Sobolev spaces on an arbitrary metric space

We define Sobolev space W ^1,p for 1<p on an arbitrary metric space with finite diameter and equipped with finite, positive Borel measure. In the Euclidean case it coincides with standard Sobolev space. Several classical imbedding theorems are special cases of general results which hold in the metric case. We apply our results to weighted Sobolev space with Muckenhoupt weight.This work is supported by KBN grant no. 2 1057 91 01     

On an extension of a lattice-continuous mapping with embeddability applications

It is well known that ifX andY are completely regularT ₂ spaces, then any continuous function,f, fromX toY, has a unique continuous extension,(f), fromX toY, whereX andY are the Stone—ech compactifications ofX andY, respectively. This function plays an important role in Stone—ech Theory, especially in questions pertaining to embeddability.In this paper, we first extend this construction to general Wallman spaces, and then apply the results to extend well-known embeddability theorems.     

The singularity spectrum of Lévy processes in multifractal time

Let X=(Xt)_t?0 be a Lévy process and μ a positive Borel measure on R₊. Suppose that the integral of μ defines a continuous increasing multifractal time . Under suitable assumptions on μ, we compute the singularity spectrum of the sample paths of the process X in time μ defined as the process (XF(t))_t?0.A fundamental example consists in taking a measure μ equal to an “independent random cascade” and (independently of μ) a suitable stable Lévy process X. Then the associated process X in time μ is naturally related to the so-called fixed points of the smoothing transformation in interacting particles systems.Our results rely on recent heterogeneous ubiquity theorems.     

Rational Unfoldings of ± X k and their relation with geometric equilibrium theory

In this paper, two theorems determine the locus B (buoyancy locus) of the centre of gravity of the displaced water of a ship with any number of 2n sides, as a function of the arbitrary continuous curves that make up those sides, and that generalize the classical wall-sided case (n=1). Although the theorems may have interest from the pure geometrical point of view, they have been obtained with a view to simulate processes in biology that, in analogy to the self-righting of ships, are self-regulating, such as the neural control of body heat and the servo-control of the heart rate.     

Operator-valued positive definite kernels on tubes

In this paper we generalize the concept of an infinite positive measure on a -algebra to a vector valued setting, where we consider measures with values in the compactification of a convex coneC which can be described as the set of monoid homomorphisms of the dual coneC ^* into [0, ]. Applying these concepts to measures on the dual of a vector space leads to generalizations of Bochner's Theorem to operator valued positive definite functions on locally compact abelian groups and likewise to generalizations of Nussbaum's Theorem on positive definite functions on cones. In the latter case we use the Laplace transform to realize the corresponding Hilbert spaces by holomorphic functions on tube domains.     

Finitely subadditive and countably subadditive outer measures

Results are given comparing countably subadditive (csa) outer measures and finitely subadditive (fsa) outer measures, especially relating to regularity and measurability conditions such as (*) condition:A setE (of an arbitrary setX), is measurable ( an outer measure),ES (the collection of measurable sets) iff (X)=(E)+(E). Specific examples are given contrasting csa and fsa outer measures. In particular fsa and csa outer measures derived from finitely additive measures defined on an algebra of sets generated by a lattice of sets, are investigated in some detail.     

Infinitely Many Securities and the Fundamental Theorem of Asset Pricing

Several authors have pointed out the possible absence of martingale measures for static arbitrage free markets with an infinite number of available securities. Accordingly, the literature constructs martingale measures by generalizing the concept of arbitrage (free lunch, free lunch with bounded risk, etc.) or introducing the theory of large financial markets. This paper does not modify the definition of arbitrage and addresses the caveat by drawing on projective systems of probability measures. Thus we analyze those situations for which one can provide a projective system of σ–additive measures whose projective limit may be interpreted as a risk-neutral probability of an arbitrage free market. Hence the Fundamental Theorem of Asset Pricing is extended so that it can apply for models with infinitely many assets. Partially funded by the Spanish Ministry of Science and Education (ref: BEC2003 – 09067 –C04 – 03) and Comunidad Autonoma de Madrid (ref: s – 0505/tic/000230).     

Positive Operators à la Aumann-Shapley on Spaces of Functions on D-Lattices

Let L be a σ-complete D-lattice and BV the AL-space of all realvalued, null in zero, functions on L of bounded variation. We prove the existence of a continuous Aumann-Shapley operator on the closed subspace of BV generated by powers of nonatomic σ-additive positive modular measures on L. The integral representation of on a class of functions that correspond to measure games is also exhibited. Dedicated to Professor Paolo De Lucia on the occasion of his 70th birthday.     

Properties of a special class of doubly stochastic measures

Summary A measure on the unit squareI } I is doubly stochastic if(A } I) = (I } A) = the Lebesgue measure ofA for every Lebesgue measurable subsetA ofI = [0, 1]. By the hairpinL L ^–1, we mean the union of the graphs of an increasing homeomorphismL onI and its inverseL ^–1. By the latticework hairpin generated by a sequence {x _n :n Z} such thatx _n-1 < x_n (n Z), x _n = 0 and x _n = 1, we mean the hairpinL L ^–1 , whereL is linear on [x _n-1 ,x _n ] andL(n) =x _n-1 forn Z. In this note, a characterization of latticework hairpins which support doubly stochastic measures is given. This allows one to construct a variety of concrete examples of such measures. In particular, examples are given, disproving J. H. B. Kemperman's conjecture concerning a certain condition for the existence of doubly stochastic measures supported in hairpins.     

