On covariant functions and distributions under the action of a compact group

Let G be a compact subgroup of GLn(R) acting linearly on a finite dimensional complex vector space E. B. Malgrange has shown that the space C^∞G(Rn,E) of C^∞ and G-covariant functions is a finite module over the ring C^∞G(Rn) of C^∞ and G-invariant functions. First, we generalize this result for the Schwartz space SG(Rn,E) of G-covariant functions. Secondly, we prove that any G-covariant distribution can be decomposed into a sum of G-invariant distributions multiplied with a fixed family of G-covariant polynomials. This gives a generalization of an Oksak result proved in [4].     

The problem of harmonic analysis on the infinite-dimensional unitary group

The goal of harmonic analysis on a (noncommutative) group is to decompose the most “natural” unitary representations of this group (like the regular representation) on irreducible ones. The infinite-dimensional unitary group U(∞) is one of the basic examples of “big” groups whose irreducible representations depend on infinitely many parameters. Our aim is to explain what the harmonic analysis on U(∞) consists of.We deal with unitary representations of a reasonable class, which are in 1-1 correspondence with characters (central, positive definite, normalized functions on U(∞)). The decomposition of any representation of this class is described by a probability measure (called spectral measure) on the space of indecomposable characters. The indecomposable characters were found by Dan Voiculescu in 1976.The main result of the present paper consists in explicitly constructing a 4-parameter family of “natural” representations and computing their characters. We view these representations as a substitute of the nonexisting regular representation of U(∞). We state the problem of harmonic analysis on U(∞) as the problem of computing the spectral measures for these “natural” representations. A solution to this problem is given in the next paper (Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes, math/0109194, to appear in Ann. Math.), joint with Alexei Borodin.We also prove a few auxiliary general results. In particular, it is proved that the spectral measure of any character of U(∞) can be approximated by a sequence of (discrete) spectral measures for the restrictions of the character to the compact unitary groups U(N). This fact is a starting point for computing spectral measures.     

Pointwise Lipschitz functions on metric spaces

For a metric space X, we study the space D^∞(X) of bounded functions on X whose pointwise Lipschitz constant is uniformly bounded. D^∞(X) is compared with the space LIP^∞(X) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare D^∞(X) with the Newtonian-Sobolev space N1,∞(X). In particular, if X supports a doubling measure and satisfies a local Poincaré inequality, we obtain that D^∞(X)=N1,∞(X).     

Moderate deviations and Erdös-Rényi-Shepp laws for weighted sums

Let X₀,X₁,… be i.i.d. random variables with E(X₀)=0, E(X²₀)=1 and E(exp{tX₀})<∞ for any |t|<t₀. We prove that the weighted sums V(n)=∑_j=0^∞a_j(n)X_j, n?1 obey a moderately large deviation principle if the weights satisfy certain regularity conditions. Then we prove a new version of the Erdös-Rényi-Shepp laws for the weighted sums.     

A generalization of the Strassen converse

This note is devoted to a generalization of the Strassen converse. Let gn:R^∞→[0,∞], n?1 be a sequence of measurable functions such that, for every n?1, and for all x,y∈R^∞, where 0<C<∞ is a constant which is independent of n. Let be a sequence of i.i.d. random variables. Assume that there exist r?1 and a function ?:[0,∞)→[0,∞) with limt→∞?(t)=∞, depending only on the sequence such that lim supn→∞gn(X₁,X₂,…)=?(Er|X|) a.s. whenever Er|X|<∞ and EX=0. We prove the converse result, namely that lim supn→∞gn(X₁,X₂,…)<∞ a.s. implies Er|X|<∞ (and EX=0 if, in addition, lim supn→∞gn(c,c,…)=∞ for all c≠0). Some applications are provided to illustrate this result.     

Multiplication operators in Köthe-Bochner spaces

Let X be a Banach space and E an order continuous Banach function space over a finite measure μ. We prove that an operator T in the Köthe-Bochner space E(X) is a multiplication operator (by a function in L^∞(μ)) if and only if the equality T(g〈f,x^∗〉x)=g〈T(f),x^∗〉x holds for every g∈L^∞(μ), f∈E(X), x∈X and x^∗∈X^∗.     

