Zygmund Type Research Articles

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

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

Global Fractional Calderon–Zygmund Type Regularity

Global Fractional Calderon–Zygmund Type Regularity

Schauder and Calderón–Zygmund type estimates for fully nonlinear parabolic equations under “small ellipticity aperture” and applications

In this manuscript, we derive some Schauder estimates for viscosity solutions to non-convex fully nonlinear second-order parabolic equations of the form: ∂tu−F(x,t,D2u)=f(x,t)inQ1=B1×(−1,0],provided that the source f and the coefficients of F are Hólder continuous functions, and F enjoys a small ellipticity aperture. Furthermore, for problems with merely bounded data, we prove that such solutions are C1,Log-Lip-regular. We also obtain Calderón-Zygmund estimates for such a class of non-convex operators. Finally, we connect our results and recent estimates for fully nonlinear models in certain solution classes.

Commutators on Spaces of Homogeneous Type in Generalized Block Spaces

In this paper, we are interested in studying a generalized block space (denoted as Bφp,r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{B}^{p,r}_\\varphi $$\\end{document}) on a space of homogeneous type. We show that this space is the predual of certain generalized Morrey–Lorentz space. By duality, we obtain the Bφp,r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{B}^{p,r}_\\varphi $$\\end{document}-bound of operators of Calderón–Zygmund type. In addition, we prove a weak Hardy factorization in terms of commutators of integral operator of Calderón–Zygmund type in block spaces. Thanks to the Hardy factorization result, we obtain a characterization of functions in BMO\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{BMO}$$\\end{document} via the boundedness of commutators of homogeneous linear Calderón–Zygmund operators in the generalized block space (resp. the generalized Morrey–Lorentz space). Finally, we study a compactness characterization of commutators of Calderón–Zygmund type in generalized Morrey–Lorentz spaces.

Calderón–Zygmund Type Results for a Class of Quasilinear Elliptic Equations Involving the p(x)-Laplacian

Calderón–Zygmund Type Results for a Class of Quasilinear Elliptic Equations Involving the p(x)-Laplacian

On the von Bahr–Esseen inequality for pairwise independent random vectors in Hilbert spaces with applications to mean convergence

Abstract In this correspondence, we prove the von Bahr–Esseen moment inequality for pairwise independent random vectors in Hilbert spaces. Our constant in the von Bahr–Esseen moment inequality is better than that obtained for the real-valued random variables by Chen et al. [The von Bahr–Esseen moment inequality for pairwise independent random variables and applications, J. Math. Anal. Appl. 419 (2014), 1290–1302], and Chen and Sung [Generalized Marcinkiewicz–Zygmund type inequalities for random variables and applications, J. Math. Inequal. 10(3) (2016), 837–848]. The result is then applied to obtain mean convergence theorems for triangular arrays of rowwise and pairwise independent random vectors in Hilbert spaces. Some results in the literature are extended.

Interior a priori estimates for supersolutions of fully nonlinear subelliptic equations under geometric conditions

AbstractIn this note, we prove interior a priori first‐ and second‐order estimates for solutions of fully nonlinear degenerate elliptic inequalities structured over the vector fields of Carnot groups, under the main assumption that is semiconvex along the fields. These estimates for supersolutions are new even for linear subelliptic inequalities in nondivergence form, whereas in the nonlinear setting they do not require neither convexity nor concavity on the second derivatives. We complement the analysis exhibiting an explicit example showing that horizontal regularity of Calderón–Zygmund type for fully nonlinear subelliptic equations posed on the Heisenberg group cannot be in general expected in the range , being the homogeneous dimension of the group.

O gipersingulyarnykh operatorakh, svyazannykh s peridinamikoy

For a hypersingular integral operator of the Calderón–Zygmund type associated with problems of peridynamics, we find a Hilbert space that is taken by this operator to the space of square integrable periodic functions.

