Real analysis related to the Monge-Ampère equation

In this paper we consider a family of convex sets in , , , , satisfying certain axioms of affine invariance, and a Borel measure satisfying a doubling condition with respect to the family The axioms are modelled on the properties of the solutions of the real Monge-Ampère equation. The purpose of the paper is to show a variant of the Calderón-Zygmund decomposition in terms of the members of This is achieved by showing first a Besicovitch-type covering lemma for the family and then using the doubling property of the measure The decomposition is motivated by the study of the properties of the linearized Monge-Ampère equation. We show certain applications to maximal functions, and we prove a John and Nirenberg-type inequality for functions with bounded mean oscillation with respect to

     

Covering the integers by arithmetic sequences. II

Let ( be a system of arithmetic sequences where and . For system will be called an (exact) -cover of if every integer is covered by at least (exactly) times. In this paper we reveal further connections between the common differences in an (exact) -cover of and Egyptian fractions. Here are some typical results for those -covers of : (a) For any there are at least positive integers in the form where . (b) When (, either or , and for each positive integer the binomial coefficient can be written as the sum of some denominators of the rationals if forms an exact -cover of . (c) If is not an -cover of , then have at least distinct fractional parts and for each there exist such that (mod 1). If forms an exact -cover of with or () then for every and there is an such that (mod 1).

     

Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète

Let be a hyperelliptic curve of genus over a discrete valuation field . In this article we study the models of over the ring of integers of . To each Weierstrass model (that is a projective model arising from a hyperelliptic equation of with integral coefficients), one can associate a (valuation of) discriminant. Then we give a criterion for a Weierstrass model to have minimal discriminant. We show also that in the most cases, the minimal regular model of over dominates every minimal Weierstrass model. Some classical facts concerning Weierstrass models over of elliptic curves are generalized to hyperelliptic curves, and some others are proved in this new setting.

     

Fine structure of the space of spherical minimal immersions

The space of congruence classes of full spherical minimal immersions of a given source dimension and algebraic degree is a compact convex body in a representation space of the special orthogonal group . In Ann. of Math. 93 (1971), 43--62 DoCarmo and Wallach gave a lower bound for and conjectured that the estimate was sharp. Toth resolved this ``exact dimension conjecture' positively so that all irreducible components of became known. The purpose of the present paper is to characterize each irreducible component of in terms of the spherical minimal immersions represented by the slice . Using this geometric insight, the recent examples of DeTurck and Ziller are located within .

     

The fixed-point property for simply connected plane continua

We answer a question of R. Ma\'{n}ka by proving that every simply-connected plane continuum has the fixed-point property. It follows that an arcwise-connected plane continuum has the fixed-point property if and only if its fundamental group is trivial. Let be a plane continuum with the property that every simple closed curve in bounds a disk in . Then every map of that sends each arc component into itself has a fixed point. Hence every deformation of has a fixed point. These results are corollaries to the following general theorem. If is a plane continuum, is a decomposition of , and each element of is simply connected, then every map of that sends each element of into itself has a fixed point.

     

Extremal functions for Moser's inequality

Let be a bounded smooth domain in , and a function with compact support in . Moser's inequality states that there is a constant , depending only on the dimension , such that

where is the Lebesgue measure of , and the surface area of the unit ball in . We prove in this paper that there are extremal functions for this inequality. In other words, we show that the

is attained. Earlier results include Carleson-Chang (1986, is a ball in any dimension) and Flucher (1992, is any domain in 2-dimensions).

     

A Characterization of Finitely Decidable Congruence Modular Varieties

For every finitely generated, congruence modular variety of finite type we find a finite family of finite rings such that the variety is finitely decidable if and only if is congruence permutable and residually small, all solvable congruences in finite algebras from are Abelian, each congruence above the centralizer of the monolith of a subdirectly irreducible algebra from is comparable with all congruences of , each homomorphic image of a subdirectly irreducible algebra with a non-Abelian monolith has a non-Abelian monolith, and, for each ring from , the variety of -modules is finitely decidable.

