Zygmund Operators Research Articles

Central Calderón–Zygmund operators on Herz-type Hardy spaces of variable smoothness and integrability

In this article we use the atomic decomposition of a Herz-type Hardy space of variable smoothness and integrability to obtain the boundedness of the central Calderon–Zygmund operators on Herz-type Hardy spaces with variable smoothness and integrability.

Commutators of multilinear Calderón–Zygmund operators with Dini type kernels on some function spaces

Commutators of multilinear Calderón–Zygmund operators with Dini type kernels on some function spaces

On Boundedness of Bergman Projection Operators in Banach Spaces of Holomorphic Functions in Half-Plane and Harmonic Functions in Half-Space

We present a simple proof of the boundedness of holomorphic and harmonic Bergman projection operators on a half-plane and a half-space respectively on the Orlicz space, the variable exponent Lebesgue space, and the variable exponent generalized Morrey space. The approach is based on an idea due to V. P. Zaharyuta and V. I. Yudovich (1962) to use Calderon–Zygmund operators for proving the boundedness of the Bergman projection in Lebesgue spaces on the unit disc. We also study the rate of growth of functions near the boundary in the spaces under consideration.

Real Interpolation of Compact Bilinear Operators

We establish an analog for bilinear operators of the compactness interpolation result for bounded linear operators proved by Cwikel and Cobos, Kuhn and Schonbek. We work with the assumption that $$T:(A_0+A_1) \times (B_0+B_1) \longrightarrow E_0+E_1$$ is bounded, but we also study the case when this does not hold. Applications are given to compactness of convolution operators and compactness of commutators of bilinear Calderon–Zygmund operators.

On pointwise and weighted estimates for commutators of Calderón–Zygmund operators

In recent years, it has been well understood that a Calderón–Zygmund operator T is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar pointwise estimate for the commutator [b,T] with a locally integrable function b. This result is applied into two directions. If b∈BMO, we improve several weighted weak type bounds for [b,T]. If b belongs to the weighted BMO, we obtain a quantitative form of the two-weighted bound for [b,T] due to Bloom–Holmes–Lacey–Wick.

Convex body domination and weighted estimates with matrix weights

We introduce the so called convex body valued sparse operators, which generalize the notion of sparse operators to the case of spaces of vector valued functions.We prove that Calderón–Zygmund operators as well as Haar shifts and paraproducts can be dominated by such operators. By estimating sparse operators we obtain weighted estimates with matrix weights. We get two weight A2–A∞ estimates, that in the one weight case give us the estimate‖T‖L2(W)→L2(W)≤C[W]A21/2[W]A∞≤C[W]A23/2 where T is either Calderón–Zygmund operator (with modulus of continuity satisfying the Dini condition), or a Haar shift or a paraproduct.

Bounds for Calderón–Zygmund operators with matrix A2 weights

It is well-known that dyadic martingale transforms are a good model for Calderón–Zygmund singular integral operators. In this paper we extend some results on weighted norm inequalities to vector-valued functions. We prove that if W is an A2 matrix weight, then the weighted L2-norm of a Calderón–Zygmund operator with cancellation has the same dependence on the A2 characteristic of W as the weighted L2-norm of an appropriate matrix martingale transform. Thus the question of the dependence of the norm of matrix-weighted Calderón–Zygmund operators on the A2 characteristic of the weight is reduced to the case of dyadic martingales and paraproducts. We also show a slightly different proof for the special case of Calderón–Zygmund operators with even kernel, where only scalar martingale transforms are required. We conclude the paper by proving a version of the matrix-weighted Carleson Embedding Theorem.Our method uses a Bellman function technique introduced by S. Treil to obtain the right estimates for the norm of dyadic Haar shift operators. We then apply the representation theorem of T. Hytönen to extend the result to general Calderón–Zygmund operators.

Hardy-type space estimates for multilinear commutators of Calderón–Zygmund operators on nonhomogeneous metric measure spaces

Let (X,d,μ) be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let T be a Calderón–Zygmund operator and let b→:=(b1,…,bm) be a finite family of \widetilde{RBMO}(μ) functions. In this article, the authors establish the boundedness of the multilinear commutator Tb→, generated by T and b→ from the atomic Hardy-type space H˜fin,b→,m,ρ1,q,m+1(μ) into the Lebesgue space L1(μ). The authors also prove that Tb→ is bounded from the atomic Hardy-type space H˜fin,b→,m,ρ1,q,m+2(μ) into the atomic Hardy space H˜1(μ) via the molecular characterization of H˜1(μ).

Oscillation of Generalized Differences of Hölder and Zygmund Functions

In this paper we analyze the oscillation of functions having derivatives in the Holder or Zygmund class in terms of generalized differences and prove that its growth is governed by a version of the classical Kolmogorov’s Law of the Iterated Logarithm. A better behavior is obtained for functions in the Lipschitz class via an interesting connection with Calderon–Zygmund operators.

Endpoint boundedness of commutators on spaces of homogeneous type

Let be a space of homogeneous type in the sense of Coifman and Weiss. Let be a family of sublinear operators that include the well-known Calderón–Zygmund operator. The authors prove that the commutator [b, T] is bounded from a Hardy-type subspace of to , where denotes the Lebesgue space of all integrable functions, the atomic Hardy space of Coifman and Weiss with the dual space , b is a non-constant -function and . Indeed, the space is the largest subspace in that possesses this property. The approach taken in this article adopts the bilinear decomposition theory for the product of functions in and .