Calderón–Zygmund type estimates for singular quasilinear elliptic obstacle problems with measure data

Calderón–Zygmund type estimates for singular quasilinear elliptic obstacle problems with measure data

Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of random variables with infinite moment and its applications

In this work, the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of widely orthant dependent random variables is established under mild conditions including infinite variance. As applications, some new results such as the Cesàro strong law of large numbers, the strong consistency of the least squares estimator in multiple linear regression models, and the strong consistency of the wavelet estimator in nonparametric regression models are presented. Simulation studies are also provided to support the theoretical results.

Aspects of the Kahane–Salem–Zygmund inequalities in Banach spaces

The main aim of this work is to discuss several different approaches to the celebrated Kahane–Salem–Zygmund inequalities. In particular, we prove estimates for exponential Orlicz norms of averages sup _{1le jle N}Big |sum _{i=1}^K a_i(j) gamma _iBig |,, where (a_i(j)) in ell _infty ^N, , 1 le i le K and the (gamma _i) form a sequence of real or complex subgaussian random variables. Lifting these inequalities to finite dimensional Banach spaces, we get some new Kahane–Salem–Zygmund type inequalities—in particular, for spaces of subgaussian random polynomials and multilinear forms on finite dimensional Banach spaces, and also for subgaussian random Dirichlet polynomials.

Oscillation Estimates for Truncated Singular Radon Operators

In this paper we prove uniform oscillation estimates on L^p, with pin (1,infty ), for truncated singular integrals of the Radon type associated with the Calderón–Zygmund kernel, both in continuous and discrete settings. In the discrete case we use the Ionescu–Wainger multiplier theorem and the Rademacher–Menshov inequality to handle the number-theoretic nature of the discrete singular integral. The result we obtained in the continuous setting can be seen as a generalisation of the results of Campbell, Jones, Reinhold and Wierdl for the continuous singular integrals of the Calderón–Zygmund type.

Complete moment convergence for maximum of randomly weighted sums of martingale difference sequences

In this paper, complete moment convergence for maximum of randomly weighted sums and complete convergence for randomly indexed sums of martingale difference sequences (MDS) are investigated under some proper and sufficient conditions. A Marcinkiewicz–Zygmund type strong law of large numbers (MZSLLN) for MDS is obtained. In addition, relationships among weights, weight functions and boundary functions are revealed in a sense. The results obtained in the paper generalize some corresponding ones for independent and some dependent random variables. As an application, strong consistency for estimators in a nonparametric regression model is established.

Noether’s Theorem of Some Classes of Two-Dimensonal Singular Integral Equations of the Mikhlin–Calderón–Zygmund Type over a Bounded Domain

Noether’s Theorem of Some Classes of Two-Dimensonal Singular Integral Equations of the Mikhlin–Calderón–Zygmund Type over a Bounded Domain

Calderón–Zygmund theory for asymptotically regular nonlinear elliptic problems with double obstacles

ABSTRACT We study nonlinear elliptic double obstacles problems with an asymptotically regular nonlinearity in a non-smooth domain. Here, we obtain a global Calderón–Zygmund type estimate in the setting of Lorentz spaces by making use of the perturbation method. More precisely, we approximate the solutions of asymptotically regular double obstacles problems to the solutions of regular single obstacle problems when the gradients of solutions are close to infinity.

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.

Weighted estimates for operators of fractional integration of variable order in generalized variable Hölder spaces

The paper is devoted to weighted estimates for operators of fractional integration of variable order of Bergman type in generalized variable Hölder spaces of holomorphic functions on the unit disc \( {\mathbb {D}} \). Due to the choice of the weight, we can include in consideration the case when the real part of the complex power of the operator degenerates. We prove the estimates of Zygmund type for the modulus of continuity, and then we obtain the corresponding weighted boundedness result.

