Representation of the compact-valued sublinear operator by basis selectors packet and sub-invertibility

2017 ◽  
Author(s):  
Svetlana Smirnova ◽  
Igor Orlov
Keyword(s):  
Sublinear Operator

scholarly journals Weak Orlicz–Hardy martingale spaces

2015 ◽  
Vol 26 (08) ◽  
pp. 1550062 ◽  
Author(s):  
Yong Jiao ◽  
Lian Wu ◽  
Lihua Peng
Keyword(s):  
Atomic Decomposition ◽  
Type Inequality ◽  
Sublinear Operator ◽  
Sufficient Condition ◽  
Concave Functions ◽  
Atomic Decompositions ◽  
Stochastic Basis ◽  
Decomposition Theorems

In this paper, several weak Orlicz–Hardy martingale spaces associated with concave functions are introduced, and some weak atomic decomposition theorems for them are established. With the help of weak atomic decompositions, a sufficient condition for a sublinear operator defined on the weak Orlicz–Hardy martingale spaces to be bounded is given. Further, we investigate the duality of weak Orlicz–Hardy martingale spaces and obtain a new John–Nirenberg type inequality when the stochastic basis is regular. These results can be regarded as weak versions of the Orlicz–Hardy martingale spaces due to Miyamoto, Nakai and Sadasue.

scholarly journals Atomic decompositions of Lorentz martingale spaces and applications

2009 ◽  
Vol 7 (2) ◽  
pp. 153-166 ◽  
Author(s):  
Jiao Yong ◽  
Peng Lihua ◽  
Liu Peide
Keyword(s):  
Atomic Decomposition ◽  
Weak Type ◽  
Interpolation Theorem ◽  
Sublinear Operator ◽  
Sufficient Condition ◽  
Atomic Decompositions ◽  
Type Interpolation ◽  
Marcinkiewicz Interpolation Theorem ◽  
Decomposition Theorems

In the paper we present three atomic decomposition theorems of Lorentz martingale spaces. With the help of atomic decomposition we obtain a sufficient condition for sublinear operator defined on Lorentz martingale spaces to be bounded. Using this sufficient condition, we investigate some inequalities on Lorentz martingale spaces. Finally we discuss the restricted weak-type interpolation, and prove the classical Marcinkiewicz interpolation theorem in the martingale setting.

scholarly journals On the sublinear operators factoring throughLq

2004 ◽  
Vol 2004 (50) ◽  
pp. 2695-2704
Author(s):  
Lahcène Mezrag ◽  
Abdelmoumene Tiaiba
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Positive Constant ◽  
Bounded Linear Operator ◽  
Bounded Operator ◽  
Linear Operators ◽  
Sublinear Operator ◽  
Bounded Linear ◽  
Sublinear Operators

Let0<p≤q≤+∞. LetTbe a bounded sublinear operator from a Banach spaceXinto anLp(Ω,μ)and let∇Tbe the set of all linear operators≤T. In the present paper, we will show the following. LetCbe a positive constant. For alluin∇T,Cpq(u)≤C(i.e.,uadmits a factorization of the formX→u˜Lq(Ω,μ)→MguLq(Ω,μ), whereu˜is a bounded linear operator with‖u˜‖≤C,Mguis the bounded operator of multiplication byguwhich is inBLr+(Ω,μ)(1/p=1/q+1/r),u=Mgu∘u˜andCpq(u)is the constant ofq-convexity ofu) if and only ifTadmits the same factorization; This is under the supposition that{gu}u∈∇Tis latticially bounded. Without this condition this equivalence is not true in general.

scholarly journals Anisotropic Hardy Spaces of Musielak-Orlicz Type with Applications to Boundedness of Sublinear Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-19 ◽  
Author(s):  
Baode Li ◽  
Dachun Yang ◽  
Wen Yuan
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Orlicz Function ◽  
Real Variable ◽  
Sublinear Operator ◽  
Atomic Decompositions ◽  
Anisotropic Hardy Space ◽  
Sublinear Operators ◽  
Uniformly Bounded

Letφ:ℝn×[0,∞)→[0,∞)be a Musielak-Orlicz function andAan expansive dilation. In this paper, the authors introduce the anisotropic Hardy space of Musielak-Orlicz type,HAφ(ℝn), via the grand maximal function. The authors then obtain some real-variable characterizations ofHAφ(ℝn)in terms of the radial, the nontangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy spaceHAp(ℝn)withp∈(0,1]and are new even for its weighted variant. Finally, the authors characterize these spaces by anisotropic atomic decompositions. The authors also obtain the finite atomic decomposition characterization ofHAφ(ℝn), and, as an application, the authors prove that, for a given admissible triplet(φ,q,s), ifTis a sublinear operator and maps all(φ,q,s)-atoms withq<∞(or all continuous(φ,q,s)-atoms withq=∞) into uniformly bounded elements of some quasi-Banach spacesℬ, thenTuniquely extends to a bounded sublinear operator fromHAφ(ℝn)toℬ. These results are new even for anisotropic Orlicz-Hardy spaces onℝn.

