Uniformly contractive semigroups and submarkovian semigroups (I)

Let (M, ℱ,d, x) be a σ-finite measure space with a σ-field ℱ countably generated. We call a linear mapT uniformly contractive if which maps measurable functions onM to measurable functions and If a linear mapT which maps measurable functions onM to measurable functions has positivity property, namely,Tf≧0 forf≧0, we call it a submarkovian operator. In this article we prove     

Abstract Lebesgue-Radon-Nikodym theorems

Summary In this paper generalizations of the classical Lebesgue-Radon-Nikodym type decomposition of additive set functions are obtained for pairs of vector measures when both measures take values in possibly different Banach spaces. Some applications of these results are made to (i) the representation of wearly compact operators on the spaces of integrable scalar functions relative to a vector measure to an arbitrary Banach space, and (ii) a problem of comparison of measures in inference theory. The abstract conditional expectations of operator valued strongly measurable and integrable random variables on a σ-finite space are briefly treated. Supported, in part, under the NSF Grants GP-1349 and GP-5921.     

Differentiability of p-Harmonic Functions on Metric Measure Spaces

We study p-harmonic functions on metric measure spaces, which are formulated as minimizers to certain energy functionals. For spaces supporting a p-Poincaré inequality, we show that such functions satisfy an infinitesmal Lipschitz condition almost everywhere. This result is essentially sharp, since there are examples of metric spaces and p-harmonic functions that fail to be locally Lipschitz continuous on them. As a consequence of our main theorem, we show that p-harmonic functions also satisfy a generalized differentiability property almost everywhere, in the sense of Cheeger’s measurable differentiable structures.     

The existence of invariantσ-finite measures for a group of transformations

SupposeG is a group of measurable transformations of aσ-finite measure space (X, A, m). The main result of this paper gives necessary and sufficient conditions for the existence of aG-invariant,σ-finite measure defined onA and dominating the measurem in the sense of absolute continuity. An example is also given of aσ-finite nonatomic measure space (X, A, m) together with a countable groupG of its measurable transformations such that noG-invariant,σ-finite nonatomic measure exists onA. Whether the Lebesgue space ([0, 1],L, λ) provides such an example, depends on set-theoretic assumptions. This paper was written while the author was visiting the Technische Universitat Berlin as a research fellow of the Alexander von Humboldt Foundation.     

On the approximation of measurable functions by continuous functions

We present a characterization of the completed Borel measure spaces for which every measurable function, with values in a separable Frechet space, is the almost everywhere limit of a sequence of continuous functions. From this characterization one can easily obtain results that have appeared recently in the literature, in a more general form. We also examine what happens when the range is a subset of an arbitrary Banach space, and show that this case often reduces to the separable case.     

A minimax theorem for irreducible compact operators inL p-spaces

Let (X, Σ, μ) be a σ-finite measure space,T a compact irreducible (positive, linear) operator onL ^p (μ) (1≦p<+∞). It is shown that the spectral radiusr ofT is characterized by the minimax property {fx196-1} where ∑₀ denotes the ring of sets of finite measure and whereQ denotes the set of all, almost everywhere positive functions inL ^p. Moreover, ifr>0 then equality on either side is assumed ifff is the (essentially unique) positive eigenfunction ofT. Various refinements are given in terms of corresponding relations for irreducible finite rank operators approximatingT. Dedicated to H. G. Tillmann on his 60th birthday     

First return map and invariant measures

We give sufficient conditions for the existence of absolutely continuous invariant measures, finite or σ-finite, for maps on the interval. We givea priori bound for the number of different ergodic measures. The results are obtained via the first return map.     

On continuous restrictions of measurable multilinear mappings

This article deals with measurablemultilinear mappings on Fréchet spaces and analogs of two properties which are equivalent for a measurable (with respect to gaussian measure) linear functional: (i) there exists a sequence of continuous linear functions converging to the functional almost everywhere; (ii) there exists a compactly embedded Banach space X of full measure such that the functional is continuous on it. We show that these properties for multilinear functions defined on a power of the space X are not equivalent; but property (ii) is equivalent to the apparently stronger condition that the compactly embedded subspace is a power of the subspace embedded in X.     

Measurable Categories

We develop the theory of categories of measurable fields of Hilbert spaces and bounded fields of operators. We examine classes of functors and natural transformations with good measure theoretic properties, providing in the end a rigorous construction for the bicategory used in [3] and [4] as the basis for a representation theory of (Lie) 2-groups. Two important technical results are established along the way: first it is shown that all invertible additive bounded functors (and thus a fortiori all invertible *-functors) between categories of measurable fields of Hilbert spaces are induced by invertible measurable transformations between the underlying Borel spaces and second we establish the distributivity of Hilbert space tensor product over direct integrals over Lusin spaces with respect to σ-finite measures. The paper concludes with a general definition of measurable bicategories.     

Semigroups,Potential Spaces and Applications to (S)PDE

The paper studies perturbed semilinear parabolic partial (pseudo-) differential equations on σ-finite measure spaces under low smoothness assumptions. We obtain results on existence, uniqueness and regularity. The hypotheses are formulated in terms of the semigroup, regularity is measured by means of abstract potential spaces. Being a priori analytic, our results allow to investigate related stochastic partial differential equations in the almost sure pathwise sense. For example we can study (fractional) semilinear heat equations driven by fractional Brownian noises on metric measure spaces.     

