General Measure Spaces Research Articles

On the equivalence of parabolic Harnack inequalities and heat kernel estimates

We prove the equivalence of parabolic Harnack inequalities and sub-Gaussian heat kernel estimates in a general metric measure space with a local regular Dirichlet form.

GENERALIZED FUZZY VALUED θ-CHOQUET INTEGRALS AND THEIR DOUBLE-NULL ASYMPTOTIC ADDITIVITY

The generalized fuzzy valued -Choquet integrals will be estab- lished for the given -integrable fuzzy valued functions on a general fuzzy measure space, and the convergence theorems of this kind of fuzzy valued in- tegral are being discussed. Furthermore, the whole of integrals is regarded as a fuzzy valued set function on measurable space, the double-null asymptotic additivity and pseudo-double-null asymptotic additivity of the fuzzy valued set functions formed are studied when the fuzzy measure satises autocontinuity from above (below).

A stochastic interpretation of propositional dynamic logic: expressivity

AbstractWe propose a probabilistic interpretation of Propositional Dynamic Logic (PDL). We show that logical and behavioral equivalence are equivalent over general measurable spaces. This is done first for the fragment of straight line programs and then extended to cater for the nondeterministic nature of choice and iteration, expanded to PDL as a whole. Bisimilarity is also discussed and shown to be equivalent to logical and behavioral equivalence, provided the base spaces are Polish spaces. We adapt techniques from coalgebraic stochastic logic and point out some connections to Souslin's operation from descriptive set theory. This leads to a discussion of complete stochastic Kripke models and model completion, which permits an adequate treatment of the test operator.

Characterizations of Orlicz-Sobolev Spaces by Means of Generalized Orlicz-Poincaré Inequalities

Let Φ be anN-function. We show that a functionu∈LΦ(ℝn)belongs to the Orlicz-Sobolev spaceW1,Φ(ℝn)if and only if it satisfies the (generalized) Φ-Poincaré inequality. Under more restrictive assumptions on Φ, an analog of the result holds in a general metric measure space setting.

On the lattice oscillator-type Kirkwood–Salsburg equation with attractive many-body potentials

We consider a lattice oscillator-type Kirkwood–Salsburg (KS) equation with general one-body phase measurable space and many-body interaction potentials. For special choices of the measurable space, its solutions describe grand-canonical equilibrium states of lattice equilibrium classical and quantum linear oscillator systems. We prove the existence of the solution of the symmetrized KS equation for manybody interaction potentials which are either attractive (nonpositive) and finite-range or infinite-range and repulsive (positive). The proposed procedure of symmetrization of the KS equation is new and based on the superstability of many-body potentials.

Unprovability of the logical characterization of bisimulation

We quickly review labelled Markov processes ( LMP) and provide a counterexample showing that in general measurable spaces, event bisimilarity and state bisimilarity differ in LMP. This shows that the Hennessy–Milner logic proposed by Desharnais does not characterize state bisimulation in non-analytic measurable spaces. Furthermore we show that, under current foundations of Mathematics, such logical characterization is unprovable for spaces that are projections of a coanalytic set. Underlying this construction there is a proof that stationary Markov processes over general measurable spaces do not have semi-pullbacks.

Coalgebraic logic over general measurable spaces – a survey

In this survey we discuss the generalisation of stochastic Kripke models for general modal logics through predicate liftings for functors over general measurable spaces. We derive results on expressivity and show that selection arguments allow us to incorporate the discussion of bisimilarity, provided the underlying spaces are assumed to be standard Borel.

Backward Stochastic Difference Equations for a Single Jump Process

We define Backward Stochastic Difference Equations related to a discrete finite time single jump process. We prove the existence and uniqueness of solutions under some assumptions. A comparison theorem for these solutions is also given. Applications to the theory of nonlinear expectations are then investigated. In this paper the single jump process takes values in a general measurable space where as previous work has considered the situation where the noise is a finite state Markov chain, so the state space is finite.

