Calderon Zygmund Research Articles

Weighted estimates for the multilinear maximal function on the upper half‐spaces

AbstractFor a general dyadic grid, we give a Calderón–Zygmund type decomposition, which is the principle fact about the multilinear maximal function on the upper half‐spaces. Using the decomposition, we study the boundedness of . We obtain a natural extension to the multilinear setting of Muckenhoupt's weak‐type characterization. We also partially obtain characterizations of Muckenhoupt's strong‐type inequalities with one weight. Assuming the reverse Hölder's condition, we get a multilinear analogue of Sawyer's two weight theorem. Moreover, we also get Hytönen–Pérez type weighted estimates.

Compact bilinear commutators: the quasi-Banach space case

We present the details of an extension of known results about the compactness of the commutators of bilinear Calderon–Zygmund operators with multiplication by appropriate functions in the John–Nirenberg space, to the case when the target of the operators is a quasi-Banach Lebesgue space.

Inequalities for the Extended Best Polynomial Approximation Operator in Orlicz Spaces

In this paper we pursue the study of the best approximation operator extended from LΦ to Lφ, where φ denotes the derivative of the function Φ. We get pointwise convergence for the coefficients of the extended best approximation polynomials for a wide class of function f, closely related to the Calderon–Zygmund class t (x) which had been introduced in 1961. We also obtain weak and strong type inequalities for a maximal operator related to the extended best polynomial approximation and a norm convergence result for the coefficients is derived. In most of these results, we have to consider Matuszewska–Orlicz indices for the function φ.

Atomic and Littlewood–Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

Let $$\vec {a}:=(a_1,\ldots ,a_n)\in [1,\infty )^n$$ , $$\vec {p}:=(p_1,\ldots ,p_n)\in (0,\infty )^n$$ and $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ be the anisotropic mixed-norm Hardy space associated with $$\vec {a}$$ defined via the non-tangential grand maximal function. In this article, via first establishing a Calderon–Zygmund decomposition and a discrete Calderon reproducing formula, the authors then characterize $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ , respectively, by means of atoms, the Lusin area function, the Littlewood–Paley g-function or $$g_{\lambda }^*$$ -function. The obtained Littlewood–Paley g-function characterization of $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ coincidentally confirms a conjecture proposed by Hart et al. (Trans Am Math Soc, https://doi.org/10.1090/tran/7312 , 2017). Applying the aforementioned Calderon–Zygmund decomposition as well as the atomic characterization of $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ , the authors establish a finite atomic characterization of $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ , which further induces a criterion on the boundedness of sublinear operators from $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of anisotropic Calderon–Zygmund operators from $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ to itself [or to the mixed-norm Lebesgue space $$L^{\vec {p}}(\mathbb {R}^n)$$ ]. The obtained atomic characterizations of $$H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$$ and boundedness of anisotropic Calderon–Zygmund operators on these Hardy-type spaces positively answer two questions mentioned by Cleanthous et al. (J Geom Anal 27:2758–2787, 2017). All these results are new even for the isotropic mixed-norm Hardy spaces on $$\mathbb {R}^n$$ .

Nondoubling Calderón–Zygmund theory: a dyadic approach

Given a measure $$\mu $$ of polynomial growth, we refine a deep result by David and Mattila to construct an atomic martingale filtration of $$\mathrm {supp}(\mu )$$ which provides the right framework for a dyadic form of nondoubling harmonic analysis. Despite this filtration being highly irregular, its atoms are comparable to balls in the given metric—which in turn are all doubling—and satisfy a weaker but crucial form of regularity. Our dyadic formulation is effective to address three basic questions: Our martingale RBMO space preserves the crucial properties of Tolsa’s original definition and reveals its interpolation behavior with the $$L_p$$ scale in the category of Banach spaces, unknown so far. On the other hand, due to some known obstructions for Haar shifts and related concepts over nondoubling measures, our pointwise domination theorem via sparsity naturally deviates from its doubling analogue. In a different direction, matrix-valued harmonic analysis over noncommutative $$L_p$$ spaces has recently produced profound applications. Our analogue for nondoubling measures was expected for quite some time. Finally, we also find a dyadic form of the Calderon–Zygmund decomposition which unifies those by Tolsa and Lopez-Sanchez/Martell/Parcet.

