consider the continuity properties of these operators. Then, by the Frechet–Kolmogorov–Riesz–Tsuji theorem, the non-compactness properties of these dyadic operators will be studied. Moreover, we show that their commutators are compact with $\operatorname{CMO}$ functions, which is quite different from the non-compactness properties of these dyadic operators. These results are similar to those for Calderon–Zygmund singular integral operators.

Estimates of commutators on Herz-type spaces with variable exponent and applications

Let [b, T] be the commutator generated by b and T, where $$b\in \mathrm {BMO}({\mathbb {R}}^{n})$$ and T is a Calderon–Zygmund singular integral operator. In this paper, the authors establish some strong type and weak type boundedness estimates including the $$L\log L$$ type inequality for [b, T] on the Herz-type spaces with variable exponent. Meanwhile, the similar results for the commutators $$[b,I_l]$$ of fractional integral operator are also obtained. As applications, we consider the regularity in the Herz-type spaces with variable exponent of strong solutions to nondivergence elliptic equations with $$\mathrm {VMO}$$ coefficients.

Boundedness of Rough Operators on Grand Variable Herz Spaces

In this paper, we prove the boundedness of Calderon-Zygmund singular integral operators TΩ on grand Herz spaces with variable exponent under some conditions.

Boundedness of Calderon-Zygmund singular integral operators from l1(Lp) into weak - l1(Lp)

Boundedness of Calderon-Zygmund singular integral operators from l1(Lp) into weak - l1(Lp)

Logistic Neumann problems with discontinuous coefficients

The purpose of this paper is for the first time to provide a careful and accessible exposition of the study of the existence of positive solutions of semilinear Neumann problems for diffusive logistic equations with discontinuous coefficients, which models population dynamics in environments with spatial heterogeneity. A biological interpretation of our main result is that when the environment has an impassable boundary and is on the average unfavorable, then high diffusion rates have the same effect (that is, the ultimate extinction of the population) as they always have when the boundary is deadly; but if the boundary is impassable and the environment is on the average neutral or favorable, then the population can persist, no matter what its rate of diffusion. The approach here is based on explicit representation formulas for the solutions of the Neumann problem and also on the $$L^{p}$$ boundedness of Calderon–Zygmund singular integral operators appearing in those representation formulas. That is why we consider the case where the space dimension is greater than 2. Moreover, we make use of an $$L^{p}$$ variant of an estimate for the Green operator of the Neumann problem introduced in the study of Feller semigroups.

The Dirichlet problem for the uniformly elliptic equation in generalized weighted Morrey spaces

Abstract In this paper, we obtain generalized weighted Sobolev-Morrey estimates with weights from the Muckenhoupt class Ap by establishing boundedness of several important operators in harmonic analysis such as Hardy-Littlewood operators and Calderon-Zygmund singular integral operators in generalized weighted Morrey spaces. As a consequence, a priori estimates for the weak solutions Dirichlet boundary problem uniformly elliptic equations of higher order in generalized weighted Sobolev-Morrey spaces in a smooth bounded domain Ω ⊂ ℝn are obtained.

Calderón–Zygmund Operators on Homogeneous Product Lipschitz Spaces

The purpose of this paper is to establish a necessary and sufficient condition for the boundedness of product Calderon–Zygmund singular integral operators introduced by Journe on the product Lipschitz spaces. The key idea used in this paper is to develop the Littlewood–Paley theory for the product spaces which includes the characterization of a special product Besov space and a density argument for the product Lipschitz spaces in the weak sense.

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

Let Ω ∈ Ls(Sn−1) (s > 1) 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 Calderon-Zygmund singular integral operator TΩ and its commutator [b, TΩ] on Herz-Morrey spaces with variable exponent.

Inhomogeneous Lipschitz spaces of variable order and their applications

In this article, the authors first give a Littlewood–Paley characterization for inhomogeneous Lipschitz spaces of variable order with the help of inhomogeneous Calderon identity and almost-orthogonality estimates. As applications, the boundedness of inhomogeneous Calderon–Zygmund singular integral operators of order (ϵ,σ) on these spaces has been presented. Finally, we note that a class of pseudodifferential operators Ta∈OpS1,10 are continuous on the inhomogeneous Lipschitz spaces of variable order as a corollary. We may observe that those operators are not, in general, continuous in L2.

English

We obtain the boundedness of Calderon-Zygmund singular integral operators T of non-convolution type on Hardy spaces H p (X) for 1/(1 + e) < p ⩽ 1, where X is a space of homogeneous type in the sense of Coifman and Weiss (1971), and e is the regularity exponent of the kernel of the singular integral operator T. Our approach relies on the discrete Littlewood-Paley-Stein theory and discrete Calderon’s identity. The crucial feature of our proof is to avoid atomic decomposition and molecular theory in contrast to what was used in the literature.

Norm inequalities for higher-order commutators of one-sided oscillatory singular integrals

In the present paper, we study the weighted norm inequalities for higher-order commutators formed by a class of one-sided oscillatory singular integrals and $BMO$ functions. We obtain that the boundedness of these commutators can be deduced by that of one-sided Calderon-Zygmund singular integral operators.

