Zygmund Operators Research Articles

Linear operators and their commutators generated by Calderón–Zygmund operators on generalized Morrey spaces associated with ball Banach function spaces

Linear operators and their commutators generated by Calderón–Zygmund operators on generalized Morrey spaces associated with ball Banach function spaces

Ap Weights in Directionally (γ,m) Limited Space and Applications

Let (X,d) be a directionally (γ,m)-limited space with every γ∈(0,∞). In this setting, we aim to study an analogue of the classical theory of Ap(μ) weights. As an application, we establish some weighted estimates for the Hardy–Littlewood maximal operator. Then, we introduce the relationship between directionally (γ,m)-limited spaceand geometric doubling. Finally, we obtain the weighted norm inequalities of the Calderón–Zygmund operator and commutator in non-homogeneous space.

Calderón—Zygmund Operators and Commutators in Spaces of Homogeneous Type: Weighted Inequalities

The recent proof of the sharp weighted bound for Calderón-Zygmund operators has led to much investigation in sharp mixed bounds for operators and commutators, that is, a sharp weighted bound that is a product of at least two different Ap weight constants. The reason why these are sought after is that the product will be strictly smaller than the original one-constant bound. We prove a variety of these bounds in spaces of homogeneous type, using the new techniques of Lerner, for both operators and commutators.

Inhomogeneous cancellation conditions and Calderón–Zygmund type operators on [formula omitted

In this work we present a new approach to molecules on Goldberg’s local Hardy spaces hp(Rn), 0<p≤1, 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.

Calderón–Zygmund operators and their commutators on generalized weighted Orlicz–Morrey spaces

In this paper, we obtain the necessary and sufficient conditions for the weak/strong boundedness of the Calderón–Zygmund operators in generalized weighted Orlicz–Morrey spaces. We also study the boundedness of the commutators of Calderón–Zygmund operators on these spaces. Moreover, the boundedness of Calderón–Zygmund operators in the vector-valued setting is given.

Necessary and sufficient conditions for boundedness of commutators associated with Calderón–Zygmund operators on slice spaces

Necessary and sufficient conditions for boundedness of commutators associated with Calderón–Zygmund operators on slice spaces

Wavelet representation of singular integral operators

This article develops a novel approach to the representation of singular integral operators of Calderón–Zygmund type in terms of continuous model operators, in both the classical and the bi-parametric setting. The representation is realized as a finite sum of averages of wavelet projections of either cancellative or noncancellative type, which are themselves Calderón–Zygmund operators. Both properties are out of reach for the established dyadic-probabilistic technique. Unlike their dyadic counterparts, our representation reflects the additional kernel smoothness of the operator being analyzed. Our representation formulas lead naturally to a new family of T(1) theorems on weighted Sobolev spaces whose smoothness index is naturally related to kernel smoothness. In the one parameter case, we obtain the Sobolev space analogue of the \(A_2\) theorem; that is, sharp dependence of the Sobolev norm of T on the weight characteristic is obtained in the full range of exponents. In the bi-parametric setting, where local average sparse domination is not generally available, we obtain quantitative \(A_p\) estimates which are best known, and sharp in the range \(\max \{p,p'\}\ge 3\) for the fully cancellative case.

The Dyadic Representation Theorem Using Smooth Wavelets with Compact Support

The representation of a general Calderón–Zygmund operator in terms of dyadic Haar shift operators first appeared as a tool to prove the A_2 theorem, and it has found a number of other applications. In this paper we prove a new dyadic representation theorem by using smooth compactly supported wavelets in place of Haar functions. A key advantage of this is that we achieve a faster decay of the expansion when the kernel of the general Calderón–Zygmund operator has additional smoothness.

{text {UMD}}-Extensions of Calderón–Zygmund Operators with Mild Kernel Regularity

Armed with new methods we revisit a result of Figiel concerning the {text {UMD}}-extensions of linear Calderón–Zygmund operators with mild kernel regularity and then extend our new proof to the multilinear setting improving recent {text {UMD}}-valued estimates of multilinear singular integrals.

Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Littlewood—Paley Characterizations with Applications to Boundedness of Calderón—Zygmund Operators

Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Littlewood—Paley Characterizations with Applications to Boundedness of Calderón—Zygmund Operators

Mixed inequalities for commutators with multilinear symbol

We prove mixed inequalities for commutators of Calderón–Zygmund operators (CZO) with multilinear symbols. Concretely, let $$m\in {\mathbb {N}}$$ and $${\mathbf {b}}=(b_1,b_2,\ldots , b_m)$$ be a vectorial symbol such that each component $$b_i\in \mathrm {Osc}_{\mathrm {exp}\, L^{r_i}}$$ , with $$r_i\ge 1$$ . If $$u\in A_1$$ and $$v\in A_\infty (u)$$ we prove that the inequality $$\begin{aligned} uv\left( \left\{ x\in {\mathbb {R}}^n: \frac{|T_{\mathbf {b}}(fv)(x)|}{v(x)}>t\right\} \right) \le C\int _{{\mathbb {R}}^n}\Phi \left( \Vert {\mathbf {b}}\Vert \frac{|f(x)|}{t}\right) u(x)v(x)\,dx \end{aligned}$$ holds for every $$t>0$$ , where $$\Phi (t)=t(1+\log ^+t)^r$$ , with $$1/r=\sum _{i=1}^m 1/r_i$$ . We also consider operators of convolution type with kernels satisfying less regularity properties than CZO. In this setting, we give a Coifman type inequality for the associated commutators with multilinear symbol. This result allows us to deduce the $$L^p(w)$$ -boundedness of these operators when $$1<p<\infty $$ and $$w\in A_p$$ . As a consequence, we can obtain the desired mixed inequality in this context.

A T1 theorem for general Calderón—Zygmund operators with comparable doubling weights, and optimal cancellation conditions

A T1 theorem for general Calderón—Zygmund operators with comparable doubling weights, and optimal cancellation conditions

Singular and Fractional Integral Operators on Weighted Local Morrey Spaces

We obtain a characterization of the weighted inequalities for the Riesz transforms on weighted local Morrey spaces. The condition is sufficient for the boundedness on the same spaces of all Calderón–Zygmund operators suitably defined on the functions of the space. In the case of the fractional maximal operator and the fractional integral we obtain a characterization valid for exponents satisfying the Sobolev relation. For power weights we get sharp results for these operators in the usual versions of weighted Morrey spaces, neither restricted to the Sobolev relation of the exponents nor to the one-weighted setting.

Weak Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Decompositions, Real Interpolation, and Calderón–Zygmund Operators

Let $$({\mathbb {X}},d,\mu )$$ be a space of homogeneous type in the sense of Coifman and Weiss, and $$X({\mathbb {X}})$$ a ball quasi-Banach function space on $${\mathbb {X}}$$ . In this article, the authors introduce the weak Hardy space $$WH_X({\mathbb {X}})$$ associated with $$X({\mathbb {X}})$$ via the grand maximal function, and characterize $$WH_X({\mathbb {X}})$$ by other maximal functions and atoms. The authors then apply these characterizations to obtain the real interpolation and the boundedness of Calderón–Zygmund operators in the critical case. The main novelties of this article exist in that the authors use the Aoki–Rolewicz theorem and both the dyadic system and the exponential decay of approximations of the identity on $${\mathbb {X}}$$ , which closely connect with the geometrical properties of $${\mathbb {X}}$$ , to overcome the difficulties caused by the deficiency of both the triangle inequality of $$\Vert \cdot \Vert _{X({\mathbb {X}})}$$ and the reverse doubling assumption of the measure $$\mu $$ under consideration, and also use the relation between the convexification of $$X({\mathbb {X}})$$ and the weak ball quasi-Banach function space $$WX({\mathbb {X}})$$ associated with $$X({\mathbb {X}})$$ to prove that the infinite summation of atoms converges in the space of distributions on $${\mathbb {X}}$$ . Moreover, all these results have a wide range of generality and, particularly, even when they are applied to the weighted Lebesgue space, the Orlicz space, and the variable Lebesgue space, the obtained results are also new and, actually, some of them are new even on RD-spaces (namely, spaces of homogeneous type satisfying the additional reverse doubling condition).

