Zygmund Type Research Articles

Marcinkiewicz–Zygmund type results in multivariate domains

We investigate Marcinkiewicz–Zygmund type inequalities for multivariate polynomials on various compact domains in $${\mathbb{R}^d}$$ . These inequalities provide a basic tool for the discretization of the L p norm and are widely used in the study of the convergence properties of Fourier series, interpolation processes and orthogonal expansions. Recently Marcinkiewicz–Zygmund type inequalities were verified for univariate polynomials for the general class of doubling weights, and for multivariate polynomials on the ball and sphere with doubling weights. The main goal of the present paper is to extend these considerations to more general multidimensional domains, which in particular include polytopes, cones, spherical sectors, toruses, etc. Our approach will rely on application of various polynomial inequalities, such as Bernstein–Markov, Schur and Videnskii type estimates, and also using symmetry and rotation in order to generate results on new domains.

Upper and lower bounds of large deviations for some dependent sequences

Let $${\{X_{n}, n\geq1\}}$$ be a sequence of random variables with $${S_n=\sum_{i=1}^nX_i}$$ and $${M_n=\max \{X_1,X_2,\ldots, X_n\}}$$ . Under some suitable conditions, we establish the upper bound of large deviations for $${S_n}$$ and $${M_n}$$ based on some dependent sequences including acceptable random variables, widely acceptable random variables and a class of random variables that satisfies the Marcinkiewicz–Zygmund type inequality and Rosenthal type inequality. In addition, the lower bound of large deviations for some dependent sequences is also obtained. The results obtained in the paper generalize and improve some corresponding ones for independent random variables and negatively associated random variables.

Baum–Katz type theorems with exact threshold

Let be either a sequence of arbitrary random variables, or a martingale difference sequence, or a centred sequence with a suitable level of negative dependence. We prove Baum–Katz type theorems by only assuming that the variables satisfy a uniform moment bound condition. We also prove that this condition is best possible even for sequences of centred, independent random variables. This leads to Marcinkiewicz–Zygmund type strong laws of large numbers with estimate for the rate of convergence.

Weighted version of strong law of large numbers for a class of random variables and its applications

In this paper, the single index weighted version of Marcinkiewicz–Zygmund type strong law of large numbers and the double index weighted version of Marcinkiewicz–Zygmund type strong law of large numbers are investigated successively for a class of random variables, which extends the classical results for independent and identically distributed random variables. As applications of the results, we further study the strong consistency for the weighted estimator in the nonparametric regression model and the least square estimators in the simple linear errors-in-variables model. Moreover, we also present some numerical study to verify the validity of our results.

Global weighted Lorentz estimates to nonlinear parabolic equations over nonsmooth domains

An elaborate method is developed to establish a global Calderón–Zygmund type theory in weighted Lorentz spaces Lω(γ,q)(ΩT) for zero Cauchy–Dirichlet problem of nonlinear parabolic equations of p-Laplacian type with p>2dd+2. Here, we assume that the nonlinearity a=a(t,x,ξ) is merely measurable in the time variable t, and has a mall BMO semi-norm in the spatial variables x uniformly in ξ variables, while the boundary ∂Ω of bounded domain Ω is flat in the sense of Reifenberg. Our result not only develops nonlinear Calderón–Zygmund theory under weak regularity assumptions on the datum, but also extends the related framework from Lebesgue spaces to refined weighted Lorentz spaces.

Weak and strong laws of large numbers for arrays of rowwise END random variables and their applications

In the paper, the Marcinkiewicz–Zygmund type moment inequality for extended negatively dependent (END, in short) random variables is established. Under some suitable conditions of uniform integrability, the $$L_r$$ convergence, weak law of large numbers and strong law of large numbers for usual normed sums and weighted sums of arrays of rowwise END random variables are investigated by using the Marcinkiewicz–Zygmund type moment inequality. In addition, some applications of the $$L_r$$ convergence, weak and strong laws of large numbers to nonparametric regression models based on END errors are provided. The results obtained in the paper generalize or improve some corresponding ones for negatively associated random variables and negatively orthant dependent random variables.

Calderón–Zygmund estimates for ω-minimizers

We consider ω-minimizers of functionals with p-growth, and prove higher integrability of the gradient of ω-minimizers by obtaining Calderón–Zygmund type estimates under a sharp condition on ω(⋅). In addition, we deal with similar results for spherical quasi-minimizers.

Global gradient estimates for the borderline case of double phase problems with BMO coefficients in nonsmooth domains

We consider a double phase problem with BMO coefficient in divergence form on a bounded nonsmooth domain. The problem under consideration is characterized by the fact that both ellipticity and growth switch between a type of polynomial and a type of logarithm according to the position, which describes a feature of strongly anisotropic materials. We obtain the global Calderón–Zygmund type estimates for the distributional solution in the case that the associated nonlinearity has a small BMO and the boundary of the domain is sufficiently flat in the Reifenberg sense.

