coefficients

Let be a system of real smooth vector fields, satisfying Hörmander's condition in some bounded domain (). We consider the differential operator

where the coefficients are real valued, bounded measurable functions, satisfying the uniform ellipticity condition:

for a.e. , every , some constant . Moreover, we assume that the coefficients belong to the space VMO (``Vanishing Mean Oscillation'), defined with respect to the subelliptic metric induced by the vector fields . We prove the following local -estimate:

for every , . We also prove the local Hölder continuity for solutions to for any with large enough. Finally, we prove -estimates for higher order derivatives of , whenever and the coefficients are more regular.

     

Baire and

Let be a locally compact Hausdorff space and let be the Banach space of all bounded complex Radon measures on . Let and be the -rings generated by the compact subsets and by the compact subsets of , respectively. The members of are called Baire sets of and those of are called -Borel sets of (since they are precisely the -bounded Borel sets of ). Identifying with the Banach space of all Borel regular complex measures on , in this note we characterize weakly compact subsets of in terms of the Baire and -Borel restrictions of the members of . These characterizations permit us to give a generalization of a theorem of Dieudonné which is stronger and more natural than that given by Grothendieck.

     

Willmore Functionals

We prove some integral inequalities for immersed tori in the three sphere. The functionals considered are generalizations of the Willmore functional.

     

A counterexample machine

We present a general method for constructing families of measure preserving transformations which are and loosely Bernoulli with various ergodic theoretical properties. For example, we construct two transformations which are weakly isomorphic but not isomorphic, and a transformation with no roots. Ornstein's isomorphism theorem says families of Bernoulli shifts cannot have these properties. The construction uses a combination of properties from maps constructed by Ornstein and Shields, and Rudolph, and reduces the question of isomorphism of two transformations to the conjugacy of two related permutations.

     

-algebras that are only weakly semiprojective

We show that the -algebra of continuous functions on the Cantor set is a weakly semiprojective -algebra that is not semiprojective.

     

Operator ideal norms on

Let be a real number such that and its conjugate exponent . We prove that for an operator defined on with values in a Banach space, the image of the unit ball determines whether belongs to any operator ideal and its operator ideal norm. We also show that this result fails to be true in the remaining cases of . Finally we prove that when the result holds in finite dimension, the map which associates to the image of the unit ball the operator ideal norm is continuous with respect to the Hausdorff metric.

     

The

ROSSER and SCHOENFELD have used the fact that the first 3,500,000 zeros of the RIEMANN zeta function lie on the critical line to give estimates on and . With an improvement of the above result by BRENTet al., we are able to improve these estimates and to show that the prime is greater than for . We give further results without proof.

     

estimates for oscillatory integral operators

An endpoint boundedness result is established for a class of oscillatory integral operators.

     

Ultradifferentiable functions on lines in

It is well known that a function whose restriction to every line in is real analytic must itself be real analytic. In this note we study whether this property of real analytic functions is also possessed by some other subclasses of functions. We prove that if is ultradifferentiable corresponding to a sequence on every line in some `uniform way', then is ultradifferentiable corresponding to the sequence

     

convergence of Picard's successive approximations

A simple, elementary proof of the existence, uniqueness, and
smoothness of solutions to ordinary differential equations is given. In fact, it is shown that for a differential equation of class , the successive approximations of Picard converge in the -sense.

     

The -Lagrangian of a convex function

At a given point , a convex function is differentiable in a certain subspace (the subspace along which has 0-breadth). This property opens the way to defining a suitably restricted second derivative of at . We do this via an intermediate function, convex on . We call this function the -Lagrangian; it coincides with the ordinary Lagrangian in composite cases: exact penalty, semidefinite programming. Also, we use this new theory to design a conceptual pattern for superlinearly convergent minimization algorithms. Finally, we establish a connection with the Moreau-Yosida regularization.

     

-identities on associative algebras

Let be an algebra over a field and a finite group of automorphisms and anti-automorphisms of . We prove that if satisfies an essential -polynomial identity of degree , then the -codimensions of are exponentially bounded and satisfies a polynomial identity whose degree is bounded by an explicit function of . As a consequence we show that if is an algebra with involution satisfying a -polynomial identity of degree , then the -codimensions of are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity.

     

Products on -modules

Elmendorf, Kriz, Mandell and May have used their technology of modules over highly structured ring spectra to give new constructions of -modules such as , and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over that are concentrated in degrees divisible by ; this guarantees that various obstruction groups are trivial. We extend these results to the cases where or the homotopy groups are allowed to be nonzero in all even degrees; in this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity, and that the obstructions to commutativity are given by a certain power operation; this was inspired by parallel results of Mironov in Baas-Sullivan theory. We use formal group theory to derive various formulae for this power operation, and deduce a number of results about realising -local -modules as -modules.

     

On type basic hypergeometric orthogonal polynomials

The five parameter family of Koornwinder's multivariable analogues of the Askey-Wilson polynomials is studied with four parameters generically complex. The Koornwinder polynomials form an orthogonal system with respect to an explicit (in general complex) measure. A partly discrete orthogonality measure is obtained by shifting the contour to the torus while picking up residues. A parameter domain is given for which the partly discrete orthogonality measure is positive. The orthogonality relations and norm evaluations for multivariable -Racah polynomials and multivariable big and little -Jacobi polynomials are proved by taking suitable limits in the orthogonality relations for the Koornwinder polynomials. In particular new proofs of several well-known -analogues of the Selberg integral are obtained.

     

Metrizable and -metrizable betweenness spaces

It is proved that the theory of the class of all betweenness spaces metrizable by real-valued metrics does not coincide with the theory of the class of all betweenness spaces metrizable by metrics taking values in any ordered field. This solves a problem raised by Mendris and Zlatov{s}.

     

Level sets of a typical function

We determine the level set structure of a typical function.

     

A maximum principle for finite energy

We show a maximum principle for -harmonic maps with -finite energy. As an application we can generalize a non-existence theorem for harmonic maps with finite Dirichlet integral by Schoen and Yau to those maps.

     

An definition of interpolating Blaschke products

We give a new characterization of interpolating Blaschke products in terms of -norms of their reciprocals. We also obtain a characterization of finite unions of interpolating sequences.

     

Convexity properties of holomorphic mappings in

Not many convex mappings on the unit ball in for are known. We introduce two families of mappings, which we believe are actually identical, that both contain the convex mappings. These families which we have named the ``Quasi-Convex Mappings, Types A and B' seem to be natural generalizations of the convex mappings in the plane. It is much easier to check whether a function is in one of these classes than to check for convexity. We show that the upper and lower bounds on the growth rate of such mappings is the same as for the convex mappings.