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).

     

Packing dimension and Cartesian products

We show that for any analytic set in , its packing dimension can be represented as , where the supremum is over all compact sets in , and denotes Hausdorff dimension. (The lower bound on packing dimension was proved by Tricot in 1982.) Moreover, the supremum above is attained, at least if . In contrast, we show that the dual quantity , is at least the ``lower packing dimension' of , but can be strictly greater. (The lower packing dimension is greater than or equal to the Hausdorff dimension.)

     

Packing measure of the sample paths of fractional Brownian motion

Let be a fractional Brownian motion of index in If , then there exists a positive finite constant such that with probability 1,

where and - is the -packing measure of .

     

Properties of extremal functions for some nonlinear functionals on Dirichlet spaces

Let be the set of holomorphic functions on the unit disc with and Dirichlet integral not exceeding one, and let be the set of complex-valued harmonic functions on the unit disc with and Dirichlet integral not exceeding one. For a (semi)continuous function , define the nonlinear functional on or by . We study the existence and regularity of extremal functions for these functionals, as well as the weak semicontinuity properties of the functionals. We also state a number of open problems.

     

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 .

     

Cesàro Summability of Two-dimensional Walsh-Fourier Series

We introduce p-quasi-local operators and the two-dimensional
dyadic Hardy spaces defined by the dyadic squares. It is proved that, if a sublinear operator is p-quasi-local and bounded from to , then it is also bounded from to . As an application it is shown that the maximal operator of the Cesàro means of a martingale is bounded from to and is of weak type (1,1) provided that the supremum in the maximal operator is taken over a positive cone. So we obtain the dyadic analogue of a summability result with respect to two-dimensional trigonometric Fourier series due to Marcinkievicz and Zygmund; more exactly, the Cesàro means of a function converge a.e. to the function in question, provided again that the limit is taken over a positive cone. Finally, it is verified that if we take the supremum in a cone, but for two-powers, only, then the maximal operator of the Cesàro means is bounded from to for every .

     

Package deal theorems and splitting orders in dimension 1

Let be a module-finite algebra over a commutative noetherian ring of Krull dimension 1. We determine when a collection of finitely generated modules over the localizations , at maximal ideals of , is the family of all localizations of a finitely generated -module . When is semilocal we also determine which finitely generated modules over the -adic completion of are completions of finitely generated -modules.

If is an -order in a semisimple artinian ring, but not contained in a maximal such order, several of the basic tools of integral representation theory behave differently than in the classical situation. The theme of this paper is to develop ways of dealing with this, as in the case of localizations and completions mentioned above. In addition, we introduce a type of order called a ``splitting order' of that can replace maximal orders in many situations in which maximal orders do not exist.

     

Cohomological dimension and metrizable spaces. II

The main result of the first part of the paper is a generalization of the classical result of Menger-Urysohn : . Theorem. Suppose are subsets of a metrizable space and and are CW complexes. If is an absolute extensor for and is an absolute extensor for , then the join is an absolute extensor for .

As an application we prove the following analogue of the Menger-Urysohn Theorem for cohomological dimension: Theorem. Suppose are subsets of a metrizable space. Then

for any ring with unity and

for any abelian group .

The second part of the paper is devoted to the question of existence of universal spaces: Theorem. Suppose is a sequence of CW complexes homotopy dominated by finite CW complexes. Then

a.

Given a separable, metrizable space such that , , there exists a metrizable compactification of such that , .

b.

There is a universal space of the class of all compact metrizable spaces such that for all .

c.

There is a completely metrizable and separable space such that for all with the property that any completely metrizable and separable space with for all embeds in as a closed subset.

     

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.

     

Topological entropy of standard type monotone twist maps

We study invariant measures of families of monotone twist maps with periodic Morse potential . We prove that there exist a constant such that the topological entropy satisfies . In particular, for . We show also that there exist arbitrary large such that has nonuniformly hyperbolic invariant measures with positive metric entropy. For large , the measures are hyperbolic and, for a class of potentials which includes , the Lyapunov exponent of the map with invariant measure grows monotonically with .

     

The local dimensions of the Bernoulli convolution associated with the golden number

Let be a sequence of i.i.d. random variables each taking values of 1 and with equal probability. For satisfying the equation , let be the probability measure induced by . For any in the range of , let

be the local dimension of at whenever the limit exists. We prove that

where , are respectively the maximum and minimum values of the local dimensions. If , then is the golden number, and the approximate numerical values are and .

     

Sharp upper bound for the first non-zero Neumann eigenvalue for bounded domains in rank-1 symmetric spaces

In this paper, we prove that for a bounded domain in a rank- symmetric space, the first non-zero Neumann eigenvalue where denotes the geodesic ball of radius such that

and equality holds iff . This result generalises the works of Szego, Weinberger and Ashbaugh-Benguria for bounded domains in the spaces of constant curvature.

     

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 .

     

Immersed

We give two congruence formulas concerning the number of non-trivial double point circles and arcs of a smooth map with generic singularities --- the Whitney umbrellas --- of an -manifold into , which generalize the formulas by Szücs for an immersion with normal crossings. Then they are applied to give a new geometric proof of the congruence formula due to Mahowald and Lannes concerning the normal Euler number of an immersed -manifold in . We also study generic projections of an embedded -manifold in into and prove an elimination theorem of Whitney umbrella points of opposite signs, which is a direct generalization of a recent result of Carter and Saito concerning embedded surfaces in . The problem of lifting a map into to an embedding into is also studied.

     

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).

     

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.

     

Poincaré's closed geodesic on a convex surface

We present a new proof for the existence of a simple closed geodesic on a convex surface . This result is due originally to Poincaré. The proof uses the -dimensional Riemannian manifold of piecewise geodesic closed curves on with a fixed number of corners, chosen sufficiently large. In we consider a submanifold formed by those elements of which are simple regular and divide into two parts of equal total curvature . The main burden of the proof is to show that the energy integral , restricted to , assumes its infimum. At the end we give some indications of how our methods yield a new proof also for the existence of three simple closed geodesics on .

     

Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities

We prove various results on the existence and multiplicity of harmonic and subharmonic solutions to the second order nonautonomous equation , as or where is a smooth function defined on a open interval The hypotheses we assume on the nonlinearity allow us to cover the case (or ) and having superlinear growth at infinity, as well as the case (or ) and having a singularity in (respectively in ). Applications are given also to situations like (including the so-called ``jumping nonlinearities'). Our results are connected to the periodic Ambrosetti - Prodi problem and related problems arising from the Lazer - McKenna suspension bridges model.