The geometry of p-convex intersection bodies

Busemann's theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper we provide a version of Busemann's theorem for p-convex bodies. We show that the intersection body of a p-convex body is q-convex for certain q. Furthermore, we discuss the sharpness of the previous result by constructing an appropriate example. This example is also used to show that IK, the intersection body of K, can be much farther away from the Euclidean ball than K. Finally, we extend these theorems to some general measure spaces with log-concave and s-concave measures.

Inequivalent measures of noncompactness

Two homogeneous measures of noncompactness β and γ on an infinite dimensional Banach space X are called “equivalent” if there exist positive constants b and c such that bβ(S) ≤ γ(S) ≤ cβ(S) for all bounded sets \({S\subset X}\). If such constants do not exist, the measures of noncompactness are “inequivalent.” We ask a foundational question which apparently has not previously been considered: For what infinite dimensional Banach spaces do there exist inequivalent measures of noncompactness on X? We provide here the first examples of inequivalent measures of noncompactness. We prove that such inequivalent measures exist if X is a Hilbert space; or if (Ω, Σ, μ) is a general measure space, 1 ≤ p ≤ ∞, and X = Lp(Ω, Σ, μ); or if K is a compact Hausdorff space and X = C(K); or if K is a compact metric space, 0 < λ ≤ 1, and X = C0,λ(K), the Banach space of Holder continuous functions with Holder exponent λ. We also prove the existence of such inequivalent measures of noncompactness if Ω is an open subset of \({\mathbb{R}^n}\) and X is the Sobolev space Wm,p(Ω). Our motivation comes from questions about existence of eigenvectors of homogeneous, continuous, order-preserving cone maps f : C→C and from the closely related issue of giving the proper definition of the “cone essential spectral radius” of such maps. These questions are considered in the companion paper [28]; see, also, [27].

Coalgebraic Logic over Measurable Spaces: Behavioral and Logical Equivalence

We study the relationship between logical and behavioral equivalence for coalgebras on general measurable spaces. Modal logics are interpreted in these coalgebras using predicate liftings. Prominent examples include stochastic relations and labelled Markov transition systems and corresponding Hennessy–Milner type logics. Local versions of logical and behavioral equivalence are introduced and it is shown that these notions coincide for a wide class of functors. We relate these notions to the corresponding global ones common in model checking. Throughout, we work in general measurable spaces. In contrast to previous work, no topological assumptions on the state spaces are needed.

Some new results on transition probability

In this paper, we study the basic properties of stationary transition probability of Markov processes on a general measurable space (E, ℰ), such as the continuity, maximum probability, zero point, positive probability set,standardization, and obtain a series of important results such as Continuity Theorem, Representation Theorem, Levy Theorem and so on. These results are very useful for us to study stationary tri-point transition probability on a general measurable space (E, ℰ). Our main tools such as Egoroff’s Theorem, Vitali-Hahn-Saks’s Theorem and the theory of atomic set and well-posedness of measure are also very interesting and fashionable.

Tent spaces associated with semigroups of operators

We study tent spaces on general measure spaces ( Ω , μ ) . We assume that there exists a semigroup of positive operators on L p ( Ω , μ ) satisfying a monotone property but do not assume any geometric/metric structure on Ω. The semigroup plays the same role as integrals on cones and cubes in Euclidean spaces. We then study BMO spaces on general measure spaces and get an analogue of Fefferman's H 1 –BMO duality theory. We also get a H 1 –BMO duality inequality without assuming the monotone property. All the results are proved in a more general setting, namely for noncommutative L p spaces.

Self-improving properties of inequalities of Poincaré type on measure spaces and applications

We show that the self-improving nature of Poincaré estimates persists for domains in rather general measure spaces. We consider both weak type and strong type inequalities, extending techniques of B. Franchi, C. Pérez and R. Wheeden. As an application in spaces of homogeneous type, we derive global Poincaré estimates for a class of domains with rough boundaries that we call ϕ-John domains, and we show that such domains have the requisite properties. This class includes John (or Boman) domains as well as s-John domains. Further applications appear in a companion paper.

