Zygmund Operators Research Articles

New oscillation classes and two weight bump conditions for commutators

In this paper we consider two weight bump conditions for higher order commutators. Given b and a Calderón–Zygmund operator T, define the commutator $$T^1_bf=[T,b]f= bTf-T(bf)$$ , and for $$m\ge 2$$ define the iterated commutator $$T^m_b f = [b,T_b^{m-1}]f$$ . Traditionally, commutators are defined for functions $$b\in BMO$$ , but we show that if we replace BMO by an oscillation class first introduced by Pérez (J Funct Anal 128(1):163–185, 1995), we can give a range of sufficient conditions on a pair of weights (u, v) for $$T^m_b : L^p(v) \rightarrow L^p(u)$$ to be bounded. Our results generalize work of Cruz-Uribe and Moen (Publ Mat 56(1):147–190, 2012), and more recent work by Lerner et al. (J Funct Anal 281(8):46, 2021). We also prove necessary conditions for the iterated commutators to be bounded, generalizing results of Isralowitz et al. (Commutators in the two scalar and matrix weighted setting. Preprint, 2020. http://arxiv.org/abs/2001.11182 ).

Integral operators on local Orlicz-Morrey spaces

We establish a general principle on the boundedness of operators on local Orlicz-Morrey spaces. As applications of this principle, we obtain the boundedness of the Calder?n-Zygmund operators, the nonlinear commutators of the Calder?n-Zygmund operators, the oscillatory singular integral operators, the singular integral operators with rough kernels and the Marcinkiewicz integrals on the local Orlicz-Morrey spaces.

Quantitative weighted bounds for Calderón commutators with rough kernels

We obtain a quantitative weighted bound for the Calderón commutator $\mathcal C_\Omega $ which is a typical example of a non-convolution Calderón–Zygmund operator under the condition $\Omega \in L^\infty (\mathbb S^{n-1})$; this is the best known quantita

Boundedness of Calderón–Zygmund operators on special John–Nirenberg–Campanato and Hardy-type spaces via congruent cubes

Boundedness of Calderón–Zygmund operators on special John–Nirenberg–Campanato and Hardy-type spaces via congruent cubes

Compactness of commutator of Riesz transforms in the two weight setting

We characterize the compactness of commutators in the Bloom setting. Namely, for a suitably non-degenerate Calderón–Zygmund operator T, and a pair of weights σ,ω∈Ap, the commutator [T,b] is compact from Lp(σ)→Lp(ω) if and only if b∈VMOν, where ν=(σ/ω)1/p. This extends the work of the first author, Holmes and Wick. The weighted VMO spaces are different from the classical VMO space. In dimension d=1, compactly supported and smooth functions are dense in VMOν, but this need not hold in dimensions d≥2. Moreover, the commutator in the product setting with respect to little VMO space is also investigated.

The Calderon–Zygmund Operator and its Relation to Asymptotic Estimates of Ordinary Differential Operators

The Calderon–Zygmund Operator and its Relation to Asymptotic Estimates of Ordinary Differential Operators

Boundedness of commutators of $$\theta $$-type Calderón–Zygmund operators on generalized weighted Morrey spaces over RD-spaces

Boundedness of commutators of $$\theta $$-type Calderón–Zygmund operators on generalized weighted Morrey spaces over RD-spaces

Extrapolation to weighted Morrey spaces with variable exponents and applications

AbstractThis paper establishes the mapping properties of pseudo-differential operators and the Fourier integral operators on the weighted Morrey spaces with variable exponents and the weighted Triebel–Lizorkin–Morrey spaces with variable exponents. We obtain these results by extending the extrapolation theory to the weighted Morrey spaces with variable exponents. This extension also gives the mapping properties of Calderón–Zygmund operators on the weighted Hardy–Morrey spaces with variable exponents and the wavelet characterizations of the weighted Hardy–Morrey spaces with variable exponents.