Localization of André-Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces

Let K(n) be the nth Morava K-theory at a prime p, and let T(n) be the telescope of a v_n-self map of a finite complex of type n. In this paper we study the K(n)_*-homology of Ω^∞X, the 0th space of a spectrum X, and many related matters.We give a sampling of our results.Let PX be the free commutative S-algebra generated by X: it is weakly equivalent to the wedge of all the extended powers of X. We construct a natural map

s_n(X):L_T(n)P(X)→L_T(n)Σ^∞(Ω^∞X)₊     

Remarks on Ramanujam-Kawamata-Viehweg Vanishing Theorem

In this article we prove a general result on a nef vector bundle E on a projective manifold X of dimension n depending on the vector space H^n,n(X,E): It is also shown that H^n,n(X,E) = 0 for an indecomposable nef rank 2 vector bundles E on some specific type of n dimensional projective manifold X. The same vanishing shown to hold for indecomposable nef and big rank 2 vector bundles on any variety with trivial canonical bundle.     

Global solvability for first order real linear partial differential operators

F. Treves, in [17], using a notion of convexity of sets with respect to operators due to B. Malgrange and a theorem of C. Harvey, characterized globally solvable linear partial differential operators on C^∞(X), for an open subset X of Rn.Let P=L+c be a linear partial differential operator with real coefficients on a C^∞ manifold X, where L is a vector field and c is a function. If L has no critical points, J. Duistermaat and L. Hörmander, in [2], proved five equivalent conditions for global solvability of P on C^∞(X).Based on Harvey-Treves's result we prove sufficient conditions for the global solvability of P on C^∞(X), in the spirit of geometrical Duistermaat-Hörmander's characterizations, when L is zero at precisely one point. For this case, additional non-resonance type conditions on the value of c at the equilibrium point are necessary.     

The topology of continua that are approximated by disjoint subcontinua

Suppose that is a collection of disjoint subcontinua of continuum X such that limi→∞dH(Yi,X)=0 where dH is the Hausdorff metric. Then the following are true:

(1)

X is non-Suslinean.

(2)

If each Yi is chainable and X is finitely cyclic, then X is indecomposable or the union of 2 indecomposable subcontinua.

(3)

If X is G-like, then X is indecomposable.

(4)

If all lie in the same ray and X is finitely cyclic, then X is indecomposable.

     

2-Descent on elliptic curves and rational points on certain Kummer surfaces

The first part of this paper further refines the methodology for 2-descents on elliptic curves with rational 2-division points which was introduced in [J.-L. Colliot-Thélène, A.N. Skorobogatov, Peter Swinnerton-Dyer, Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points, Invent. Math. 134 (1998) 579-650]. To describe the rest, let E⁽¹⁾ and E⁽²⁾ be elliptic curves, D⁽¹⁾ and D⁽²⁾ their respective 2-coverings, and X be the Kummer surface attached to D⁽¹⁾×D⁽²⁾. In the appendix we study the Brauer-Manin obstruction to the existence of rational points on X. In the second part of the paper, in which we further assume that the two elliptic curves have all their 2-division points rational, we obtain sufficient conditions for X to contain rational points; and we consider how these conditions are related to Brauer-Manin obstructions. This second part depends on the hypothesis that the relevent Tate-Shafarevich group is finite, but it does not require Schinzel's Hypothesis.     

Gorenstein algebras of finite Cohen-Macaulay type

An Artin algebra A is said to be CM-finite if there are only finitely many isomorphism classes of indecomposable finitely generated Gorenstein-projective A-modules. Inspired by Auslander's idea on representation dimension, we prove that for 2?n<∞, A is a CM-finite n-Gorenstein algebra if and only if there is a resolving Gorenstein-projective A-module E such that gl.dimEndAop(E)?n.     

One characterization of symmetry

In this note we present one characterization of symmetry of probability distributions in Euclidean spaces which is formulated as follows. Let X and Y be independent and identically distributed random elements in a separable Euclidean space E. If Ee^h|X|<∞, h>0, then the distribution of X is symmetric if and only if E|(X−Y,t)|^p=E|(X+Y,t)|^p for some 0<p<2 and for any t∈E. The criterion is not correct when at least one of the conditions 0<p<2 or Ee^h|X|<∞ breaks.     