Maximal singular integral operators acting on noncommutative $$L_p$$-spaces

In this paper, we study the boundedness theory for maximal Calderón–Zygmund operators acting on noncommutative $$L_p$$ -spaces. Our first result is a criterion for the weak type (1, 1) estimate of noncommutative maximal Calderón–Zygmund operators; as an application, we obtain the weak type (1, 1) estimates of operator-valued maximal singular integrals of convolution type under proper regularity conditions. These are the first noncommutative maximal inequalities for families of truly non-positive linear operators. For homogeneous singular integrals, the strong type (p, p) ( $$1<p<\infty $$ ) maximal estimates are shown to be true even for rough kernels. As a byproduct of the criterion, we obtain the noncommutative weak type (1, 1) estimate for Calderón–Zygmund operators with integral regularity condition that is slightly stronger than the Hörmander condition; this provides somewhat an affirmative answer to an open question in the noncommutative Calderón–Zygmund theory.

The Hodge star operator and the Beltrami equation

An essentially unique homeomorphic solution to the Beltrami equation with measurable coefficients was found in the 1930s by Morrey. The most well-known proof from the 1960s uses the theory of Calderón–Zygmund and singular integral operators in $$L^p(\mathbb {C})$$ . We will present an alternative method to solve the Beltrami equation using the Hodge star operator and standard elliptic PDE theory. We will also discuss a different method to prove the regularity of the solution. This approach is partially based on work by Dittmar.

Quantitative weighted bounds for composite operators on spaces of homogeneous type

Let T1 and T2 be Calderón–Zygmund operators, and let T1,b denote the commutator of T1 and the BMO function b. By establishing bisublinear sparse dominations, we obtain the quantitative weighted bounds for composite operators T1T2 and T1,bT2 on Lp(X,ω), with ω∈Ap(X).

Some applications of the dual spaces of Hardy-amalgam spaces

Thanks to the characterizations of the dual spaces of the Hardy-amalgam spaces $$\mathcal H^{(q,p)}$$ and $$\mathcal{H}_{\mathrm{loc}}^{(q,p)}$$ for $$0<q\leq1$$ and $$q\leq p<\infty$$ , obtained in one of our recent papers, we prove that the inclusion of $$\mathcal H^{(1,p)}$$ in $$(L^1,\ell^p)$$ for $$1\leq p<\infty$$ is strict, and more generally, that the one of $$\mathcal H^{(q,p)}$$ in $$\mathcal{H}_{\mathrm{loc}}^{(q,p)}$$ for $$0<q\leq1$$ and $${q\leq p<\infty}$$ , but also for $$0<p<q\leq1$$ , is strict. Moreover, as other applications, we obtain some results of boundedness of Calderón–Zygmund and convolution operators, generalizing those known in the setting of the spaces $$\mathcal H^1$$ and $$\mathrm{BMO}(\mathbb{R}^d)$$ , and more generally, in the setting of $$\mathcal H^q$$ and its dual space.

Multiplier conditions for Boundedness into Hardy spaces

In the present work we find useful and explicit necessary and sufficient conditions for linear and multilinear multiplier operators of Coifman–Meyer type, finite sum of products of Calderón–Zygmund operators, and also of intermediate types to be bounded from a product of Lebesgue or Hardy spaces into a Hardy space. These conditions state that the symbols of the multipliers

BOUNDEDNESS AND COMPACTNESS OF CAUCHY-TYPE INTEGRAL COMMUTATOR ON WEIGHTED MORREY SPACES

AbstractIn this paper we study boundedness and compactness characterizations of the commutators of Cauchy type integrals on bounded strongly pseudoconvex domains D in $\mathbb C^{n}$ with boundaries $bD$ satisfying the minimum regularity condition $C^{2}$ , based on the recent results of Lanzani–Stein and Duong et al. We point out that in this setting the Cauchy type integral is the sum of the essential part which is a Calderón–Zygmund operator and a remainder which is no longer a Calderón–Zygmund operator. We show that the commutator is bounded on the weighted Morrey space $L_{v}^{p,\kappa }(bD)$ ( $v\in A_{p}, 1&lt;p&lt;\infty $ ) if and only if b is in the BMO space on $bD$ . Moreover, the commutator is compact on the weighted Morrey space $L_{v}^{p,\kappa }(bD)$ ( $v\in A_{p}, 1&lt;p&lt;\infty $ ) if and only if b is in the VMO space on $bD$ .