Global gradient estimates for non-uniformly elliptic equations

We consider a nonlinear and non-uniformly elliptic problem in divergence form on a bounded domain. The problem under consideration is characterized by the fact that its ellipticity rate and growth radically change with the position, which provides a model for describing a feature of strongly anisotropic materials. We establish the global Calderon–Zygmund type estimates for the distributional solution in the case that the boundary of the domain is of class \(C^{1,\beta }\) for some \(\beta >0\).

Weighted Morrey spaces on non-homogeneous metric measure spaces

Let (X,d,μ) be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. In this setting, the authors introduce the weighted Morrey space and the weighted weak Morrey space, and show several properties of these spaces. As applications, some weighted weak type estimates for the Hardy–Littlewood maximal operator and the Calderón–Zygmund type operator are established.

Essential norms of weighted differentiation composition operators between Zygmund type spaces and Bloch type spaces

We study boundedness of weighted differentiation composition operators Dk?,u between Zygmund type spaces Z? and Bloch type spaces ?. We also give essential norm estimates of such operators in different cases of k ? N and 0 < ?,? < ?. Applying our essential norm estimates, we get necessary and sufficient conditions for the compactness of these operators.

Calderón–Zygmund decompositions in tent spaces and weak-type endpoint bounds for two quadratic functionals of Stein and Fefferman–Stein

We prove some Calderón–Zygmund type decompositions for functions in the tent spaces introduced by Coifman, Meyer and Stein. These decompositions will be useful in the operator theory on tent spaces developed in the author’s 2015 thesis. As an application

Essential Norms of Volterra Type Operators between Zygmund Type Spaces

We investigate the boundedness of some Volterra type operators between Zygmund type spaces. Then, we give the essential norms of such operators in terms of g,φ, their derivatives, and the nth power φn of φ.

Essential norm of generalized composition operators on Zygmund type spaces and Bloch type spaces

Essential norm of generalized composition operators on Zygmund type spaces and Bloch type spaces

Weighted composition operators from Zygmund type spaces into Bloch type spaces

Weighted composition operators from Zygmund type spaces into Bloch type spaces

Strong and weak consistency of LS estimators in the EV regression model with negatively superadditive-dependent errors

In this paper, the strong laws of large numbers for partial sums and weighted sums of negatively superadditive-dependent (NSD, in short) random variables are presented, especially the Marcinkiewicz–Zygmund type strong law of large numbers. Using these strong laws of large numbers, we further investigate the strong consistency and weak consistency of the LS estimators in the EV regression model with NSD errors, which generalize and improve the corresponding ones for negatively associated random variables. Finally, a simulation is carried out to study the numerical performance of the strong consistency result that we established.

Regularity estimates for quasilinear elliptic equations with variable growth involving measure data

We investigate a quasilinear elliptic equation with variable growth in a bounded nonsmooth domain involving a signed Radon measure. We obtain an optimal global Calderón–Zygmund type estimate for such a measure data problem, by proving that the gradient of a very weak solution to the problem is as globally integrable as the first order maximal function of the associated measure, up to a correct power, under minimal regularity requirements on the nonlinearity, the variable exponent and the boundary of the domain.

Some Zygmund type inequalities for the sth derivative of polynomials

For a polynomial P(z) of degree n having no zeros in |z| < 1, it was recently proved in [9] that $$\left| {{z^s}{P^{\left( s \right)}}\left( z \right) + \beta \frac{{n\left( {n - 1} \right)...\left( {n - s + 1} \right)}}{{{2^s}}}P\left( z \right)} \right| \leqslant \frac{{n\left( {n - 1} \right)...\left( {n - s + 1} \right)}}{2}\left( {\left| {1 + \frac{\beta }{{{2^s}}}} \right| + \left| {\frac{\beta }{{{2^s}}}} \right|} \right)\mathop {\max }\limits_{\left| z \right| = 1} \left| {P\left( z \right)} \right|$$ for every β ∈ C with |β| ≤ 1, 1 ≤ s ≤ n and |z| = 1. In this paper, we obtain the Lp mean extension of the above and other related results for the sth derivative of polynomials.

Polynomials with polar derivatives

In this paper, we present an integral inequality for the polar derivative of polynomials. Our theorem includes as special cases several interesting generalizations of some Zygmund type inequalities for polynomials.

Characterizations for the Riesz potential and its commutators on generalized Orlicz-Morrey spaces

In the present paper, we shall give a characterization for the Spanne and Adams type boundedness of the Riesz potential and its commutators on the generalized Orlicz-Morrey spaces, respectively. Also we give criteria for the weak versions of Spanne and Adams type boundedness of the Riesz potential on the generalized Orlicz-Morrey spaces. In all the cases the conditions for the boundedness are given in terms of Zygmund type integral inequalities involving the Young function $\Phi(u)$ and the function $\varphi(x,r)$ defining the space.