An intermediate Baum-Katz theorem

We extend the classical Hsu-Robbins-Erd?s theorem to the case when all moments exist, but the moment generating function does not, viz., we assume that Eexp{(log⁺|X|)^α}<∞ for some α>1. We also present multi-index versions of the same and of a related result due to Lanzinger in which the assumption is that Eexp{|X|^α}<∞ for some α∈(0,1).     

Integral representations for a class of operators

Let X be a completely regular Hausdorff space, E Hausdorff a quasi-complete locally convex space and Cb(X,E) all E-valued bounded continuous functions on X with strict topologies βt, , . We prove that a linear continuous mapping T:Cb(X,E)→E arises from a scalar measure μ∈(Cb^′(X),βz)(z=t,∞,τ) if and only if g(T(f))=0 whenever g○f=0 for any f∈Cb(X,E), g∈E^′.     

Proper asymptotic unitary equivalence in KK-theory and projection lifting from the corona algebra

In this paper we generalize the notion of essential codimension of Brown, Douglas, and Fillmore using KK-theory and prove a result which asserts that there is a unitary of the form ‘identity + compact’ which gives the unitary equivalence of two projections if the ‘essential codimension’ of two projections vanishes for certain C^∗-algebras employing the proper asymptotic unitary equivalence of KK-theory found by M. Dadarlat and S. Eilers. We also apply our result to study the projections in the corona algebra of C(X)⊗B where X is [0,1], (−∞,∞), [0,∞), and [0,1]/{0,1}.     

Convexity of the bounds induced by Markov's inequality

X is a nonnegative random variable such that EX^t < ∞ for 0≤ t < λ ≤ ∞. The (l??) quantile of the distribution of X is bounded above by [?^?1 EX^t]^1?t. We show that there exist positive ?₁ ≥ ?₂ such that for all 0 <?≤?₁ the function g(t) = [?^-1EX^t]^1?t is log-convex in [0, c] and such that for all 0 < ? ≤ ?₂ the function log g(t) is nonincreasing in [0, c].     

On the symmetric hyperspace of the circle

By X(n), n?1, we denote the n-th symmetric hyperspace of a metric space X as the space of non-empty finite subsets of X with at most n elements endowed with the Hausdorff metric. In this paper we shall describe the n-th symmetric hyperspace S¹(n) as a compactification of an open cone over ΣDn−2, here Dn−2 is the higher-dimensional dunce hat introduced by Andersen, Marjanovi? and Schori (1993) [2] if n is even, and Dn−2 has the homotopy type of Sn−2 if n is odd (see Andersen et al. (1993) [2]). Then we can determine the homotopy type of S¹(n) and detect several topological properties of S¹(n).     

On one-point metrizable extensions of locally compact metrizable spaces

For a non-compact metrizable space X, let E(X) be the set of all one-point metrizable extensions of X, and when X is locally compact, let EK(X) denote the set of all locally compact elements of E(X) and be the order-anti-isomorphism (onto its image) defined in [M. Henriksen, L. Janos, R.G. Woods, Properties of one-point completions of a non-compact metrizable space, Comment. Math. Univ. Carolin. 46 (2005) 105-123; in short HJW]. By definition λ(Y)=_?n<ωclβX(Un∩X)\X, where Y=X∪{p}∈E(X) and {Un}_n<ω is an open base at p in Y. We characterize the elements of the image of λ as exactly those non-empty zero-sets of βX which miss X, and the elements of the image of EK(X) under λ, as those which are moreover clopen in βX\X. This answers a question of [HJW]. We then study the relation between E(X) and EK(X) and their order structures, and introduce a subset ES(X) of E(X). We conclude with some theorems on the cardinality of the sets E(X) and EK(X), and some open questions.     

Extreme points of Banach lattices related to conditional expectations

Let (X,F,μ) be a complete probability space, B a sub-σ-algebra, and Φ the probabilistic conditional expectation operator determined by B. Let K be the Banach lattice {f∈L¹(X,F,μ):‖Φ(|f|)_‖∞<∞} with the norm ‖f‖=‖Φ(|f|)_‖∞. We prove the following theorems:

(1)

The closed unit ball of K contains an extreme point if and only if there is a localizing set E for B such that supp(Φ(χE))=X.

(2)

Suppose that there is n∈N such that f?nΦ(f) for all positive f in L^∞(X,F,μ). Then K has the uniformly λ-property and every element f in the complex K with is a convex combination of at most 2n extreme points in the closed unit ball of K.