Convex curves, Radon transforms and convolution operators defined by singular measures

Let be a convex curve in the plane and let be the arc-length measure of Let us rotate by an angle and let be the corresponding measure. Let . Then This is optimal for an arbitrary . Depending on the curvature of , this estimate can be improved by introducing mixed-norm estimates of the form where and are conjugate exponents.     

Asymptotic behavior of Fourier transforms of self-similar measures

Let be a self-similar probability measure on satisfying where 0$"> and Let be the Fourier transform of A necessary and sufficient condition for to approach zero at infinity is given. In particular, if and for then 0$"> if and only if is a PV-number and is not a factor of . This generalizes the corresponding theorem of Erdös and Salem for the case

     

Hilbert modules over a class of semicrossed products

Given the disk algebra and an automorphism , there is associated a non-self-adjoint operator algebra called the semicrossed product of with . Buske and Peters showed that there is a one-to-one correspondence between the contractive Hilbert modules over and pairs of contractions and on satisfying . In this paper, we show that the orthogonally projective and Shilov Hilbert modules over correspond to pairs of isometries on satisfying . The problem of commutant lifting for is left open, but some related results are presented.     

A lifting theorem for symmetric commutants

Let be bounded operators on a Hilbert space such that . Given a symmetry on , i.e., , we define the -symmetric commutant of to be the operator space

In this paper we obtain lifting theorems for symmetric commutants. The result extends the Sz.-Nagy-Foias commutant lifting theorem (), the anticommutant lifting theorem of Sebestyén ( ), and the noncommutative commutant lifting theorem ( ). Sarason's interpolation theorem for is extended to symmetric commutants on Fock spaces.     

Compact operators whose real and imaginary parts are positive

Let be a compact operator on a Hilbert space such that the operators and are positive. Let be the singular values of and the eigenvalues of , all enumerated in decreasing order. We show that the sequence is majorised by . An important consequence is that, when is less than or equal to , and when this inequality is reversed.

     

The Postnikov Tower and the Steenrod problem

The Steenrod problem asks: given a -module, when does there exist a Moore space realizing the module? By using the equivariant Postnikov Tower, it is shown that a -module is -realizable if and only if it is -realizable for all -Sylow subgroups , for all primes .

     

A dichotomy theorem for subsets of the power set of the natural numbers

The following dichotomy is established for any pair , of hereditary families of finite subsets of : Given , an infinite subset of , there exists an infinite subset of so that either , or , where denotes the set of all finite subsets of .

     

Existence and uniqueness theorems for singular anisotropic quasilinear elliptic boundary value problems

On bounded domains we consider the anisotropic problems in with 1$"> and on and in with and on . Moreover, we generalize these boundary value problems to space-dimensions 2$">. Under geometric conditions on and monotonicity assumption on we prove existence and uniqueness of positive solutions.     

Double exponential sums over thin sets

We estimate double exponential sums of the form

where is of multiplicative order modulo the prime and and are arbitrary subsets of the residue ring modulo . In the special case , our bound is nontrivial for with any fixed 0$">, while if in addition we have it is nontrivial for .

     

A note on branching theorems

Let be a complex, simply connected semisimple analytic group with a closed connected reductive subgroup. Suppose is an irreducible holomorphic -module and an irreducible holomorphic -module. We prove that Hom possesses the structure of an irreducible -module whenever is . Moreover, for all and if and only if is commutative.

     

Holomorphic perturbation of Fourier coefficients

Let be the unit circle, let be a Banach space continuously embedded in and suppose that is a Banach -module under convolution. We show that if and is holomorphic in a neighbourhood of with and then

     

Subdivision schemes for iterated function systems

We identify iterated function systems and regular Borel measures such that the matrix subdivision process relative to a finite family converges if and only if satisfies certain spectral properties.     

Les transformations de Cremona stellaires

On construit explicitement toutes les transformations de Cremona de qui satisfont à la propriété suivante: il existe tels que les droites par sont envoyées sur les droites par . On caractérise de plusieurs manières ces transformations et pour chaque entier non-négatif on donne des formules pour la dimension de l'ensemble constitué de celles qui ont degré .

ABSTRACT. We construct the Cremona transformations of satisfying the following property: there exist such that the image of all straight lines through are straight lines through . We characterise these transformations, and for all non-negative integer we give a formula for the dimension of the set of those whose degree is .

     

Diagonal type conditions on group C-algebras

Let be a locally compact group with and its enveloping and reduced C-algebras respectively. We show that if is residually finite dimensional, then is maximally almost periodic, and is residually finite dimensional if and only if is both amenable and maximally almost periodic. Letting be the left regular representation of , we show that a certain quasidiagonality condition on implies that is amenable.

     

The asymptotic behavior of a class of nonlinear delay difference equations

The asymptotic behavior of difference equations of type 0, \end{equation*}">is studied, where and each are continuous real functions with decreasing and increasing. Results include sufficient conditions for permanence, oscillations and global attractivity.

     

Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy

In this paper we consider the long time behavior of solutions of the initial value problem for semi-linear wave equations of the form

Here 0.$">

We prove that if m \ge 1,$"> then for any 0$"> there are choices of initial data from the energy space with initial energy such that the solution blows up in finite time. If we replace by , where is a sufficiently slowly decreasing function, an analogous result holds.

     

A new construction of semi-free actions on Menger manifolds

A new construction of semi-free actions on Menger manifolds is presented. As an application we prove a theorem about simultaneous coexistence of countably many semi-free actions of compact metric zero-dimensional groups with the prescribed fixed-point sets: Let be a compact metric zero-dimensional group, represented as the direct product of subgroups , a -manifold and (resp., ) its pseudo-interior (resp., pseudo-boundary). Then, given closed subsets of , there exists a -action on such that (1) and are invariant subsets of ; and (2) each is the fixed point set of any element .

     

Parameter dependence of solutions of partial differential equations in spaces of real analytic functions

Let be an open set and let denote the class of real analytic functions on . It is proved that for every surjective linear partial differential operator and every family depending holomorphically on there is a solution family depending on in the same way such that The result is a consequence of a characterization of Fréchet spaces such that the class of ``weakly' real analytic -valued functions coincides with the analogous class defined via Taylor series. An example shows that the analogous assertions need not be valid if is replaced by another set.

     

Conformal metrics and Ricci tensors in the pseudo-Euclidean space

We consider constant symmetric tensors on , , and we study the problem of finding metrics conformal to the pseudo-Euclidean metric such that . We show that such tensors are determined by the diagonal elements and we obtain explicitly the metrics . As a consequence of these results we get solutions globally defined on for the equation Moreover, we show that for certain unbounded functions defined on , there are metrics conformal to the pseudo-Euclidean metric with scalar curvature .