On Some Laws of Large Numbers for Uncertain Random Variables

Baoding Liu created uncertainty theory to describe the information represented by human language. In turn, Yuhan Liu founded chance theory for modelling phenomena where both uncertainty and randomness are present. The first theory involves an uncertain measure and variable, whereas the second one introduces the notions of a chance measure and an uncertain random variable. Laws of large numbers (LLNs) are important theorems within both theories. In this paper, we prove a law of large numbers (LLN) for uncertain random variables being continuous functions of pairwise independent, identically distributed random variables and regular, independent, identically distributed uncertain variables, which is a generalisation of a previously proved version of LLN, where the independence of random variables was assumed. Moreover, we prove the Marcinkiewicz–Zygmund type LLN in the case of uncertain random variables. The proved version of the Marcinkiewicz–Zygmund type theorem reflects the difference between probability and chance theory. Furthermore, we obtain the Chow type LLN for delayed sums of uncertain random variables and formulate counterparts of the last two theorems for uncertain variables. Finally, we provide illustrative examples of applications of the proved theorems. All the proved theorems can be applied for uncertain random variables being functions of symmetrically or asymmetrically distributed random variables, and symmetrical or asymmetrical uncertain variables. Furthermore, in some special cases, under the assumption of symmetry of the random and uncertain variables, the limits in the first and the third theorem have forms of symmetrical uncertain variables.

Regularity results for solutions to a class of obstacle problems

In this paper we prove some regularity properties of solutions to variational inequalities of the form ∫Ω〈A(x,u,Du),D(φ−u)〉dx≥∫ΩB(x,u,Du)(φ−u)dx,∀φ∈Kψ(Ω).Here Ω is a bounded open set of Rn, n≥2, the function ψ:Ω→[−∞,+∞), called obstacle, belongs to the Sobolev class W1,p(Ω) and Kψ(Ω)={w∈W1,p(Ω):w≥ψq.o. inΩ} is the class of the admissible functions. First we establish a local Calderòn–Zygmund type estimate proving that the gradient of the solutions is as integrable as the gradient of the obstacle in the scale of Lebesgue spaces Lpq, for every q∈(1,∞), provided the partial map (x,u)↦A(x,u,ξ) is Hölder continuous and B(x,u,ξ) satisfies a suitable growth condition. Next, this estimate allows us to prove that a higher differentiability in the scale of Besov spaces of the gradient of the obstacle transfers to the gradient of the solutions.

Caratheodory theorem about prime ends on Riemann surfaces

The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio--Ryazanov--Srebro--Yakubov at 2004 and then included in the known monograph of these authors in the modern mapping theory at 2009. As it was shown in the recent papers of Kovtonyuk-Petkov-Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasi--isometries. We obtain here a series of criteria in terms of dilatations for the homeomorphic extension of the mappings with finite length distortion between domains on Riemann surfaces to the completions of the domains by prime ends of Caratheodory. Here we start from the general criterion in Lemma 1 in terms of singular functional parameters and then derive on this basis many other criteria. In particular, Lemma 1 implies Theorem 1 with a criterion of the Lehto type and Corollary 1 shows that the conclusion holds, if the dilatation grows not quickly than logarithm of the hyperbolic distance at every boundary point. The next consequence in Theorem 2 gives an integral criterion of the Orlicz type and Corollary 2 says on simple integral conditions of the exponential type. Further, Theorem 3 and Remark 2 contain criteria in terms of singular integrals of the Calderon--Zygmund type. The other criterion in Theorem 4 is the existence of a dominant for the dilatation in the class FMO (functions with finite mean oscillation), i.e., having a finite mean deviation from its mean value over infinitesimal discs centered at boundary points. In other words, the latter means that such a dominant has a finite dispersion over the given infinitesimal discs. In particular, the latter leads to Corollary 3 on a dominant in the well--known class BMO (bounded mean oscillation) by John--Nirenberg and to a simple criterion in Corollary 4 on finiteness of the average of the dilatation over infinitesimal disks centered at boundary points.