Spectrum and Singular Integrals on a New Weighted Function Space

We introduce a new function space that is much finer than the classical Lebesgue spaces, and investigate the continuity and the discontinuity of the Calderon–Zygmund operators on the new weighted spaces. In order to identify the continuity of singular integrals, we prove an interpolation theorem on the new spaces and propose a concept of the spectrum of base functions.

T-minima and application to the convergence of some integral functionals with infinite energy minima

We consider integral functionals J as in (1). We study the Calderon–Zygmund theory for infinite energy minima u (T-minima) of J and the stability of the T-minima with respect to the convergence in (22), with V=W01,p(Ω).

Boundedness of the Bergman Projection and Some Properties of Bergman Type Spaces

We give a simple proof of the boundedness of Bergman projection in various Banach spaces of functions on the unit disc in the complex plain. The approach of the paper is based on the idea of Zaharyuta and Yudovich (Uspekhi Mat Nauk 19(2):139–142, 1964) where the boundedness of the Bergman projection in Lebesgue spaces was proved using Calderon–Zygmund operators. We exploit this approach and treat the cases of variable exponent Lebesgue space, Orlicz space and variable exponent generalized Morrey spaces. In the case of variable exponent Lebesgue space the boundedness result is known, so in that case we provide a simpler proof, whereas the other cases are new. The major idea of this paper is to show that the approach can be applied to a wide range of function spaces. We also study the rate of growth of functions near the boundary in spaces under consideration and their approximation by mollified dilations.

Nonlinear Calderón–Zygmund inequalities for maps

Being motivated by the problem of deducing $$\mathsf {L}^{p}$$ -bounds on the second fundamental form of an isometric immersion from $$\mathsf {L}^{p}$$ -bounds on its mean curvature vector field, we prove a nonlinear Calderon–Zygmund inequality for maps between complete (possibly noncompact) Riemannian manifolds.

Calderón–Zygmund theory for degenerate parabolic systems involving a generalized p-Laplacian type

We study a local Calderon–Zygmund theory for degenerate parabolic systems involving a generalized p-Laplacian to obtain the interior integrability of the gradient.

Extrapolation results in grand Lebesgue spaces defined on product sets

Extrapolation results in weighted grand Lebesgue spaces defined with respect to product measure \(\mu \times \nu \) on \(X\times Y\), where \((X, d, \mu )\) and \((Y, \rho , \nu )\) are spaces of homogeneous type, are obtained. As applications of the derived results we prove new one-weight estimates for multiple integral operators such as strong maximal, Calderon–Zygmund and fractional integral operators with product kernels in these spaces.

A1-Regularity and Boundedness of Riesz Transforms in Banach Lattices of Measurable Functions

Let X be a Banach lattice of measurable functions on ℝn × Ω having the Fatou property. We show that the boundedness of all Riesz transforms Rj in X is equivalent to the boundedness of the Hardy–Littlewood maximal operator M in both X and X′, and thus to the boundedness of all Calderon–Zygmund operators in X. We also prove a result for the case of operators between lattices: If Y ⊃ X is a Banach lattice with the Fatou property such that the maximal operator is bounded in Y ′, then the boundedness of all Riesz transforms from X to Y is equivalent to the boundedness of the maximal operator from X to Y , and thus to the boundedness of all Calderon–Zygmund operators from X to Y .

Nonlinear harmonic analysis of integral operators in weighted grand Lebesgue spaces and applications

In this article, we give a boundedness criterion for Cauchy singular integral operators in generalized weighted grand Lebesgue spaces. We establish a necessary and sufficient condition for the couple of weights and curves ensuring boundedness of integral operators generated by the Cauchy singular integral defined on a rectifiable curve. We characterize both weak and strong type weighted inequalities. Similar problems for Calderon–Zygmund singular integrals defined on measured quasimetric space and for maximal functions defined on curves are treated. Finally, as an application, we establish existence and uniqueness, and we exhibit the explicit solution to a boundary value problem for analytic functions in the class of Cauchy-type integrals with densities in weighted grand Lebesgue spaces.