     

The Morse spectrum of linear flows on vector bundles

For a linear flow on a vector bundle a spectrum can be defined in the following way: For a chain recurrent component on the projective bundle consider the exponential growth rates associated with (finite time) -chains in , and define the Morse spectrum over as the limits of these growth rates as and . The Morse spectrum of is then the union over all components . This spectrum is a synthesis of the topological approach of Selgrade and Salamon/Zehnder with the spectral concepts based on exponential growth rates, such as the Oseledec spectrum or the dichotomy spectrum of Sacker/Sell. It turns out that contains all Lyapunov exponents of for arbitrary initial values, and the are closed intervals, whose boundary points are actually Lyapunov exponents. Using the fact that is cohomologous to a subflow of a smooth linear flow on a trivial bundle, one can prove integral representations of all Morse and all Lyapunov exponents via smooth ergodic theory. A comparison with other spectral concepts shows that, in general, the Morse spectrum is contained in the topological spectrum and the dichotomy spectrum, but the spectral sets agree if the induced flow on the base space is chain recurrent. However, even in this case, the associated subbundle decompositions of may be finer for the Morse spectrum than for the dynamical spectrum. If one can show that the (closure of the) Floquet spectrum (i.e. the Lyapunov spectrum based on periodic trajectories in ) agrees with the Morse spectrum, then one obtains equality for the Floquet, the entire Oseledec, the Lyapunov, and the Morse spectrum. We present an example (flows induced by vector fields with hyperbolic chain recurrent components on the projective bundle) where this fact can be shown using a version of Bowen's Shadowing Lemma.

     

On polarized surfaces

Let be a smooth projective surface over and an ample Cartier divisor on . If the Kodaira dimension or , the author proved , where . If , then the author studied with . In this paper, we study the polarized surface with , , and .

     

The Bergman kernel function of some Reinhardt domains

The boundary behavior of the Bergman Kernel function of some Reinhardt domains is studied. Upper and lower bounds for the Bergman kernel function are found at the diagonal points . Let be the Reinhardt domain

where , ; and let be the Bergman kernel function of . Then there exist two positive constants and and a function such that

holds for every . Here

and is the defining function for . The constants and depend only on and , not on .

     

Seifert manifolds with fiber spherical space forms

We study the Seifert fiber spaces modeled on the product space . Such spaces are ``fiber bundles' with singularities. The regular fibers are spherical space-forms of , while singular fibers are finite quotients of regular fibers. For each of possible space-form groups of , we obtain a criterion for a group extension of to act on as weakly -equivariant maps, which gives rise to a Seifert fiber space modeled on with weakly -equivariant maps as the universal group. In the course of proving our main results, we also obtain an explicit formula for for a cocompact crystallographic or Fuchsian group . Most of our methods for apply to compact Lie groups with discrete center, and we state some of our results in this general context.

     

Regularity theory and traces of -harmonic functions

In this paper we discuss two different topics concerning -
harmonic functions. These are weak solutions of the partial differential equation

where for some fixed , the function is bounded and for a.e. . First, we present a new approach to the regularity of -harmonic functions for . Secondly, we establish results on the existence of nontangential limits for -harmonic functions in the Sobolev space , for some , where is the unit ball in . Here is allowed to be different from .

     

Abstract functions with continuous differences and Namioka spaces

Let be a semigroup and a topological space. Let be an Abelian topological group. The right differences of a function are defined by for . Let be continuous at the identity of for all in a neighbourhood of . We give conditions on or range under which is continuous for any topological space . We also seek conditions on under which we conclude that is continuous at for arbitrary . This led us to introduce new classes of semigroups containing all complete metric and locally countably compact quasitopological groups. In this paper we study these classes and explore their relation with Namioka spaces.

     

Stratified transversality by isotopy

For an abstract stratified set or a -regular stratification, hence for any -, - or -regular stratification, we prove that after stratified isotopy of , a stratified subspace of , or a stratified map , can be made transverse to a fixed stratified map .