Totally dissipative measures for the shift and conformal σ-finite measures for the stable holonomies

In this paper we investigate some results of ergodic theory with infinite measures for a subshift of finite type. We give an explicit way to construct σ-finite measures which are quasi-invariant by the stable holonomy and equivalent to the conditional measures of some σ-invariant measure. These σ-invariant measures are totally dissipative, σ-finite but satisfy a Birkhoff Ergodic-like Theorem. The constructions are done for the symbolic case, but can be extended for uniformly hyperbolic flows or diffeomorphisms.     

Measurable rigidity of the cohomological equation for linear cocycles over hyperbolic systems

We show that any measurable solution of the cohomological equation for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a Hölder solution. More generally, we show that every measurable invariant conformal structure for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a continuous invariant conformal structure. We also use the main theorem to show that a linear cocycle is conformal if none of its iterates preserve a measurable family of proper subspaces of R^d. We use this to characterize closed negatively curved Riemannian manifolds of constant negative curvature by irreducibility of the action of the geodesic flow on the unstable bundle.     

A Note on the characterization of Shannon-Wiener’s measure of information

Summary In [1] the author treated a characterization problem of the ShannonWiener measure of information for continuous probability distributions defined over an abstract measure space (R, S, m), wherem is a σ-finite measure over a σ-field S of subsets ofR, whose rangeM(S) is such thatM(S)=[0, ∞]. This condition on the range of the basic measure, however, can slightly be altered such thatM(S)=[0,1], and this modification is useful for characterization of the Kullback-Leibler mean information. In the present paper, it is shown that the characterization procedure of [1] can be applicable to continuous probability distributions defined on a finite measure space.     

Invariant measures for non-primitive tiling substitutions

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpiński gasket and carpet” tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.     

On the Definition of Relative Pressure for Factor Maps on Shifts of Finite Type

The relationship between two natural definitions of the relativepressure function, for a locally constant potential functionand a factor map from a shift of finite type, is clarified byshowing that they coincide almost everywhere with respect toevery invariant measure. With a suitable extension of one ofthe definitions, the same holds true for any continuous potentialfunction. 2000 Mathematics Subject Classification 37A35, 37B40(primary), 37B10 (secondary).     

Selezioni continue o misurabili

We give a characterization of set-valued mappings from a topological (measure) space into the class of real non-empty intervals admitting a continuous (measurable) selection. As an application we obtain a characterization of set-valued functions defined on IR ⁿ admitting an approximately continuous or an approximately continuous and almost everywhere continuous selection.     

Counter-examples to the Dunford–Schwartz pointwise ergodic theorem on $$\varvec{L^1+L^\infty }$$ L 1 + L ∞

Extending a result by Chilin and Litvinov, we show by construction that given any $$\sigma $$ -finite infinite measure space $$(\Omega ,\mathcal {A}, \mu )$$ and a function $$f\in L^1(\Omega )+L^\infty (\Omega )$$ with $$\mu (\{|f|>\varepsilon \})=\infty $$ for some $$\varepsilon >0$$ , there exists a Dunford–Schwartz operator T over $$(\Omega ,\mathcal {A}, \mu )$$ such that $$\frac{1}{N}\sum _{n=1}^N (T^nf)(x)$$ fails to converge for almost every $$x\in \Omega $$ . In addition, for each operator we construct, the set of functions for which pointwise convergence fails almost everywhere is residual in $$L^1(\Omega )+L^\infty (\Omega )$$ .     

Group-universal unary varieties

LetV be a variety of unary algebras and letM(V) be the monoid of all unary polynomials ofV. Then every group appears as the automorphism group of an algebraA∈V if and only if the left ideals ofM(V) do not form an inclusion-ordered chain. The support of the National Research Council of Canada is gratefully acknowledged. Presented by J. Mycielski.     

In some symmetric spaces monotonicity properties can be reduced to the cone of rearrangements

Geometric properties being the rearrangement counterparts of strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity in some symmetric spaces are considered. The relationships between strict monotonicity, upper local uniform monotonicity restricted to rearrangements and classical monotonicity properties (sometimes under some additional assumptions) are showed. It is proved that order continuity and lower uniform monotonicity properties for rearrangements of symmetric spaces together are equivalent to the classical lower local uniform monotonicity for any symmetric space over a \({\sigma}\)-finite complete and non-atomic measure space. It is also showed that in the case of order continuous symmetric spaces over a \({\sigma}\)-finite and complete measure space, upper local uniform monotonicity and its rearrangement counterpart shortly called ULUM* coincide. As an application of this result, in the case of a non-atomic complete finite measure a new proof of the theorem which is already known in the literature, giving the characterization of upper local uniform monotonicity of Orlicz–Lorentz spaces, is presented. Finally, it is proved that every rotund and reflexive space X such that both X and X* have the Kadec-Klee property is locally uniformly rotund. Some other results are also given in the first part of Sect. 2.     

CM newforms with rational coefficients

We classify newforms with rational Fourier coefficients and complex multiplication for fixed weight up to twisting. Under the extended Riemann hypothesis for odd real Dirichlet characters, these newforms are finite in number. We produce tables for weights 3 and 4, where finiteness holds unconditionally. I am indepted to K. Hulek for his continuous interest and encouragement. Partial support by the DFG Schwerpunkt 1094 “Globale Methoden in der komplexen Geometrie” is gratefully acknowledged. My thanks go also to the referee for helpful comments. Part of the revising took place while I enjoyed the hospitality of the Dipartimento di Matematica “Frederico Enriques” of Milano University. Funding from the network Arithmetic Algebraic Geometry, a Marie Curie Research Training Network, is gratefully acknowledged. I particularly thank M. Bertolini and B. van Geemen. The final version was prepared while I was funded by DFG under grant Schu 2266/2-2.