L P estimates for the maximal singular integral in terms of the singular integral

This paper continues the study, initiated in [MOV] and [MOPV], of the problem of controlling the maximal singular integral T* f by the singular integral Tf. Here, T is a smooth homogeneous Calderon-Zygmund singular integral operator of convolution type. We consider two forms of control, namely, in the weighted Lp(ω) norm and via pointwise estimates of T* f by M(Tf ) or M2(Tf), where M is the Hardy-Littlewood maximal operator and M2 = M po M its iteration. The novelty with respect to the aforementioned works lies in the fact that here p is different from 2 and the Lp space is weighted.

相关于微分算子的(Musielak-)Orlicz-Hardy空间的实变理论及其应用

As is well-known, because of their important applications in several branches of mathematics, such as harmonic analysis and partial differential equations, the theory of Calderon-Zygmund singular integral operators and the real-variable theory of various function spaces, which could characterize the boundedness of those operators, turn into one of the main contents of modern harmonic analysis. However, the real-variable theory of classical function spaces has been no longer suitable for characterizing the boundedness of singular integral operators associated with some more general differential operators than Laplace operators. Thus, for different operators, it has become one of the very active research fields of harmonic analysis in recent years to develop the real-variable theory of function spaces, which are suitable to those operators and could characterize the boundedness of singular integral operators associated with those operators. In this article, we study the realvariable theory of (Musielak-)Orlicz-Hardy spaces associated with some differential operators, including secondorder divergence form elliptic operators and Schrodinger operators as special cases, on n -dimensional Euclidean space R n , strongly domains of R n or metric spaces with doubling measure, and its applications to the boundedness of operators.

𝐿^{𝑝} bounds for the commutators of singular integrals and maximal singular integrals with rough kernels

The commutator of convolution type Calderon-Zygmund singular integral operators with rough kernels p . v . Ω ( x ) | x | n p.v. \frac {\Omega (x)}{|x|^n} are studied. The authors established the L p ( 1 &gt; p &gt; ∞ ) L^p\,(1&gt;p&gt;\infty ) boundedness of the commutators of singular integrals and maximal singular integrals with the kernel condition which is different from the condition Ω ∈ H 1 ( S n − 1 ) . \Omega \in H^1(S^{n-1}).

The boundedness of composition operators on Triebel-Lizorkin and Besov Spaces with different homogeneities

In this paper, we introduce new Triebel-Lizorkin and Besov Spaces associated with the different homogeneities of two singular integral operators. We then establish the boundedness of composition of two Calderon-Zygmund singular integral operators with different homogeneities on these Triebel-Lizorkin and Besov spaces.

Φ-Admissible singular operators and their commutators on vanishing generalized Orlicz-Morrey spaces

We study the boundedness of Φ-admissible singular operators and their commutators on vanishing generalized Orlicz-Morrey spaces including their weak versions. These conditions are satisfied by most of the operators in harmonic analysis, such as the Hardy-Littlewood maximal operator, the Calderon-Zygmund singular integral operator and so on. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on weights without assuming any monotonicity property of on r. MSC:42B20, 42B25, 42B35, 46E30.

Estimates for vector-valued commutators on weighted Morrey space and applications

In this paper, we introduce weighted vector-valued Morrey spaces and obtain some estimates for vector-valued commutators on these spaces. Applications to Calderon-Zygmund singular integral operators, oscillatory singular integral operators and parabolic difference equations are considered.

Some estimates for commutators of Calder&amp;oacute;n-Zygmund operators on the weighted Morrey spaces

Let T be a Calderon-Zygmund singular integral operator. In this paper, we will use a uni ed approach to show some boundedness properties of commutators [ b , T ] on the weighted Morrey spaces L p,k ( w ) under appropriate conditions on the weight w , where the symbol b belongs to weighted BMO or Lipschitz space or weighted Lipschitz space

On the boundedness of the maximal operator and singular integral operators in generalized Morrey spaces

In the paper we find conditions on the pair (!1, !2) which ensure the bounded- ness of the maximal operator and the Calderon-Zygmund singular integral operators from one generalized Morrey space Mp,!1 to another Mp,!2, 1 < p < ∞, and from the space M1,!1 to the weak space W M1,!2. As applications, we get some estimates for uniformly elliptic operators on generalized Morrey spaces.

Estimates for the maximal singular integral in terms of the singular integral: the case of even kernels

Let T be a smooth homogeneous Calder on-Zygmund singular integral operator in R n . In this paper we study the problem of controlling the maximal singular integral T ? f by the singular integral Tf. The most basic form of control one may consider is the estimate of the L 2 (R n ) norm of T ? f by a constant times the L 2 (R n ) norm of Tf. We show that if T is an even higher order Riesz transform, then one has the stronger pointwise inequality T ? f(x) C M(Tf)(x), where C is a constant and M is the Hardy-Littlewood maximal operator. We prove that the L 2 estimate of T ? by T is equivalent, for even smooth homogeneous Calder on-Zygmund operators, to the pointwise inequality between T ? and M(T ). Our main result characterizes the L 2 and pointwise inequalities in terms of an algebraic condition expressed in terms of the kernel ( x) jxjn of T , where is an