Multilinear Calderón-Zygmund operators on Hardy spaces

Boundedness of the Hardy-Littlewood Maximal Operator, Fractional Integral Operators, and Calderon-Zygmund Operators on Generalized Weighted Morrey Spaces

In this paper we investigate the boundedness of classical operators, namely the Hardy-Littlewood maximal operator, fractional integral operators, and Calderon-Zygmund operators, on generalized weighted Morrey spaces and generalized weighted weak Morrey spaces. We prove that the operators are bounded on these spaces, under certain assumptions.

Inhomogeneous cancellation conditions and Calderón-Zygmund type operators on $h^p$

In this work we present a new approach to molecules on Goldberg's local Hardy spaces $h^p(\mathbb{R}^n)$, $0<p\leq1$, assuming an appropriate cancellation condition. As applications, we prove a version of Hardy's inequality and improved continuity results for inhomogeneous Calderón-Zygmund operators on these spaces.

Maximal and Calderon-Zygmund operators on the local variable Morrey-Lorentz spaces and some applications

In this paper, we give the definition of local variable Morrey Lorentz spaces which are a new class of functions. Also, we prove the boundedness of the Hardy Littlewood maximal operator M and Calderon Zygmund operators T on these spaces. Finally, we apply these results to the Bochner Riesz operator, identity approximation and the Marcinkiewicz operator on these spaces.

Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators in nonhomogeneous spaces

In the paper we consider Calderón-Zygmund operators in nonhomogeneous spaces. We are going to prove the analogs of classical results for homogeneous spaces. Namely, we prove that a Calderón-Zygmund operator is of weak type if it is bounded in $L^2$. We also prove several versions of Cotlar's inequality for maximal singular operator. One version of Cotlar's inequality (a simpler one) is proved in Euclidean setting, another one in a more abstract setting when Besicovich covering lemma is not available. We obtain also the weak type of maximal singular operator from these inequalities.

区域上 Zygmund 空间中的 Calderón-Zygmund 算子

Calderón-Zygmund operators on Zygmund spaces on domains

本文研究有界 Lipschitz 区域上 Calderón-Zygmund 算子在一般增长函数定义的 Zygmund 空间中的有界性，建立了相应的 T(P) 定理，并给出偶算子在多项式区域中额外消除性质的新形式。

Dyadic representation and boundedness of non-homogeneous Calderón--Zygmund operators with mild kernel regularity

We prove a new dyadic representation theorem with applications to the $T(1)$ and $A_2$ theorems. In particular, we obtain the non-homogeneous $T(1)$ theorem under weaker kernel regularity than the earlier approaches.

Boundedness of commutators generated by m-th Calderón-Zygmund type singular integrals and local Campanato functions on generalized local Morrey spaces

Let $T_m$ be the $m$-th Calderón-Zygmund type singular integral. In the paper, we consider the boundedness of $T_m$ on the generalized product local Morrey spaces $LM_{p_1,φ_1}^{\{x_0\}}\times LM_{p_2,φ_2}^{\{x_0\}}\times\dots\times LM_{p_m,φ_m}^{\{x_0\}}.$ And, the boundedness of the commutators of $T_m$ with local Campanato functions is obtained, also.

On the continuity of strongly singular Calderón-Zygmund-type operators on Hardy spaces

In this work, we establish results on the continuity of strongly singular Calderón-Zygmund operators of type $σ$ on Hardy spaces $H^p(\mathbb{R}^n)$ for $0<p\leq 1$ assuming a weaker $L^{s}-$type Hörmander condition on the kernel. Operators of this type include appropriated classes of pseudodifferential operators $OpS^{m}_{σ,b}(\mathbb{R}^n)$ and operators associated to standard $δ$-kernels of type $σ$ introduced by Álvarez and Milman. As application, we show that strongly singular Calderón-Zygmund operators are bounded from $H^{p}_{w}(\mathbb{R}^n)$ to $L^{p}_{w}(\mathbb{R}^n)$, where $w$ bel…

Bilinear $θ$-type Calderón-Zygmund operators and its commutator on generalized weighted Morrey spaces over RD-spaces

An RD-space $\mathcal{X}$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in $\mathcal{X}$. In this setting, the authors establish the boundedness of bilinear $θ$-type Calderón-Zygmund operator $T_θ$ and its commutator $[b_1,b_2,T_θ]$ generated by the function $b_1,b_2\in BMO(μ)$ and $T_θ$ on generalized weighted Morrey space $\mathcal{M}^{p,ϕ}(ω)$ and generalized weighted weak Morrey space $W\mathcal{M}^{p,ϕ}(ω)$ over RD-spaces.