Calderón–Zygmund Estimate for Homogenization of Steady State Stokes Systems in Nonsmooth Domains

This paper investigates the higher integrability in homogenization theory for a generalized steady state Stokes system in divergence form with discontinuous coefficients in a bounded nonsmooth domain. We obtain a global and uniform Calderon–Zygmund estimate by essentially proving that both the gradient of the weak solution and its associated pressure are as integrable as the nonhomogeneous term under BMO smallness of the rapidly oscillating periodic coefficients and sufficient flatness of the boundary in the Reifenberg sense. The result improves previous works either concerned with nonhomogenization problem or focused on weakening the regularity requirements on both the coefficients and the boundary to the homogenization of such Stokes systems.

Commutators of Calderón–Zygmund and generalized fractional integral operators on generalized Morrey spaces

We consider the commutators [b, T] and $$[b,I_{\rho }]$$ , where T is a Calderon–Zygmund operator, $$I_{\rho }$$ is a generalized fractional integral operator and b is a function in generalized Campanato spaces with variable growth condition. We give necessary and sufficient conditions for the boundedness of the commutator on generalized Morrey spaces with variable growth condition.

Weighted Lorentz estimates for parabolic equations with non-standard growth on rough domains

The main aim of this paper is to prove the Calderon–Zygmund estimates for a general nonlinear parabolic equation of p(x, t)-Laplacian type in the weighted Lorentz spaces. Note that we only require some mild conditions on the nonlinearity of coefficients and the underlying domain. The result for these nonlinear parabolic equations is new even in the particular case when the growth p(x, t) is a constant.

Interior Calderón–Zygmund estimates for solutions to general parabolic equations of p-Laplacian type

We study general parabolic equations of the form $$u_t = \text{ div }\,\mathbf {A}(x,t, u,D u) +\text{ div }\,(|\mathbf {F}|^{p-2} \mathbf {F})+ f$$ whose principal part depends on the solution itself. The vector field $$\mathbf {A}$$ is assumed to have small mean oscillation in x, measurable in t, Lipschitz continuous in u, and its growth in Du is like the p-Laplace operator. We establish interior Calderon–Zygmund estimates for locally bounded weak solutions to the equations when $$p>2n/(n+2)$$ . This is achieved by employing a perturbation method together with developing a two-parameter technique and a new compactness argument. We also make crucial use of the intrinsic geometry method by DiBenedetto (Degenerate parabolic equations, Springer, New York, 1993) and the maximal function free approach by Acerbi and Mingione (Duke Math J 136(2):285–320, 2007).

Boundedness of sublinear operators in weighted grand Morrey spaces

The boundedness of sublinear integral operators in grand Morrey spaces defined by means of measures generated by the Muckenhoupt weights is established. The operators under consideration involve operators of Harmonic Analysis such as Hardy–Littlewood and fractional maximal operators, Calderon–Zygmund operators, potential operators etc.

Multilinear singular integral operators with generalized kernels and their multilinear commutators

In this paper, the authors study a class of multilinear singular integral operators with generalized kernels and their multilinear commutators with BMO functions. By establishing the sharp maximal estimates, the boundedness on product of weighted Lebesgue spaces and product of variable exponent Lebesgue spaces is obtained, respectively. Moreover, the endpoint estimate of this class of mutilinear singular integral operators is also established. These results can improve the corresponding known results of classical multilinear Calderon–Zygmund operators and multilinear Calderon–Zygmund operators with Dini type kernels.

Multilinear Commutators of Singular Integral Operators in Variable Exponent Herz-Type Spaces

In this paper, we study the boundedness of multilinear commutators of Hardy–Littlewood maximal operators in variable exponent Herz and Herz–Morrey spaces, which in turn are used to obtain the boundedness for a large class of the multilinear commutators related to sublinear operators. Moreover, based on the atomic decomposition and on generalization of the BMO norm, we study the boundedness of multilinear commutators of singular integral operators with Calderon–Zygmund kernels in variable exponent Herz-type Hardy spaces.