scholarly journals Boundedness of a Class of Sublinear Operators and Their Commutators on Generalized Morrey Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Vagif S. Guliyev ◽  
Seymur S. Aliyev ◽  
Turhan Karaman
Keyword(s):  
Pseudodifferential Operators ◽  
Sufficient Conditions ◽  
Morrey Spaces ◽  
Sublinear Operator ◽  
Generalized Morrey Space ◽  
Riesz Operator ◽  
Generalized Morrey Spaces ◽  
Marcinkiewicz Operator ◽  
Sublinear Operators ◽  
Type Integral

The authors study the boundedness for a large class of sublinear operatorTgenerated by Calderón-Zygmund operator on generalized Morrey spacesMp,φ. As an application of this result, the boundedness of the commutator of sublinear operatorsTaon generalized Morrey spaces is obtained. In the casea∈BMO(ℝn),1<p<∞andTais a sublinear operator, we find the sufficient conditions on the pair (φ1,φ2) which ensures the boundedness of the operatorTafrom one generalized Morrey spaceMp,φ1to anotherMp,φ2. In all cases, the conditions for the boundedness ofTaare given in terms of Zygmund-type integral inequalities on (φ1,φ2), which do not assume any assumption on monotonicity ofφ1,φ2inr. Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator, and Bochner-Riesz operator.

Quantitative approximation by shift invariant multivariate sublinear-Choquet operators

2018 ◽  
Vol 24 (2) ◽  
pp. 115-126 ◽  
Author(s):  
George A. Anastassiou
Keyword(s):  
Choquet Integral ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Sublinear Operator ◽  
Type Operator ◽  
Multivariate Approximation ◽  
Integral Type ◽  
Type Function ◽  
Global Smoothness ◽  
Shift Invariance

Abstract A very general multivariate positive sublinear Choquet integral type operator is given through a convolution-like iteration of another multivariate general positive sublinear operator with a multivariate scaling type function. For it, sufficient conditions are given for shift invariance, preservation of global smoothness, convergence to the unit with rates. Furthermore, two examples of very general multivariate specialized operators are presented fulfilling all the above properties; the higher order of multivariate approximation of these operators is also studied.

Dynamics of a 3D Benjamin–Bona–Mahony equations with sublinear operator

2020 ◽  
Vol 121 (1) ◽  
pp. 75-100
Author(s):  
Mingxia Zhao ◽  
Xin-Guang Yang ◽  
Xingjie Yan ◽  
Xiaona Cui
Keyword(s):  
Shallow Water ◽  
Energy Equation ◽  
Three Dimensional ◽  
Solution Process ◽  
Sublinear Operator ◽  
Sufficient Condition ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Pullback Attractors ◽  
Long Time

This paper is concerned with the tempered pullback dynamics for a three dimensional Benjamin–Bona–Mahony equations with sublinear operator on bounded domain, which describes the long time behavior for long waves model in shallow water with friction. By virtue of a new retarded Gronwall inequality, and using the energy equation method from J.M. Ball (Disc. Cont. Dyn. Syst. 10 (2004) 31–52) to achieve asymptotic compactness for solution process, the minimal family of pullback attractors has been obtained, which reduces a single trajectory under a sufficient condition.

scholarly journals Uncertainty orders on the sublinear expectation space

2016 ◽  
Vol 14 (1) ◽  
pp. 247-259
Author(s):  
Dejian Tian ◽  
Long Jiang
Keyword(s):  
Risk Measures ◽  
Sublinear Operator ◽  
The Other ◽  
Probability Measures ◽  
Space Theory ◽  
Random Vectors ◽  
Sublinear Expectation ◽  
Normal Distributions ◽  
Robust Representation ◽  
Other Hand

AbstractIn this paper, we introduce some definitions of uncertainty orders for random vectors in a sublinear expectation space. We all know that, under some continuity conditions, each sublinear expectation 𝔼 has a robust representation as the supremum of a family of probability measures. We describe uncertainty orders from two different viewpoints. One is from sublinear operator viewpoint. After giving definitions such as monotonic orders, convex orders and increasing convex orders, we use these uncertainty orders to derive characterizations for maximal distributions, G-normal distributions and G-distributions, which are the most important random vectors in the sublinear expectation space theory. On the other hand, we also establish some uncertainty orders’ characterizations from the viewpoint of probability measures and build some connections with the theory of risk measures.

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