Sublinear Operators Research Articles

Atomic decomposition of martingale weighted Lorentz spaces with two-parameter and applications

We introduce martingale weighted two-parameter Lorentz spaces and establish atomic decomposition theorems. As an application of atomic decomposition we obtain a sufficient condition for sublinear operators defined on martingale weighted Lorentz spaces to be bounded. Moreover, some interpolation properties with a function parameter of those spaces are obtained.

Bounded operators on vector-valued weak Orlicz martingale spaces

This paper is devoted to studying the boundedness of sublinear operators on vector-valued weak Orlicz martingale spaces. These results closely depend on the geometrical properties of the Banach space in which the martingales take values. Also the results obtained here extend the corresponding known results from scalar-valued setting to vector-valued setting.

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.

Endpoint estimates for commutators of sublinear operators in the Morrey-type spaces

Let $[b,\mathcal T_\alpha]~(0\leq\alpha<n)$ be the commutators generated by $BMO(\mathbb R^n)$ functions and a class of sublinear operators satisfying certain size conditions. The aim of this paper is to study the endpoint estimates of these commutators in the weighted Morrey spaces and in the generalized Morrey spaces, under the assumptions that $[b,\mathcal T_\alpha]~(0\leq\alpha<n)$ satisfy (weighted or unweighted) endpoint inequalities on $\mathbb R^n$ and on bounded domains. Furthermore, as applications of our main results, we will obtain, in the endpoint case, the boundedness properties of many important operators in classical harmonic analysis on the weighted Morrey and the generalized Morrey spaces.

An approximate Hahn-Banach-Lagrange theorem

In this paper, we proved a new extended version of the Hahn-Banach-Lagrange theorem that is valid in the absence of a qualification condition and is called an approximate Hahn-Banach-Lagrange theorem. This result, in special cases, gives rise to approximate sandwich and approximate Hahn-Banach theorems. These results extend the Hahn-Banach-Lagrange theorem, the sandwich theorem in [18], and the celebrated Hahn-Banach theorem. The mentioned results extend the original ones into two features: Firstly, they extend the original versions to the case with extended sublinear functions (i.e., the sublinear functions that possibly possess extended real values). Secondly, they are topological versions which held without any qualification condition. Next, we showed that our approximate Hahn-Banach-Lagrange theorem was actually equivalent to the asymptotic Farkas-type results that were established recently [10]. This result, together with the results [5, 16], give us a general picture on the equivalence of the Farkas lemma and the Hahn-Banach theorem, from the original version to their corresponding extensions and in either non-asymptotic or asymptotic forms.

Atomic decompositions of vector-valued weak Musielak–Orlicz martingale spaces and applications

In this paper, some vector-valued weak martingale Hardy spaces of Musielak–Orlicz type are introduced, and several atomic decompositions are established for them. As applications of the atomic decompositions, a sufficient condition for sublinear operators defined on vector-valued weak Musielak–Orlicz martingale spaces to be bounded is given. Using the sufficient condition, some martingale inequalities are deduced. These results closely depend on the geometrical properties of the Banach space in which the martingales take values. The results obtained here generalize the corresponding known results of scalar-valued and vector-valued weak martingale Hardy spaces.

$B_w^u$-function Spaces and Their Interpolation

We introduce $B_w^u$-function spaces which unify Lebesgue, Morrey-Campanato, Lipschitz, $B^p$, $\mathrm{CMO}$, local Morrey-type spaces, etc., and investigate the interpolation property of $B_w^u$-function spaces. We also apply it to the boundedness of linear and sublinear operators, for example, the Hardy-Littlewood maximal and fractional maximal operators, singular and fractional integral operators with rough kernel, the Littlewood-Paley operator, Marcinkiewicz operator, and so on.

Balancing between over-weighting and under-weighting in supervised term weighting

Supervised term weighting could improve the performance of text categorization. A way proven to be effective is to assign larger weight to terms with more imbalanced distributions across categories. This paper shows that supervised term weighting should not just assign large weights to imbalanced terms, but should also control the trade-off between over-weighting and under-weighting. Over-weighting, a new concept proposed in this paper, is caused by the improper handling of singular terms and too large ratios between term weights. To prevent over-weighting, we present three regularization techniques: add-one smoothing, sublinear scaling and bias term. Add-one smoothing is used to handle singular terms. Sublinear scaling and bias term shrink the ratios between term weights. However, if sublinear functions scale down term weights too much, or the bias term is too large, under-weighting would occur and harm the performance. It is therefore critical to balance between over-weighting and under-weighting. Inspired by this insight, we also propose a new supervised term weighting scheme, regularized entropy (re). Our re employs entropy to measure term distribution, and introduces the bias term to control over-weighting and under-weighting. Empirical evaluations on topical and sentiment classification datasets indicate that sublinear scaling and bias term greatly influence the performance of supervised term weighting, and our re enjoys the best results in comparison with existing schemes.

On the Cohen p-nuclear positive sublinear operators

In the present paper, we will introduce the concept of Cohen p-nuclear positive sublinear operators. We give an analogue to “Pietsch’s domination theorem” and we study some properties concerning this notion.