Weighted boundedness of the Hardy-Littlewood maximal and Calderón-Zygmund operators on Orlicz-Morrey and weak Orlicz-Morrey spaces

For the Hardy-Littlewood maximal and Calderón-Zygmund operators, the weighted boundedness on the Lebesgue spaces are well known. We extend these to the Orlicz-Morrey spaces. Moreover, we prove the weighted boundedness on the weak Orlicz-Morrey spaces. To do this we show the weak-weak modular inequality. The Orlicz-Morrey space and its weak version contain weighted Orlicz, Morrey and Lebesgue spaces and their weak versions as special cases. Then we also get the boundedness for these function spaces as corollaries.

A_1-regularity and boundedness of Calderon-Zygmund operators

The Coifman-Fefferman inequality implies quite easily that a Calderon-Zygmund operator $T$ acts boundedly in a Banach lattice $X$ on $\mathbb R^n$ if the Hardy-Littlewood maximal operator $M$ is bounded in both $X$ and $X'$. We discuss this phenomenon in some detail and establish a converse result under the assumption that $X$ is $p$-convex and $q$-concave with some $1 < p, q < \infty$ and satisfies the Fatou property: if a linear operator $T$ is bounded in $X$ and $T$ is nondegenerate in a certain sense (for example, if $T$ is a Riesz transform) then $M$ has to be bounded in both $X$ and $X'$…

$\mathbf A_1$-regularity and boundedness of Calderón-Zygmund operators. II

A proof is given for the "only if" part of the result stated in the previous paper of the series that a suitably nondegenerate Calderón-Zygmund operator $T$ is bounded in a Banach lattice $X$ on $\mathbb R^n$ if and only if the Hardy-Littlewood maximal operator $M$ is bounded in both $X$ and $X'$, under the assumption that $X$ has the Fatou property and $X$ is $p$-convex and $q$-concave with some $1 < p, q < \infty$. We also get rid of a fixed point theorem in the proof of the main lemma and give an improved version of an earlier result concerning the divisibility of $\mathrm {BMO}$-regularity…

球 Banach 函数空间上关于多重线性 Calderón-Zygmund 算子的 Hardy 空间弱分解

Weak Factorizations of the Hardy Spaces in Terms of Multilinear Calderón-Zygmund Operators on Ball Banach Function Spaces

本文在球 Banach 函数空间上基于多重线性 Calderón-Zygmund 算子建立了经典 Hardy 空间的弱分解，并通过该算子生成交换子的有界性获得了 BMO 空间的新刻画，结果可应用于多种函数空间。

The $L^2$-boundedness of the variational Calderón-Zygmund operators

In this paper, we verify the $L^2$-boundedness for the jump functions and variations of Calderón-Zygmund singular integral operators with the underlying kernels satisfying \begin{align*}\int_{\varepsilon\leq |x-y|\leq N} K(x,y)dy=\int_{\varepsilon\leq |x-y|\leq N}K(x,y)dx=0\; \forall 0<\varepsilon\leq N<\infty,\end{align*} in addition to some proper size and smooth conditions. This result should be the first general criteria for the variational inequalities for kernels not necessarily of convolution type. The $L^2$-boundedness assumption that we verified here is also the starting point of the …

The boundedness of certain sublinear operators with rough kernel generated by Calderón-Zygmund operators and their commutators on generalized weighted Morrey spaces

The aim of this paper is to get the boundedness of certain sublinear operators with rough kernel generated by Calderón-Zygmund operators on the generalized weighted Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. We also prove that the commutator operators formed by BMO functions and certain sublinear operators with rough kernel are also bounded on the generalized weighted Morrey spaces. Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.

