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.

     

There are no denting points in the unit ball of

For an infinite compact set and for any Banach space we show that the unit ball of the space of -valued functions that are continuous when is equipped with the weak topology, has no denting points.

     

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.

     

The growth theorem of convex mappings on the unit ball in

Let be an arbitrary norm on . Let be a normalized biholomorphic convex mapping on the unit ball in with respect to the norm . We will give an upper bound of the growth of .

     

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

     

Identity principles for commuting holomorphic self-maps of the unit disc

Let f,g be two commuting holomorphic self-maps of the unit disc. If f and g agree at the common Wolff point up to a certain order of derivatives (no more than 3 if the Wolff point is on the unit circle), then f≡g.     

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.

     

Support functionals and smooth points in abstract spaces

By presenting some properties of support functionals in abstract spaces, we get some sufficient and necessary conditions for smooth points in abstract (function) spaces. Moreover, the notion of the smallest support semi-norm is introduced and an explicit form for this functional in abstract function spaces is also given.

     

Willmore Functionals

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

     

Level sets of a typical function

We determine the level set structure of a typical function.

     

Nonexpansive, -continuous antirepresentations have common fixed points

Let be a closed convex subset of a Banach (dual Banach) space . By we denote an antirepresentation of a semitopological semigroup as nonexpansive mappings on . Suppose that the mapping is jointly continuous when has the weak (weak*) topology and the Banach space of bounded right uniformly continuous functions on has a right invariant mean. If is weakly compact (for some the set is weakly* compact) and norm separable, then has a common fixed point in .

     

Tautness and complete hyperbolicity of domains in

We prove that the existence of a local peak holomorphic function at each boundary point of an unbounded domain and at infinity implies the complete hyperbolicity of this domain, and we give a link between local tautness and global tautness of a domain. We end the note with some examples of taut and complete hyperbolic domains arising from the study of domains with noncompact automorphisms group.

     

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.

     

Involutions with

Let be a smooth involution on a closed -dimensional manifold such that all Stiefel-Whitney classes of the tangent bundle to each component of the fixed point set of vanish in positive dimension. In this paper, we estimate the least possible lower bound of dim if does not bound.

     

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.

     

Homoclinic points of algebraic -actions

Let be an action of by continuous automorphisms of a compact abelian group . A point in is called homoclinic for if as . We study the set of homoclinic points for , which is a subgroup of . If is expansive, then is at most countable. Our main results are that if is expansive, then (1) is nontrivial if and only if has positive entropy and (2) is nontrivial and dense in if and only if has completely positive entropy. In many important cases is generated by a fundamental homoclinic point which can be computed explicitly using Fourier analysis. Homoclinic points for expansive actions must decay to zero exponentially fast, and we use this to establish strong specification properties for such actions. This provides an extensive class of examples of -actions to which Ruelle's thermodynamic formalism applies. The paper concludes with a series of examples which highlight the crucial role of expansiveness in our main results.

     

Trigonometric moment problems for arbitrary finite subsets of

We consider finite subsets satisfying the extension property, i.e. the property that every collection of complex numbers which is positive-definite on is the restriction to of the Fourier coefficients of some positive measure on . A simple algebraic condition on the set of trigonometric polynomials with non-zero coefficients restricted to is shown to imply the failure of the extension property for . This condition is used to characterize the one-dimensional sets satisfying the extension property and to provide many examples of sets failing to satisfy it in higher dimensions. Another condition, in terms of unitary matrices, is investigated and is shown to be equivalent to the extension property. New two-dimensional examples of sets satisfying the extension property are given as well as explicit examples of collections for which the extension property fails.

     

Ample divisors on the blow up of at points

Fix integers with and ; if assume . Let be general points of the complex projective space and let be the blow up of at with exceptional divisors , . Set . Here we prove that the divisor is ample if and only if , i.e. if and only if .

     

estimates for oscillatory integral operators

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