Interpolation of weighted and vector-valued Hardy spaces
Serguei V.
Kisliakov;
Quan Hua
Xu
1-34
Abstract: Real and complex interpolation methods, when applied to the couple $ ({H^{{p_0}}}({E_0};{w_0}),{H^{{p_1}}}({E_1};{w_1}))$, give what is expected if $ {E_0}$ and ${E_1}$ are quasi-Banach lattices of measurable functions satisfying certain mild conditions and if $\log (w_0^{1/{p_0}}w_1^{ - 1/{p_1}}) \in {\text{BMO}}\,({w_0},{w_1}$ being weights on the unit circle). The last condition is in fact necessary. (It is expected, of course, that the resulting spaces coincide with the subspaces of analytic functions in the corresponding interpolation spaces for the couple $ ({L^{{p_0}}}({E_0};{w_0}),{L^{{p_1}}}({E_1};{w_1})).)$
Some cubic modular identities of Ramanujan
J. M.
Borwein;
P. B.
Borwein;
F. G.
Garvan
35-47
Abstract: There is a beautiful cubic analogue of Jacobi's fundamental theta function identity: $\theta _3^4 = \theta _4^4 + \theta _2^4$. It is $\displaystyle {\left({\sum\limits_{n,m = - \infty }^\infty {{q^{{n^2} + nm + {m... ...+ (n + \frac{1}{3})(m + \frac{1}{3}) + {{(m + \frac{1}{3})}^2}}}} } \right)^3}.$ Here $\omega = \exp (2\pi i/3)$. In this note we provide an elementary proof of this identity and of a related identity due to Ramanujan. We also indicate how to discover and prove such identities symbolically.
Generalized Casson invariants for ${\rm SO}(3),\;{\rm U}(2),\;{\rm Spin}(4),$ and ${\rm SO}(4)$
Cynthia L.
Curtis
49-86
Abstract: We investigate Casson-type invariants corresponding to the low-rank groups $ {\text{SO}}(3)$, ${\text{SU}}(2) \times {S^1}$, ${\text{U}}(2)$, $ {\text{Spin}}(4)$ and ${\text{SO}}(4)$. The invariants are defined following an approach similar to those of K. Walker and S. Cappell, R. Lee, and E. Miller. We obtain a description for each of the invariants in terms of the $ {\text{SU}}(2)$-invariant. Thus, all of them may be calculated using formulae for the $ {\text{SU}}(2)$-invariant. In defining these invariants, we offer methods which should prove useful for studying the invariants for other non-simply-connected groups once the invariants for the simply-connected covering groups are known.
Intersections of analytically and geometrically finite subgroups of Kleinian groups
James W.
Anderson
87-98
Abstract: We consider the intersection of pairs of subgroups of a Kleinian group of the second kind K whose limit sets intersect, where one subgroup G is analytically finite and the other J is geometrically finite, possibly infinite cyclic. In the case that J is infinite cyclic generated by M, we show that either some power of M lies in G or there is a doubly cusped parabolic element Q of G with the same fixed point as M. In the case that J is nonelementary, we show that the intersection of the limit sets of G and J is equal to the limit set of the intersection $G \cap J$ union with a subset of the rank 2 parabolic fixed points of K. Hence, in both cases, the limit set of the intersection is essentially equal to the intersection of the limit sets. The main facts used in the proof are results of Beardon and Pommerenke [4] and Canary [6], which yield that the Poincaré metric on the ordinary set of an analytically finite Kleinian group G is comparable to the Euclidean distance to the limit set of G.
On superquadratic elliptic systems
Djairo G.
de Figueiredo;
Patricio L.
Felmer
99-116
Abstract: In this article we study the existence of solutions for the elliptic system \begin{displaymath}\begin{array}{*{20}{c}} { - \Delta u = \frac{{\partial H}}{{\... ...quad v = 0\quad {\text{on}}\;\partial \Omega .} \end{array} \end{displaymath} where $\Omega$ is a bounded open subset of ${\mathbb{R}^N}$ with smooth boundary $\partial \Omega$, and the function $ H:{\mathbb{R}^2} \times \bar \Omega \to \mathbb{R}$, is of class ${C^1}$. We assume the function H has a superquadratic behavior that includes a Hamiltonian of the form $\displaystyle H(u,v) = \vert u{\vert^\alpha } + \vert v{\vert^\beta }\quad {\te... ... \frac{1}{\alpha } + \frac{1}{\beta } < 1\;{\text{with}}\;\alpha > 1,\beta > 1.$ We obtain existence of nontrivial solutions using a variational approach through a version of the Generalized Mountain Pass Theorem. Existence of positive solutions is also discussed.
On transformation group $C\sp *$-algebras with continuous trace
Siegfried
Echterhoff
117-133
Abstract: In this paper we answer some questions posed by Dana Williams in [19] concerning the problem under which conditions the transformation group ${C^ \ast }$-algebra ${C^\ast}(G,\Omega )$ of a locally compact transformation group (G, $\Omega$) has continuous trace. One consequence will be, for compact G, that ${C^\ast}(G,\Omega )$ has continuous trace if and only if the stabilizer map is continuous. We also give a complete solution to the problem if G is discrete.