Molecular Characterizations of Variable Anisotropic Hardy Spaces with Applications to Boundedness of Calderón-Zygmund Operators

Let $p(\cdot):\ \mathbb{R}^n\to(0,\infty]$ be a variable exponent function satisfying the globally log-Hölder continuous condition and $A$ a general expansive matrix on $\mathbb{R}^n$. Let $H_A^{p(\cdot)}(\mathbb{R}^n)$ be the variable anisotropic Hardy space associated with $A$ defined via the non-tangential grand maximal function. In this article, via the known atomic characterization of $H_A^{p(\cdot)}(\mathbb{R}^n)$, the author establishes its molecular characterization with the known best possible decay of molecules. As an application, the author obtains a criterion on the boundedness of …

Calderón-Zygmund operators on RBMO

Let $μ$ be an $n$-dimensional finite positive measure on $\mathbb{R}^m$. We obtain a $T1$ condition sufficient for the boundedness of Calderón-Zygmund operators on $\textrm{RBMO}(μ)$, the regular BMO space of Tolsa.

Hardy spaces, Regularized BMO spaces and the boundedness of Calderón-Zygmund operators on non-homogeneous spaces

One defines a non-homogeneous space $(X, μ)$ as a metric space equipped with a non-doubling measure $μ$ so that the volume of the ball with center $x$, radius $r$ has an upper bound of the form $r^n$ for some $n> 0$. The aim of this paper is to study the boundedness of a Calderón-Zygmund operator $T$ as well as the boundedness of certain related singular integrals associated with $T$ on various function spaces on $(X, μ)$ such as the Hardy spaces, the $L^p$ spaces and the regularized BMO spaces. This article thus extends the work of X. Tolsa \cite{T1} on the non-homogeneous space $(\mathbb R^n…

关于某些指数增长 Lie 群的 Calderón-Zygmund 理论

Calderón-Zygmund theory on some Lie groups of exponential growth

本文研究半直积指数增长 Lie 群，证明其关于流测度与度量具有 Hebisch–Steger 意义下的 Calderón-Zygmund 性质，并建立自然极大算子的弱型 $(1,1)$ 估计。

若干广义Hardy-Morrey空间中Calderón-Zygmund算子的范数不等式

Norm inequalities for Calderon-Zygmund operators in some generalized Hardy-Morrey spaces

该研究利用广义Hardy-Morrey空间的分子刻画，推导了Calderón-Zygmund算子及其交换子在这类空间中的范数不等式，为相关算子的有界性分析提供了依据。

Product generalized local Morrey spaces and commutators of multi-sublinear operators generated by multilinear Calderón-Zygmund operators and local Campanato functions

The aim of this paper is to get the boundedness of the commutators of multi-sublinear operators generated by local campanato functions and multilinear Calderón-Zygmund operators on the product generalized local Morrey spaces.

Uniform rectifiability, Calderon-Zygmund operators with odd kernel, and quasiorthogonality

In this paper we study some questions in connection with uniform rectifiability and the $L^2$ boundedness of Calderon-Zygmund operators. We show that uniform rectifiability can be characterized in terms of some new adimensional coefficients which are related to the Jones' $β$ numbers. We also use these new coefficients to prove that n-dimensional Calderon-Zygmund operators with odd kernel of type $C^2$ are bounded in $L^2(μ)$ if $μ$ is an n-dimensional uniformly rectifiable measure.

Lipschitz spaces and Calderon-Zygmund operators associated to non-doubling measures

In the setting of $\R^d$ with an $n-$dimensional measure $μ,$ we give several characterizations of Lipschitz spaces in terms of mean oscillations involving $μ.$ We also show that Lipschitz spaces are preserved by those Calderon-Zygmund operators $T$ associated to the measure $μ$ for which T(1) is the Lipschitz class $0.$

Weak boundedness of Calderón-Zygmund operators on noncommutative $L_1$-spaces

In 2008, J. Parcet showed the $(1,1)$ weak-boundedness of Calderón-Zygmund operators acting on functions taking values in a von Neumann algebra. We propose a simplified version of his proof using the same tools : Cuculescu's projections and a pseudo-localisation theorem. This will unable us to recover the $L_p$-boundedness of Calderón-Zygmund operators with Hilbert valued kernels acting on operator valued functions for $1 < p < \infty$ and an $L_p$-pseudo-localisation result of P. Hytönen.