On Ahlfors–Beurling Operator

We investigate regularity properties of solutions of Beltrami equation expressed in terms of moduli of continuity. In particular, we prove that a class of Calderon–Zygmund operators, including Ahlfors–Beurling operator, preserves certain type of modulus of continuity of compactly supported functions. We also prove a purely topological result which easily gives injectivity of normal solutions of Beltrami equation.

The Lp-to-Lq boundedness of commutators with applications to the Jacobian operator

Supplying the missing necessary conditions, we complete the characterisation of the Lp→Lq boundedness of commutators [b,T] of pointwise multiplication and Calderón–Zygmund operators, for arbitrary pairs of 1<p,q<∞ and under minimal non-degeneracy hypotheses on T.For p≤q (and especially p=q), this extends a long line of results under more restrictive assumptions on T. In particular, we answer a recent question of Lerner, Ombrosi, and Rivera-Ríos by showing that b∈BMO is necessary for the Lp-boundedness of [b,T] for any non-zero homogeneous singular integral T. We also deal with iterated commutators and weighted spaces.For p>q, our results are new even for special classical operators with smooth kernels. As an application, we show that every f∈Lp(Rd) can be represented as a convergent series of normalised Jacobians Ju=det⁡∇u of u∈W˙1,dp(Rd)d. This extends, from p=1 to p>1, a result of Coifman, Lions, Meyer and Semmes about J:W˙1,d(Rd)d→H1(Rd), and supports a conjecture of Iwaniec about the solvability of the equation Ju=f∈Lp(Rd).

Extrapolation of compactness on weighted spaces: Bilinear operators

In a previous paper, we obtained several “compact versions” of Rubio de Francia’s weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calderón–Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers. More general versions of these results are recently due to Cao, Olivo and Yabuta (arXiv:2011.13191), whose approach depends on developing weighted versions of the Fréchet–Kolmogorov criterion of compactness, whereas we avoid this by relying on “softer” tools, which might have an independent interest in view of further extensions of the method.

Molecular Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

Let p→∈(0,∞)n be an exponent vector and A be a general expansive matrix on Rn. Let HAp→(Rn) be the anisotropic mixed-norm Hardy spaces associated with A defined via the non-tangential grand maximal function. In this article, using the known atomic characterization of HAp→(Rn), the authors characterize this Hardy space via molecules with the best possible known decay. As an application, the authors establish a criterion on the boundedness of linear operators from HAp→(Rn) to itself, which is used to explore the boundedness of anisotropic Calderón–Zygmund operators on HAp→(Rn). In addition, the boundedness of anisotropic Calderón–Zygmund operators from HAp→(Rn) to the mixed-norm Lebesgue space Lp→(Rn) is also presented. The obtained boundedness of these operators positively answers a question mentioned by Cleanthous et al. All of these results are new, even for isotropic mixed-norm Hardy spaces on Rn.

Off-Diagonal Estimates for Bi-Commutators

Abstract We study the bi-commutators $[T_1, [b, T_2]]$ of pointwise multiplication and Calderón–Zygmund operators and characterize their $L^{p_1}L^{p_2} \to L^{q_1}L^{q_2}$ boundedness for several off-diagonal regimes of the mixed-norm integrability exponents $(p_1,p_2)\neq (q_1,q_2)$. The strategy is based on a bi-parameter version of the recent approximate weak factorization method.

Sparse domination results for compactness on weighted spaces

By means of appropriate sparse bounds, we deduce compactness on weighted $$L^p(w)$$ spaces, $$1<p<\infty$$ , for all Calderón–Zygmund operators having compact extensions on $$L^2({\mathbb {R}}^n)$$ . Similar methods lead to new results on boundedness and compactness of Haar multipliers on weighted spaces. In particular, we prove weighted bounds for weights in a class strictly larger than the typical $$A_p$$ class.