Bounded holomorphic functions on bounded symmetric domains
Joel M.
Cohen;
Flavia
Colonna
135-156
Abstract: Let D be a bounded homogeneous domain in ${\mathbb{C}^n}$, and let $\Delta$ denote the open unit disk. If $z \in D$ and $f:D \to \Delta$ is holomorphic, then ${\beta _f}(z)$ is defined as the maximum ratio $\vert{\nabla _z}(f)x\vert/{H_z}{(x,\bar x)^{1/2}}$, where x is a nonzero vector in ${\mathbb{C}^n}$ and ${H_z}$ is the Bergman metric on D. The number ${\beta _f}(z)$ represents the maximum dilation of f at z. The set consisting of all ${\beta _f}(z)$ for $z \in D$ and $f:D \to \Delta$ holomorphic, is known to be bounded. We let ${c_D}$, be its least upper bound. In this work we calculate ${c_D}$ for all bounded symmetric domains having no exceptional factors and give indication on how to handle the general case. In addition we describe the extremal functions (that is, the holomorphic functions f for which $ {\beta _f} = {c_D}$) when D contains $\Delta$ as a factor, and show that the class of extremal functions is very large when $\Delta$ is not a factor of D.
The hexagonal packing lemma and Rodin Sullivan conjecture
Dov
Aharonov
157-167
Abstract: The Hexagonal Packing Lemma of Rodin and Sullivan [6] states that ${s_n} \to 0$ as $n \to \infty$. Rodin and Sullivan conjectured that $ {s_n} = O(1/n)$. This has been proved by Z-Xu He [2]. Earlier, the present author proved the conjecture under some additional restrictions [1]. In the following we are able to remove these restrictions, and thus give an alternative proof of the RS conjecture. The proof is based on our previous article [1]. It is completely different from the proof of He, and it is mainly based on discrete potential theory, as developed by Rodin for the hexagonal case [4].
Amenable relations for endomorphisms
J. M.
Hawkins
169-191
Abstract: We give necessary and sufficient conditions for an endomorphism to admit an equivalent invariant $\sigma$-finite measure in terms of a generalized Perron-Frobenius operator. The assumptions are that the endomorphism is nonsingular (preserves sets of measure zero), conservative, and finite-to-1. We study two orbit equivalence relations associated to an endomorphism, and their connections to nonsingularity, ergodicity, and exactness. We also discuss Radon-Nikodym derivative cocycles for the relations and the endomorphism, and relate these to the Jacobian of the endomorphism.
Multivariate orthogonal polynomials and operator theory
Yuan
Xu
193-202
Abstract: The multivariate orthogonal polynomials are related to a family of commuting selfadjoint operators. The spectral theorem for these operators is used to prove that a polynomial sequence satisfying a vector-matrix form of the three-term relation is orthonormal with a determinate measure.
Noetherian properties of skew polynomial rings with binomial relations
Tatiana
Gateva-Ivanova
203-219
Abstract: In this work we study standard finitely presented associative algebras over a fixed field K. A restricted class of skew polynomial rings with quadratic relations considered in an earlier work of M. Artin and W. Schelter will be studied. We call them binomial skew polynomial algebras. We establish necessary and sufficient conditions for such an algebra to be a Noetherian domain.
Product recurrence and distal points
J.
Auslander;
H.
Furstenberg
221-232
Abstract: Recurrence is studied in the context of actions of compact semigroups on compact spaces. (An important case is the action of the Stone-Čech compactification of an acting group.) If the semigroup E acts on the space X and F is a closed subsemigroup of E, then x in X is said to be F-recurrent if $px = x$ for some $p \in F$, and product F-recurrent if whenever y is an F-recurrent point (in some space Y on which E acts) the point (x, y) in the product system is F-recurrent. The main result is that, under certain conditions, a point is product F-recurrent if and only if it is a distal point.
Amenability and the structure of the algebras $A\sb p(G)$
Brian
Forrest
233-243
Abstract: A number of characterizations are given of the class of amenable locally compact groups in terms of the ideal structure of the algebras ${A_p}(G)$. An almost connected group is amenable if and only if for some $ 1 < p < \infty$ and some closed ideal I of ${A_p}(G)$, I has a bounded approximate identity. Furthermore, G is amenable if and only if every derivation of ${A_p}(G)$ into a Banach ${A_p}(G)$-bimodule is continuous.
Microlocal analysis of some isospectral deformations
F.
Marhuenda
245-275
Abstract: We study the microlocal structure of the examples of isospectral deformations of Riemannian manifolds given by D. DeTurck and C. Gordon in [DeT-Gl]. The Schwartz kernel of the intertwining operators considered by them may be written as an oscillatory integral with a singular phase function and product type amplitude. In certain instances, we identify them as belonging to the space of Fourier integral operators associated with various pairwise intersecting Lagrangians. After formulating a class of operators incorporating the most relevant features of the operators above, we establish a composition calculus for this class and show that is not necessary to introduce new Lagrangians in the composition.