Rearrangement transformations on general measure spaces

For a general set transformation R between two measure spaces, we define the rearrangement of a measurable function by means of the Layer's cake formula. We study some functional properties of the Lorentz spaces defined in terms of R, giving a unified approach to the classical rearrangement, Steiner's symmetrization, the multidimensional case, and the discrete setting of trees.

Harmonicity of Quasiconformal Measures and Poisson Boundaries of Hyperbolic Spaces

We consider a group Γ of isometries acting on a proper (not necessarily geodesic) δ -hyperbolic space X. For any continuous α-quasiconformal measure ν on ∂X assigning full measure to Λ r , the radial limit set of Γ, we produce a (nontrivial) measure μ on Γ for which ν is stationary. This means that the limit set together with ν forms a μ-boundary and ν is harmonic with respect to the random walk induced by μ. As a basic example, take $$X = {\mathbb{H}}^n$$ and Γ to be any geometrically finite Kleinian group with ν a Patterson-Sullivan measure for Γ. In the case when X is a CAT(−1) space and Γ is discrete with quasiconvex action, we show that (Λ r , ν) is the Poisson boundary for μ. In the course of the proofs, we establish sufficient conditions for a set of continuous functions to form a positive basis, either in the L 1 or L ∞ norm, for the space of uniformly positive lower-semicontinuous functions on a general metric measure space.

Harmonicity of Gibbs measures

We show that any continuous measure ν in the class of a generalized Gibbs stream on the boundary of a CAT(−κ) group G arises as a harmonic measure for a random walk on G. Under an additional mild hypothesis on G and for ν, Hölder equivalent to a Gibbs measure, we show that (∂G,ν) arises as a Poisson boundary for a random walk on G. We also prove a new approximation theorem for general metric measure spaces giving quite flexible conditions for a set of functions to be a positive basis for the cone of positive continuous functions

Extrapolation results for q &lt; 1

Given a sublinear operator T such that \({T:\ell\log\ell \rightarrow X}\) is bounded, it can be shown that \({T:\ell^q\rightarrow X}\) is bounded, with constant C/(1−q), for every 0 < q < 1. In this paper, we study the converse result, not only for sequence spaces, but for general measure spaces proving that, if T : Lq(μ) → X is bounded, with constant C/(1−q), for every \({q_0\le q < 1}\) and X is Banach, then T : L log (1/L)(μ) → X is bounded. Moreover, this result is optimal. We also show that things are quite different if the Banach condition on X is dropped.

Some Notes on Function Spaces and Dirichlet Forms on Self-Similar Sets

Abstract Grigor’yan, Hu and Lau [10] introduced sub-Gaussian heat kernels on general metric measure spaces and defined a family of function spaces to characterize the domain of associated Dirichlet forms. In this paper, we will improve their results about norm equivalence. As an application, we construct self-similar Dirichlet forms on a class of self-similar sets containing the Sierpiński gaskets and carpets. Then we prove the Poincaré inequality and give effective resistance estimates by the self-similarity. Consequently, we have a new equivalent characterization of heat kernel estimates through function spaces with strong recurrent condition.

Towards an L p Potential Theory for Sub–Markovian Semigroups: Kernels and Capacities

We study in fairly general measure spaces (X, μ) the (non–linear) potential theory of Lp sub–Markovian semigroups which are given by kernels having a density with respect to the underlying measure. In terms of mapping properties of the operators we provide sufficient conditions for the existence (and regularity) of such densities. We give various (dual) representations for several associated capacities and, in the corresponding abstract Bessel potential spaces, we study the role of the truncation property. Examples are discussed in the case of ℝn, where abstract Bessel potential spaces can be identified with concrete function spaces.