A two weight inequality for Calderón–Zygmund operators on spaces of homogeneous type with applications

Let (X,d,μ) be a space of homogeneous type in the sense of Coifman and Weiss, i.e. d is a quasi metric on X and μ is a positive measure satisfying the doubling condition. Suppose that u and v are two locally finite positive Borel measures on (X,d,μ). Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calderón–Zygmund operator T from L2(u) to L2(v) in terms of the A2 condition and two testing conditions. For every cube B⊂X, we have the following testing conditions, with 1B taken as the indicator of B‖T(u1B)‖L2(B,v)≤T‖1B‖L2(u),‖T⁎(v1B)‖L2(B,u)≤T‖1B‖L2(v).The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.

Maximal and Calderón–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 Calderón–Zygmund operators T on these spaces. Finally, we apply these results to the Bochner–Riesz operator , identity approximation and the Marcinkiewicz operator on the spaces .

On Function Spaces with Mixed Norms — A Survey

The targets of this article are threefold. The first one is to give a survey on the recent developments of function spaces with mixed norms, including mixed Lebesgue spaces, iterated weak Lebesgue spaces, weak mixed-norm Lebesgue spaces and mixed Morrey spaces as well as anisotropic mixed-norm Hardy spaces. The second one is to provide a detailed proof for a useful inequality about mixed Lebesgue norms and the Hardy--Littlewood maximal operator and also to improve some known results on the maximal function characterizations of anisotropic mixed-norm Hardy spaces and the boundedness of Calderon--Zygmund operators from these anisotropic mixed-norm Hardy spaces to themselves or to mixed Lebesgue spaces. The last one is to correct some errors and seal some gaps existing in the known articles.

XMO and Weighted Compact Bilinear Commutators

To study the compactness of bilinear commutators of certain bilinear Calderón–Zygmund operators which include (inhomogeneous) Coifman–Meyer bilinear Fourier multipliers and bilinear pseudodifferential operators as special examples, Torres and Xue (Rev Mat Iberoam 36:939–956, 2020) introduced a new subspace of BMO\(\,(\mathbb {R}^n)\), denoted by XMO\(\,(\mathbb {R}^n)\), and conjectured that it is just the space VMO\(\,(\mathbb {R}^n)\) introduced by D. Sarason. In this article, the authors give a negative answer to this conjecture by establishing an equivalent characterization of XMO\(\,(\mathbb {R}^n)\), which further clarifies that XMO\(\,(\mathbb {R}^n)\) is a proper subspace of VMO\(\,(\mathbb {R}^n)\). This equivalent characterization of XMO\(\,(\mathbb {R}^n)\) is formally similar to the corresponding one of CMO\(\,(\mathbb {R}^n)\) obtained by A. Uchiyama, but its proof needs some essential new techniques on dyadic cubes as well as some exquisite geometrical observations. As an application, the authors also obtain a weighted compactness result on such bilinear commutators, which optimizes the corresponding result in the unweighted setting.

Weighted endpoint estimates for the composition of Calderón–Zygmund operators on spaces of homogeneous type

Weighted endpoint estimates for the composition of Calderón–Zygmund operators on spaces of homogeneous type

Boundedness of Integral Operators in Generalized Weighted Grand Lebesgue Spaces with Non-doubling Measures

The goal of this paper is to prove the boundedness of fundamental integral operators of Harmonic Analysis in generalized weighted grand Lebesgue spaces $$L^{p),\varphi }_w$$ defined on domains in $${\mathbb {R}}^n$$ without assuming that the underlying measure $$\mu $$ is doubling. Operators under consideration involve maximal and Calderon–Zygmund operators, also commutators of singular integrals with BMO functions. Essential part of this paper is devoted to the weighted Sobolev-type inequalities in these spaces for fractional integrals with non-doubling measures satisfying the growth condition. We discuss the case when a weight function defines the absolute continuous measure of integration, or plays the role of multiplier in the definition of the norm. All these results are new even for the classical weighted grand Lebesgue spaces $$L^{p),\theta }_w$$ defined with respect to non-doubling measures. The results are obtained under the Muckenhoupt condition on weights. The results are obtained for weight functions satisfying Muckenhoupt $$A_p$$ conditions defined with respect to non-homogeneous spaces.