Ergodic theory and the Strong Law of Large Numbers on Riesz spaces

In [W.-C. Kuo, C.C.A. Labuschagne, B.A. Watson, Discrete-time stochastic processes on Riesz spaces, Indag. Math. (N.S.) 15 (3) (2004) 435-451], we introduced the concepts of conditional expectations, martingales and stopping times on Riesz spaces. Here we formulate and prove order theoretic analogues of the Birkhoff, Hopf and Wiener ergodic theorems and the Strong Law of Large Numbers on Riesz spaces (vector lattices).     

Characterizations of Riesz spaces with b-property

A Riesz space E is said to have b-property if each subset which is order bounded in E^~~ is order bounded in E. The relationship between b-property and completeness, being a retract and the absolute weak topology |σ|(E^~, E) is studied. Perfect Riesz spaces are characterized in terms of b-property. It is shown that b-property coincides with the Levi property in Dedekind complete Frechet lattices.      

Conditional expectations on Riesz spaces

Conditional expectations operators acting on Riesz spaces are shown to commute with a class of principal band projections. Using the above commutation property, conditional expectation operators on Riesz spaces are shown to be averaging operators. Here the theory of f-algebras is used when defining multiplication on the Riesz spaces. This leads to the extension of these conditional expectation operators to their so-called natural domains, i.e., maximal domains for which the operators are both averaging operators and conditional expectations. The natural domain is in many aspects analogous to L¹.     

Quadratic variation of martingales in Riesz spaces

We derive quadratic variation inequalities for discrete-time martingales, sub- and supermartingales in the measure-free setting of Riesz spaces. Our main result is a Riesz space analogue of Austin?s sample function theorem, on convergence of the quadratic variation processes of martingales.     

A characterization of an order ideal in Riesz spaces

In this paper we give a characterization of order ideals in Riesz spaces.

     

Polynomials on Riesz spaces

In this paper we investigate polynomial mappings on Riesz spaces. We give a characterization of positivity of homogeneous polynomials in terms of forward differences. Finally we prove Hahn-Banach type extension theorems for positive and regular polynomial mappings.     

Generalizations of the Riesz convergence theorem for Lorentz spaces

Summary We obtain preservation inequalities for Lipschitz constants of higher order in simultaneous approximation processes by Bernstein type operators. From such inequalities we derive the preservation of the corresponding Lipschitz spaces.     

Amarts on Riesz Spaces

The concepts of conditional expectations, martingales and stopping times were extended to the Riesz space context by Kuo, Labuschagne and Watson （Discrete time stochastic processes on Riesz spaces, Indag. Math.,15（2004）, 435-451）. Here we extend the definition of an asymptotic martingale （amart） to the Riesz spaces context, and prove that Riesz space amarts can be decomposed into the sum of a martingale and an adapted sequence convergent to zero. Consequently an amart convergence theorem is deduced.     

Pointfree Spectra of Riesz Spaces

One of the best ways of studying ordered algebraic structures is through their spectra. The three well-known spectra usually considered are the Brumfiel, Keimel, and the maximal spectra. The pointfree versions of these spectra were studied by B. Banaschewski for f-rings. Here, we give the pointfree versions of the Keimel and the maximal spectra for Riesz spaces. Moreover, we briefly mention how one can use the results of this paper to give a pointfree version of the Kakutani duality for Riesz spaces.     

Riesz product spaces and representation theory

Let {E _i:i∈I} be a family of Archimedean Riesz spaces. The Riesz product space is denoted by ∏ _i∈I E_i. The main result in this paper is the following conclusion: There exists a completely regular Hausdorff spaceX such that ∏ _i∈I E_i is Riesz isomorphic toC(X) if and only if for everyi∈I there exists a completely regular Hausdorff spaceX _i such thatE _i is Riesz isomorphic toC(X _i). Supported by the National Natural Science Foundation of China     

Riesz potential operator in continual variable exponents Herz spaces

We find conditions on the variable parameters $p\left(x\right),q\left(t\right)$ and $\alpha \left(t\right)$, defining the Herz space $H^{p(·),q(·),\alpha (·)}\left({\mathbb{R}}^n\right)$, for the validity of Sobolev type theorem for the Riesz potential operator to be bounded within the frameworks of such variable exponents Herz spaces. We deal with a “continual” version of Herz spaces (which coincides with the “discrete” one when q is constant).     

A note on Riesz spaces with property-b

We study an order boundedness property in Riesz spaces and investigate Riesz spaces and Banach lattices enjoying this property.     

Probabilistic normed Riesz spaces

In this paper, the concepts of probabilistic normed Riesz space and probabilistic Banach lattice are introduced, and their basic properties are studied. In this context, some continuity and convergence theorems are proved.     

Trudinger's exponential integrability for Riesz potentials of functions in generalized grand Morrey spaces

Our aim in this paper is to discuss Trudinger's exponential integrability for Riesz potentials of functions in generalized grand Morrey spaces. Our result will imply the boundedness of the Riesz potential operator from a grand Morrey space to a Morrey space.     

Riesz transform on locally symmetric spaces and Riemannian manifolds with a spectral gap

In this paper we study the Riesz transform on complete and connected Riemannian manifolds M with a certain spectral gap in the L² spectrum of the Laplacian. We show that on such manifolds the Riesz transform is Lp bounded for all p∈(1,∞). This generalizes a result by Mandouvalos and Marias and extends a result by Auscher, Coulhon, Duong, and Hofmann to the case where zero is an isolated point of the L² spectrum of the Laplacian.     

Sobolev's inequality for Riesz potentials of functions in Morrey spaces of integral form

Morrey spaces have become a good tool for the study of existence and regularity of solutions of partial differential equations. Our aim in this paper is to give Sobolev's inequality for Riesz potentials of functions in Morrey spaces (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Orlicz-Pettis theorem for topological Riesz spaces

A finitely additive vector measure from a -ring to a Riesz space is countably additive (exhaustive) for all Hausdorff Lebesgue topologies on the range space, or for none of them. In particular, subseries convergent series are the same for all Hausdorff Lebesgue topologies on a Riesz space.