Sublinear operators in generalized weighted Morrey spaces

Generalized weighted Morrey spaces defined on spaces of homogeneous type are introduced by using weight functions in the Muckenhoupt class. Theorems on the boundedness of a large class of sublinear operators on these spaces are presented. The classes of sublinear operators under consideration contain a whole series of important operators of harmonic analysis, such as, e.g., maximal functions, singular and fractional integrals, Bochner–Riesz means, and so on.

Support functions and subdifferentials

In this paper we study Minkowski duality, i.e. the correspondence between sublinear functions and closed convex sets in the context of dual pairs of vector spaces.

Banach envelopes of some quasi-Banach function spaces

We show that under some assumptions on the Musielak−Orlicz function generating a quasi-Banach Musielak−Orlicz function space, the Banach envelope of the weighted Cesaro−Musielak−Orlicz space generated by a certain positive sublinear operator is a weighted \(L_1\)-space.

Boundedness of certain commutators over non-homogeneous metric measure spaces

Let \((\mathcal {X},d,\mu )\) be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let T be a Calderon-Zygmund operator with kernel satisfying only the size condition and some Hormander-type condition, and \(b\in \widetilde{\mathrm{RBMO}}(\mu )\) (the regularized BMO space with the discrete coefficient). In this paper, the authors establish the boundedness of the commutator \(T_b:=bT-Tb\) generated by T and b from the atomic Hardy space \(\widetilde{H}^1(\mu )\) with the discrete coefficient into the weak Lebesgue space \(L^{1,\,\infty }(\mu )\). From this and an interpolation theorem for sublinear operators which is also proved in this paper, the authors further show that the commutator \(T_b\) is bounded on \(L^p(\mu )\) for all \(p\in (1,\infty )\). Moreover, the boundedness of the commutator generated by the generalized fractional integral \(T_\alpha \,(\alpha \in (0,1))\) and the \(\widetilde{\mathrm{RBMO}}(\mu )\) function from \(\widetilde{H}^1(\mu )\) into \(L^{1/{(1-\alpha )},\,\infty }(\mu )\) is also presented.

First and second order optimality conditions in vector optimization problems with nontransitive preference relation

We present first and second order conditions, both necessary and sufficient, for ≺-minimizers of vector-valued mappings over feasible sets with respect to a nontransitive preference relation ≺. Using an analytical representation of a preference relation ≺ in terms of a suitable family of sublinear functions, we reduce the vector optimization problem under study to a scalar inequality, from which, using the tools of variational analysis, we derive minimality conditions for the initial vector optimization problem.

Biased Predecessor Search

We consider the problem of performing predecessor searches in a bounded universe while achieving query times that depend on the distribution of queries. We obtain several data structures with various properties: in particular, we give data structures that achieve expected query times logarithmic in the entropy of the distribution of queries but with space bounded in terms of universe size, as well as data structures that use only linear space but with query times that are higher (but still sublinear) functions of the entropy. For these structures, the distribution is assumed to be known. We also consider individual query times on universe elements with general weights, as well as the case when the distribution is not known in advance.

Positively Homogeneous Functions Revisited

The paper deals with positively homogeneous functions defined on a finite-dimensional space. Our attention is mainly focused on those subspaces of positively homogeneous functions that are important in nonsmooth analysis and optimization: the subspace of continuous positively homogeneous functions, of Lipschitz continuous positively homogeneous functions, of difference sublinear functions, and of piecewise linear functions. We reproduce some known results and present a number of new ones, in particular, those that concern Lipschitz continuous positively homogeneous functions.

Reduced Pairs of Compact Convex Sets and Ordered Median Functions

We prove that in finite dimensional spaces every ordered median function is the Minkowski dual of a reduced pair of polytopes. This implies a very general theorem on the representation of an ordered median function as a uniquely determined difference of two sublinear functions up to adding and subtracting one and the same arbitrary sublinear function.

A generalization of a theorem of Arrow, Barankin and Blackwell to a nonconvex case

The paper presents a generalization of a known density theorem of Arrow, Barankin, and Blackwell for properly efficient points defined as support points of sets with respect to monotonically increasing sublinear functions. This result is shown to hold for nonconvex sets of a partially ordered reflexive Banach space.

Weighted local Hardy spaces associated to Schrödinger operators

In this paper, we characterize the weighted local Hardy spaces $h^{p}_{\rho}(\omega)$ related to the critical radius function $\rho$ and weights $\omega\in A_{\infty}^{\rho,\infty}(\mathbb{R}^{n})$ which locally behave as Muckenhoupt’s weights and actually include them, by the local vertical maximal function, the local nontangential maximal function and the atomic decomposition. Then, we establish the equivalence of the weighted local Hardy space $h^{1}_{\rho}(\omega)$ and the weighted Hardy space $H^{1}_{\mathcal{L}}(\omega)$ associated to Schrodinger operators $\mathcal{L}$ with $\omega\in A_{1}^{\rho,\infty}(\mathbb{R}^{n})$. By the atomic characterization, we also prove the existence of finite atomic decompositions associated with $h^{p}_{\rho}(\omega)$. Furthermore, we establish boundedness in $h^{p}_{\rho}(\omega)$ of quasi-Banach-valued sublinear operators.