Witt equivalence of global fields. II. Relative quadratic extensions
Kazimierz
Szymiczek
277-303
Abstract: This paper explores the consequences of the Hasse Principle for Witt equivalence of global fields in the case of relative quadratic extensions. We are primarily interested in generating the Witt equivalence classes of quadratic extensions of a given number field, and we study the structure of the class, the number of classes, and the structure of the set of classes. Along the way, we reprove several results obtained earlier in the absolute case of the rational ground field, giving unified and short proofs based on the Hasse Principle.
Semirigid spaces
Věra
Trnková
305-325
Abstract: Semirigid spaces are introduced and used as a means to construct two metrizable spaces with isomorphic monoids of continuous self-maps and nonisomorphic clones; this resolves Problem 1 in [13]. The clone of any free variety of a given type with sufficiently many constants is shown to be isomorphic to the clone of a metrizable (semirigid) space.
Periodic orbits for Hamiltonian systems in cotangent bundles
Christophe
Golé
327-347
Abstract: We prove the existence of at least $ \operatorname{cl}(M)$ periodic orbits for certain time-dependent Hamiltonian systems on the cotangent bundle of an arbitrary compact manifold M. These Hamiltonians are not necessarily convex but they satisfy a certain boundary condition given by a Riemannian metric on M. We discretize the variational problem by decomposing the time-1 map into a product of "symplectic twist maps". A second theorem deals with homotopically non-trivial orbits of negative curvature.
Pseudocircles in dynamical systems
Judy A.
Kennedy;
James A.
Yorke
349-366
Abstract: We construct an example of a $ {C^\infty }$ map on a 3-manifold which has an invariant set with an uncountable number of components, each of which is a pseudocircle. Furthermore, any map which is sufficiently close (in the ${C^1}$-metric) to the constructed map has a similar set.
Calculating discriminants by higher direct images
Jerzy
Weyman
367-389
Abstract: The author uses the homological algebra to construct for any line bundle $\mathcal{L}$ on a nonsingular projective variety X the complex $\mathbb{F}(\mathcal{L})$ whose determinant is equal to the equation of the dual variety ${X^{\text{V}}}$. This generalizes the Cayley-Koszul complexes defined by Gelfand, Kapranov and Zelevinski. The formulas for the codimension and degree of ${X^{\text{V}}}$ in terms of complexes $\mathbb{F}(\mathcal{L})$ are given. In the second part of the article the general technique is applied to classical discriminants and hyperdeterminants.
Fibrations of classifying spaces
Kenshi
Ishiguro;
Dietrich
Notbohm
391-415
Abstract: We investigate fibrations of the form $Z \to Y \to X$, where two of the three spaces are classifying spaces of compact connected Lie groups. We obtain certain finiteness conditions on the third space which make it also a classifying space. Our results allow to express some of the basic notions in group theory in terms of homotopy theory, i.e., in terms of classifying spaces. As an application we prove that every retract of the classifying space of a compact connected Lie group is again a classifying space.
On the number of singularities in meromorphic foliations on compact complex surfaces
Edoardo
Ballico
417-432
Abstract: Here we study meromorphic foliations with singularities on a smooth compact complex surface. Aim: study the locations of the singularities, using vector bundle techniques and techniques introduced in the cohomological study of projective geometry.
${\germ F}$-categories and ${\germ F}$-functors in the representation theory of Lie algebras
Ben
Cox
433-453
Abstract: The fields of algebra and representation theory contain abundant examples of functors on categories of modules over a ring. These include of course Horn, Ext, and Tor as well as the more specialized examples of completion and localization used in the setting of representation theory of a semisimple Lie algebra. In this article we let $\mathfrak{a}$ be a Lie subalgebra of a Lie algebra $\mathfrak{g}$ and $\Gamma$ be a functor on some category of $\mathfrak{a}$-modules. We then consider the following general question: For a $ \mathfrak{g}$-module E what hypotheses on $\Gamma$ and E are sufficient to insure that $\Gamma (E)$ admits a canonical structure as a $\mathfrak{g}$-module? The article offers an answer through the introduction of the notion of $\mathfrak{F}$-categories and $\mathfrak{F}$-functors. The last section of the article treats various examples of this theory.
Solutions to the quantum Yang-Baxter equation arising from pointed bialgebras
David E.
Radford
455-477
Abstract: Let $R:M \otimes M \to M \otimes M$ be a solution to the quantum Yang-Baxter equation, where M is a finite-dimensional vector space over a field k. We introduce a quotient $ {A^{{\text{red}}}}(R)$ of the bialgebra $A(R)$ constructed by Fadeev, Reshetihkin and Takhtajan, whose characteristics seem to more faithfully reflect properties R possesses as a linear operator. We characterize all R such that ${A^{{\text{red}}}}(R)$ is a pointed bialgebra, and we determine all solutions R to the quantum Yang-Baxter equation when ${A^{{\text{red}}}}(R)$ is pointed and $\dim M = 2$ (with a few technical exceptions when k has characteristic 2). Extensions of such solutions to the quantum plane are studied.