非交换弱类型(1,1)估计 for 加尔登-齐姆尤恩算子 with 一类 L_1 积分条件

Noncommutative weak type $(1,1)$ estimates for Calderón-Zygmund operators with a class of $L_1$-integral conditions

本文提出了一种新的非交换加尔登-齐姆尤恩分解方法，用于在特定 L_1-积分条件下证明非交换加尔登-齐姆尤恩算子的弱类型(1,1)有界性。

The factorizations of $H^ρ(\mathbb{R}^n)$ via multilinear Calderón-Zygmund operators on weighted Lebesgue spaces

We extend the recently much-studied Hardy factorization theorems to the weight case. The key point of this paper is to establish the factorization theorems without individual condition on the weight functions. As a direct application, we obtain the characterizations of $BMO(\mathbb{R}^n)$ space and Lipschitz spaces via the weighted boundedness of commutators of multilinear Calderón-Zygmund operators with the genuinely multilinear weights.

分子与带非交换核的Calderón-Zygmund算子在$\n\thrm{H}_1^c$上的性质

Molecules and Calderón-Zygmund operators with noncommuting kernels on $\mathrm{H}_1^c$

本研究给出半交换Hardy空间的分子刻画，并据此得到核取值于半有限von Neumann代数的Calderón-Zygmund算子的$\n\thrm{H}_1^c$型有界性估计。

Boundedness of Maximal Calderón-Zygmund Operators on Non-homogeneous Metric Measure Spaces

Let $(\cx,\,d,\,μ)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. In this paper, the authors show that for the maximal Calderón-Zygmund operator associated with a singular integral whose kernel satisfies the standard size condition and the Hörmander condition, its $L^p(μ)$ boundedness with $p\in(1,\infty)$ is equivalent to its boundedness from $L^1(μ)$ into $L^{1,\infty}(μ)$. Moreover, applying this, together with a new Cotlar type inequality, the authors show that if the Calderón-Zygmund operator $T$ is bounded on $L^2(μ…

Calderón-Zygmund operators on local hardy spaces

We give necessary and sufficient conditions for inhomogeneous Calderón-Zgymund operators to be bounded on the local hardy spaces $h^p(\mathbb{R}^n)$. We then give applications to local and truncated Riesz transforms, as well as pseudo-differential operators defined by amplitudes.

Hardy Space Estimates for Littlewood-Paley-Stein Square Functions and Calderón-Zygmund Operators

In this work, we give new sufficient conditions for a Littlewood-Paley-Stein square function and necessary and sufficient conditions for a Calderón-Zygmund operator to be bounded on Hardy spaces $H^p$ with indices smaller than $1$. New Carleson measure type conditions are defined for Littlewood-Paley-Stein operators, and we show that they are sufficient for the associated square function to be bounded from $H^p$ into $L^p$. New polynomial growth $BMO$ conditions are also introduced for Calderón-Zygmund operators. These results are applied to prove that Bony paraproducts can be constructed such…

A local $T(b)$ theorem for perfect multilinear Calderón-Zygmund operators

We prove a multilinear local $T(b)$ theorem that differs from previously considered multilinear local $T(b)$ theorems in using exclusively general testing functions $b$ as opposed to a mix of general testing functions and indicator functions. The main new feature is a set of relations between the various testing functions $b$ that to our knowledge has not been observed in the literature and is necessitated by our approach. For simplicity we restrict attention to the perfect dyadic model.

A T1 criterion for Hermite-Calderon-Zygmund operators on the BMO_H(R^n) space and applications

In this paper we establish a T1 criterion for the boundedness of Hermite-Calderon-Zygmund operators on the BMO_H(R^n) space naturally associated to the Hermite operator H. We apply this criterion in a systematic way to prove the boundedness on BMO_H(R^n) of certain harmonic analysis operators related to H (Riesz transforms, maximal operators, Littlewood-Paley g-functions and variation operators).