Hardy space estimates for limited ranges of Muckenhoupt weights

In this article, we give a full necessary and sufficient set of conditions for a Calderón–Zygmund operator to be bounded on weighted Hardy spaces Hwp where w is an Muckenhoupt weight and 0<p<∞. In fact, this result is new even when 1<p<∞ since it allows for Hwp boundedness of an operator when 1<p<q<∞ and w∈Aq, where it is possible that Hwp≠Lwp. These singular integral results are achieved by proving Littlewood–Paley–Stein square function type estimates from Hwp into Lwp for 0<p<∞ and a Muckenhoupt weight w, which are interesting results in their own right. New techniques involving A∞ weight invariant spaces are also used to prove the weighted estimates for Calderón–Zygmund operators. More precisely, we prove the following BMO type weight invariance properties: for a fixed s≥0, the weighted Sobolev-BMO spaces Is(BMOw) coincide for all w∈A∞, the weighted p=∞ type Triebel–Lizorkin spaces F˙∞,ws,2 coincide for all w∈A∞, and these two classes of spaces coincide with each other as well, all of which have comparable norms up to constants depending on an Ap character of the weight w∈A∞.

Sublinear operators with rough kernel generated by Calderón–Zygmund operators and their commutators on generalized Morrey spaces

The aim of this paper is to establish the boundedness of certain sublinear operators with rough kernel generated by Calderon–Zygmund operators and their commutators on generalized Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. The Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.

Dirichlet boundary value problems for uniformly elliptic equations in modified local generalized Sobolev–Morrey spaces

In this paper, we study the boundedness of the sublinear operators, generated by Calderon–Zygmund operators in local generalized Morrey spaces. By using these results we prove the solvability of the Dirichlet boundary value problem for a polyharmonic equation in modified local generalized Sobolev–Morrey spaces. We obtain a priori estimates for the solutions of the Dirichlet boundary value problems for the uniformly elliptic equations in modified local generalized Sobolev–Morrey spaces defined on bounded smooth domains.

Commutators of Bilinear Pseudodifferential Operators and Lipschitz Functions

Commutators of bilinear pseudodifferential operators and the operation of multiplication by a Lipschitz function are studied. The bilinear symbols of the pseudodifferential operators considered belong to classes that are shown to properly contain certain bilinear Hormander classes of symbols of order one. The corresponding commutators are proved to be bilinear Calderon–Zygmund operators.

Besov spaces via wavelets on metric spaces endowed with doubling measure, singular integral, and the T1 type theorem

The aim of this paper is twofold. We first establish the Besov spaces on metric spaces endowed with a doubling measure, via the remarkable orthonormal wavelet basis constructed recently by T. Hytönen and O. Tapiola, and characterize the dual spaces of these Besov spaces. Second, we prove the T1 type theorem for the boundedness of Calderón–Zygmund operators on these Besov spaces. Finally, we introduce a new class of Lipschitz spaces and characterize these spaces via the Littlewood–Paley theory. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Tent space boundedness via extrapolation

We study the action of operators on tent spaces such as maximal operators, Calderon–Zygmund operators, Riesz potentials. We also consider singular non-integral operators. We obtain boundedness as an application of extrapolation methods in the Banach range. In the non Banach range, boundedness results for Calderon–Zygmund operators follow by using an appropriate atomic theory. We end with some consequences on amalgam spaces.

Weighted estimates for commutators of strongly singular Calderón–Zygmund operators

In this paper, the authors establish the boundedness of commutators generated by strongly singular Calderon–Zygmund operators and weighted BMO functions on weighted Herz-type Hardy spaces. Moreover, the corresponding results for commutators generated by strongly singular Calderon–Zygmund operators and weighted Lipschitz functions can also be obtained.

Molecular characterization of anisotropic Musielak–Orlicz Hardy spaces and their applications

Let A be an expansive dilation on ℝn and φ : ℝn × [0, ∞) → [0, ∞) an anisotropic Musielak–Orlicz function. Let HAφ(ℝn) be the anisotropic Hardy space of Musielak–Orlicz type defined via the grand maximal function. In this article, the authors establish its molecular characterization via the atomic characterization of HAφ(ℝn). The molecules introduced in this article have the vanishing moments up to order s and the range of s in the isotropic case (namely, A := 2In×n) coincides with the range of well-known classical molecules and, moreover, even for the isotropic Hardy space Hp(ℝn) with p ∈ (0, 1] (in this case, A := 2In×n, φ(x, t) := tp for all x ∈ ℝn and t ∈ [0, ∞)), this molecular characterization is also new. As an application, the authors obtain the boundedness of anisotropic Calderon–Zygmund operators from HAφ(ℝn) to Lφ(ℝn) or from HAφ(ℝn) to itself.

Maximal operator and Calderon–Zygmund operators in local Morrey–Lorentz spaces

ABSTRACTIn this paper we proved the boundedness of the Hardy–Littlewood maximal operator M, the Calderon–Zygmund operators T and the maximal Calderon–Zygmund operators on the local Morrey–Lorentz spaces . Finally, we give some applications of these results.

Vector-valued maximal multilinear Calderón–Zygmund operator with nonsmooth kernel on weighted Morrey space

In this paper, vector-valued Maximal multilinear Calderon–Zygmund operator with nonsmooth kernel and its commutators generated by \(\textit{BMO}\) function are studied. The purpose of this paper is to establish that these operators are bounded on certain multiple weighted Morrey space. Moreover, we also get the multiple weighed Morrey estimate for the vector-valued Calderon–Zygmund operator and its commutator generated by \(\textit{BMO}\) function.