     

Linear additive functionals of superdiffusions and related nonlinear P.D.E.

Let be a second order elliptic differential operator in a bounded smooth domain in and let . We get necessary and sufficient conditions on measures under which there exists a positive solution of the boundary value problem

The conditions are stated both analytically (in terms of capacities related to the Green's and Poisson kernels) and probabilistically (in terms of branching measure-valued processes called -superdiffusions).

We also investigate a closely related subject --- linear additive functionals of superdiffusions. For a superdiffusion in an arbitrary domain in , we establish a 1-1 correspondence between a class of such functionals and a class of -excessive functions (which we describe in terms of their Martin integral representation). The Laplace transform of satisfies an integral equation which can be considered as a substitute for (*).

     

Lévy group action and invariant measures on

For let be defined by . We investigate permutations of , which satisfy as with for (i.e. is in the Lévy group , or for in the subspace of Cesàro-summable sequences. Our main interest are -invariant means on or equivalently -invariant probability measures on . We show that the adjoint of maps measures supported in onto a weak*-dense subset of the space of -invariant measures. We investigate the dynamical system and show that the support set of invariant measures on is the closure of the set of almost periodic points and the set of non-topologically transitive points in . Finally we consider measures which are invariant under .

     

Nonnegative Radix Representations for the Orthant

Let be a nonnegative real matrix which is expanding, i.e. with all eigenvalues , and suppose that is an integer. Let consist of exactly nonnegative vectors in . We classify all pairs such that every in the orthant has at least one radix expansion in base using digits in . The matrix must be a diagonal matrix times a permutation matrix. In addition must be similar to an integer matrix, but need not be an integer matrix. In all cases the digit set can be diagonally scaled to lie in . The proofs generalize a method of Odlyzko, previously used to classify the one--dimensional case.

     

C*-Algebras with the Approximate Positive Factorization Property

We say that a unital -algebra has the approximate positive factorization property (APFP) if every element of is a norm limit of products of positive elements of . (There is also a definition for the nonunital case.) T. Quinn has recently shown that a unital AF algebra has the APFP if and only if it has no finite dimensional quotients. This paper is a more systematic investigation of -algebras with the APFP. We prove various properties of such algebras. For example: They have connected invertible group, trivial , and stable rank 1. In the unital case, the group separates the tracial states. The APFP passes to matrix algebras, and if is an ideal in such that and have the APFP, then so does . We also give some new examples of -algebras with the APFP, including type factors and infinite-dimensional simple unital direct limits of homogeneous -algebras with slow dimension growth, real rank zero, and trivial group. Simple direct limits of homogeneous -algebras with slow dimension growth which have the APFP must have real rank zero, but we also give examples of (nonsimple) unital algebras with the APFP which do not have real rank zero. Our analysis leads to the introduction of a new concept of rank for a -algebra that may be of interest in the future.

     

Constructing product fibrations by means of a generalization of a theorem of Ganea

A theorem of Ganea shows that for the principal homotopy fibration induced from a fibration , there is a product decomposition . We will determine the conditions for a fibration to yield a product decomposition and generalize it to pushouts. Using this approach we recover some decompositions originally proved by very computational methods. The results are then applied to produce, after localization at an odd prime , homotopy decompositions for for some which include the cases . The factors of consist of the homotopy fibre of the attaching map for and combinations of spaces occurring in the Snaith stable decomposition of .

     

Powers in Finitely Generated Groups

In this paper we study the set of -powers in certain finitely generated groups . We show that, if is soluble or linear, and contains a finite index subgroup, then is nilpotent-by-finite. We also show that, if is linear and has finite index (i.e. may be covered by finitely many translations of ), then is soluble-by-finite. The proof applies invariant measures on amenable groups, number-theoretic results concerning the -unit equation, the theory of algebraic groups and strong approximation results for linear groups in arbitrary characteristic.