Calderón-Zygmund算子在RBMO上的一个刻画

A characterization of Calderón-Zygmund operators on RBMO

本文针对$\mathbb{R}^m$上非 doubling 测度引入的 RBMO 空间，利用函数 $T1$ 给出了有界 Calderón-Zygmund 算子的一个刻画。

基于非交换核的Calderón-Zygmund理论与H_1^c

Calderón-Zygmund theory with noncommuting kernels via $H_1^c$

本文提出了一种基于非交换核的Calderón-Zygmund理论新方法，通过H_1^c空间的原子描述研究终点估计问题。

Complex Powers of the Laplacian on Affine Nested Fractals as Calderón-Zygmund operators

We give the first natural examples of Calderón-Zygmund operators in the theory of analysis on post-critically finite self-similar fractals. This is achieved by showing that the purely imaginary Riesz and Bessel potentials on nested fractals with 3 or more boundary points are of this type. It follows that these operators are bounded on $L^{p}$, $1<p<\infty$ and satisfy weak 1-1 bounds. The analysis may be extended to infinite blow-ups of these fractals, and to product spaces based on the fractal or its blow-up.

On an estimate of Calderón-Zygmund operators by dyadic positive operators

Given a general dyadic grid ${\mathscr{D}}$ and a sparse family of cubes ${\mathcal S}=\{Q_j^k\}\in {\mathscr{D}}$, define a dyadic positive operator ${\mathcal A}_{\mathscr{D},{\mathcal S}}$ by $${\mathcal A}_{\mathscr{D},{\mathcal S}}f(x)=\sum_{j,k}f_{Q_j^k}χ_{Q_j^k}(x).$$ Given a Banach function space $X({\mathbb R}^n)$ and the maximal Calderón-Zygmund operator $T_{\natural}$, we show that $$\|T_{\natural}f\|_X\le c(n,T)\sup_{\mathscr{D},{\mathcal S}}\|{\mathcal A}_{\mathscr{D},{\mathcal S}}f\|_{X}.$$ This result is applied to weighted inequalities. In particular, it implies: (i) the "two-w…

Algebraic Calderón-Zygmund theory

Calderón-Zygmund theory has been traditionally developed on metric measure spaces satisfying additional regularity properties. In the lack of good metrics, we introduce a new approach for general measure spaces which admit a Markov semigroup satisfying purely algebraic assumptions. We shall construct an abstract form of "Markov metric" governing the Markov process and the naturally associated BMO spaces, which interpolate with the Lp-scale and admit endpoint inequalities for Calderón-Zygmund operators. Motivated by noncommutative harmonic analysis, this approach gives the first form of Calderó…

Boundedness properties of pseudo-differential operators and Calderón-Zygmund operators on modulation spaces

In this paper, we study the boundedness of pseudo-differential operators with symbols in $S_{ρ,δ}^m$ on the modulation spaces $M^{p,q}$. We discuss the order $m$ for the boundedness $\mathrm{Op}(S_{ρ,δ}^m) \subset \calL(M^{p,q}(\R^n))$ to be true. We also prove the existence of a Calderón-Zygmund operator which is not bounded on the modulation space $M^{p,q}$ with $q \neq 2$. This unboundedness is still true even if we assume a generalized T(1) condition. These results are induced by the unboundedness of pseudo-differential operators on $M^{p,q}$ whose symbols are of the class $S_{1,δ}^0$ with…

关于 Calderón-Zygmund 算子紧性的 $T1$ 定理

On the $T1$ theorem for compactness of Calderón-Zygmund operators

本文重新表述 Calderón-Zygmund 奇异积分算子的紧性 $T1$ 定理，证明其在 $L^2(\mathbb{R}^n)$ 上紧等价于 $T1,T^*1$ 属于 CMO 且算子弱紧。

超越Calderón-Zygmund理论的端点弱型界

Endpoint weak-type bounds beyond Calderón-Zygmund theory

本研究针对满足ℓ^q型有限范围稀疏支配的算子，推导了加权弱型(r,r)估计，改进了相关算子的界，并为Calderón-Zygmund算子等的标准稀疏支配精确端点界提供了简洁新证明，拓展了Calderón-Zygmund理论外的端点弱型界研究。