Improving dimension estimates for Furstenberg-type sets

In this paper we study the problem of estimating the generalized Hausdorff dimension of Furstenberg sets in the plane. For α∈(0,1], a set F in the plane is said to be an α-Furstenberg set if for each direction e there is a line segment ?e in the direction of e for which dimH(?e∩F)?α. It is well known that , and it is also known that these sets can have zero measure at their critical dimension. By looking at general Hausdorff measures Hh defined for doubling functions, that need not be power laws, we obtain finer estimates for the size of the more general h-Furstenberg sets. Further, this approach allow us to sharpen the known bounds on the dimension of classical Furstenberg sets.The main difficulty we had to overcome, was that if Hh(F)=0, there always exists g?h such that Hg(F)=0 (here ? refers to the natural ordering on general Hausdorff dimension functions). Hence, in order to estimate the measure of general Furstenberg sets, we have to consider dimension functions that are a true step down from the critical one. We provide rather precise estimates on the size of this step and by doing so, we can include a family of zero dimensional Furstenberg sets associated to dimension functions that grow faster than any power function at zero. With some additional growth conditions on these zero dimensional functions, we extend the known inequalities to include the endpoint α=0.     

Subordination of Regular Kernels and Abelian Theorems for Quotients

We prove general boundary limit theorems of abelian type for quotients of functions defined in the half space . The functions considered are defined as convolutions of a kernel with Borel measures defined on the boundary . Our theorems are of the form

On the magnification of cantor sets and their limit models

For aC ¹⁺ hyperbolic (cookie-cutter) Cantor setC we consider the limits of sequences of closed subsets ofR obtained by arbitrarily high magnifications around different points ofC. It is shown that a well defined set of limit models exists for the infinitesimal geometry, orscenery, in the Cantor set. IfCC} is a diffeomorphic copy ofC then the set of limit models of C is the same as that ofC. Furthermore every limit model is made of Cantor sets which areC ¹⁺ diffeomorphic withC (for some >0, (0,1)), but not all suchC ¹⁺ copies ofC occur in the limit models. We show the relation between this approach to the asymptotic structure of a Cantor set and Sullivan's scaling function. An alternative definition of a fractal is discussed.With 1 Figure     

Limits of balayage measures

LetA be a subset of a balayage space (X,W) and a measure onX. It is shown that for every sequence _n of measures such that lim_nn and lim_{n n} ^A = the limit measure is of the formf+[(1-f)]^A for some (unique) Borel function 0f1_Cb(A). Furthermore, conditions are given such that any such functionf occurs.     

The eigenvalue problem for linear and affine iterated function systems

The eigenvalue problem for a linear function L centers on solving the eigen-equation . This paper generalizes the eigenvalue problem from a single linear function to an iterated function system F consisting of possibly an infinite number of linear or affine functions. The eigen-equation becomes F(X)=λX, where λ>0 is real, X is a compact set, and F(X)=?_f∈Ff(X). The main result is that an irreducible, linear iterated function system F has a unique eigenvalue λ equal to the joint spectral radius of the functions in F and a corresponding eigenset S that is centrally symmetric, star-shaped, and full dimensional. Results of Barabanov and of Dranisnikov-Konyagin-Protasov on the joint spectral radius follow as corollaries.     

Holomorphic Functions of Exponential Growth on Abelian Coverings of a Projective Manifold

Let M be a projective manifold, p: M _G M a regular covering over M with a free Abelian transformation group G. We describe the holomorphic functions on M _G of an exponential growth with respect to the distance defined by a metric pulled back from M. As a corollary, we obtain Cartwright and Liouville-type theorems for such functions. Our approach brings together the L ₂ cohomology technique for holomorphic vector bundles on complete Kähler manifolds and the geometric properties of projective manifolds.     

An Isometric Representation of the Dual of {\user1{\mathcal{C}}}{\left( {X,\mathbb{R}} \right)}

Many structures in functional analysis are introduced as the limit of an inverse (aka projective) system of seminormed spaces [2, 3, 8]. In these situations, the dual is moreover equipped with a seminorm. Although the topology of the inverse limit is seldom metrizable, there is always a natural overlying locally convex approach structure. We provide a method for computing the adjoint of this space, by showing that the dual of a limit of locally convex approach spaces becomes a co-limit in the category of seminormed spaces. As an application we obtain an isometric representation of the dual space of real valued continuous functions on a locally compact Hausdorff space X, equipped with the compact open structure.     

Hausdorff measure of infinitely generated self-similar sets

We analyze self-similarity with respect to infinite sets of similitudes from a measure-theoretic point of view. We extend classic results for finite systems of similitudes satisfying the open set condition to the infinite case. We adopt Vitali-type techniques to approximate overlapping self-similar sets by non-overlapping self-similar sets. As an application we show that any open and bounded set with a boundary of null Lebesgue measure always contains a self-similar set generated by a countable system of similitudes and with Lebesgue measure equal to that ofA.     

On sums of real-valued functions with extremely thick graphs

We consider some properties of those functions acting from the real line R$\mathbf{R}$ into itself, whose graphs are extremely thick subsets of the Euclidean plane R²${\mathbf{R}}^2$. The structure of sums of such functions is studied and the obtained results are applied to certain measure extension problems.     

Divergence points of self-similar measures and packing dimension

Let μ be a self-similar measure in Rd. A point x∈Rd for which the limit does not exist is called a divergence point. Very recently there has been an enormous interest in investigating the fractal structure of various sets of divergence points. However, all previous work has focused exclusively on the study of the Hausdorff dimension of sets of divergence points and nothing is known about the packing dimension of sets of divergence points. In this paper we will give a systematic and detailed account of the problem of determining the packing dimensions of sets of divergence points of self-similar measures. An interesting and surprising consequence of our results is that, except for certain trivial cases, many natural sets of divergence points have distinct Hausdorff and packing dimensions.