Calderón--Zygmund 算子的平均振荡稀疏控制

Sparse domination of Calderón-Zygmund operators by mean oscillations

本文证明若 Calderón--Zygmund 算子 T 满足 T(1)=0，则其稀疏控制可改进为用局部平均振荡代替局部平均，并以此刻画点态 Sobolev 型不等式成立的充要条件为 T(1)∈L^∞，回答了 Hoang、Moen 和 Pérez 的问题。

最优尺度的矩阵加权 Besov–Triebel–Lizorkin 空间：伪微分算子、迹算子与 Calderón–Zygmund 算子的有界性

Matrix-Weighted Besov--Triebel--Lizorkin Spaces of Optimal Scale: Boundedness of Pseudo-Differential, Trace, and Calderón--Zygmund Operators

研究利用分子与小波刻画，证明伪微分、迹及 Calderón–Zygmund 算子在矩阵 $\mathcal{A}_{\infty}$ 权广义 Besov–Triebel–Lizorkin 型空间上的有界性。

Multilinear operator-valued Calderón-Zygmund theory

We develop a general theory of multilinear singular integrals with operator-valued kernels, acting on tuples of UMD Banach spaces. This, in particular, involves investigating multilinear variants of the $\mathcal R$-boundedness condition naturally arising in operator-valued theory. We proceed by establishing a suitable representation of multilinear, operator-valued singular integrals in terms of operator-valued dyadic shifts and paraproducts, and studying the boundedness of these model operators via dyadic-probabilistic Banach space-valued analysis. In the bilinear case, we obtain a $T(1)$-typ…

Multilinear Marcinkiewicz-Zygmund inequalities

We extend to the multilinear setting classical inequalities of Marcinkiewicz and Zygmund on $\ell^r$-valued extensions of linear operators. We show that for certain $1 \leq p, q_1, \dots, q_m, r \leq \infty$, there is a constant $C\geq 0$ such that for every bounded multilinear operator $T\colon L^{q_1}(μ_1) \times \cdots \times L^{q_m}(μ_m) \to L^p(ν)$ and functions $\{f_{k_1}^1\}_{k_1=1}^{n_1} \subset L^{q_1}(μ_1), \dots, \{f_{k_m}^m\}_{k_m=1}^{n_m} \subset L^{q_m}(μ_m)$, the following inequality holds \begin{equation}\label{MZ ineq abstract} (1) \quad \quad \left\Vert \left(\sum_{k_1, \dots…

Calderón-Zygmund operators associated to matrix-valued kernels

Calderón-Zygmund operators with noncommuting kernels may fail to be Lp-bounded for $p \neq 2$, even for kernels with good size and smoothness properties. Matrix-valued paraproducts, Fourier multipliers on group vNa's or noncommutative martingale transforms are frameworks where we find such difficulties. We obtain weak type estimates for perfect dyadic CZO's and cancellative Haar shifts associated to noncommuting kernels in terms of a row/column decomposition of the function. Arbitrary CZO's satisfy $H_1 \to L_1$ type estimates. In conjunction with $L_\infty \to BMO$, we get certain row/column …

BMO, H^1, and Calderon-Zygmund operators for non doubling measures

Given a Radon measure $μ$ on $R^d$, which may be non doubling, we introduce a space of type BMO with respect to this measure. It is shown that many properties that hold when $μ$ is doubling remain valid for the space BMO introduced in this paper, without assuming $μ$ doubling. For instance, Calderon-Zygmund operators which are bounded in $L^2$ are bounded from $L^\infty$ into the new BMO space. Moreover, a John-Nirenberg inequality is satisfied, and the predual of BMO is an atomic space $H^1$. Using a sharp maximal function it is proved that operators bounded from $L^\infty$ into BMO and from …

关于Calderón-Zygmund型奇异积分及其交换子的延拓

On extension of Calderón-Zygmund type singular integrals and their commutators

本文研究了Calderón-Zygmund型奇异积分及其交换子的延拓，建立了在Lipschitz、Hardy和加权空间上的一致估计，并通过取极限推导出经典情形的类似估计。