Approximation results in Hilbert cube manifolds
T. A.
Chapman
303-334
Abstract: The purpose of this paper is to answer questions of the following type for maps $ f:\,M \to \,N$ between Q-manifolds: When is f close to an approximate fibration? When is f close to a fiber bundle map? When is f close to a block bundle map? For the second and third questions a series of obstructions are encountered in the lower algebraic K-theoretic functors $ {K_{ - \,i}}$ of Bass.
Splitting criteria for $\mathfrak{g}$-modules induced from a parabolic and the Ber\v{n}ste{\u\i}n-Gel'fand-Gel'fand resolution of a finite-dimensional, irreducible $\mathfrak{g}$-module
Alvany
Rocha-Caridi
335-366
Abstract: Let $\mathcal{g}$ be a finite dimensional, complex, semisimple Lie algebra and let V be a finite dimensional, irreducible $ \mathcal{g}$-module. By computing a certain Lie algebra cohomology we show that the generalized versions of the weak and the strong Bernstein-Gelfand-Gelfand resolutions of V obtained by H. Garland and J. Lepowsky are identical. Let G be a real, connected, semisimple Lie group with finite center. As an application of the equivalence of the generalized Bernstein-Gelfand-Gelfand resolutions we obtain a complex in terms of the degenerate principal series of G, which has the same cohomology as the de Rham complex.
Algebras of automorphic forms with few generators
Philip
Wagreich
367-389
Abstract: Those finitely-generated Fuchsian groups G for which the graded algebra of automorphic forms A is generated by 2 or 3 elements are classified. In these cases the structure of A is described.
Preservation of closure in a locally convex space. I
I.
Brodsky
391-397
Abstract: This paper is concerned with the lifting of the closures of sets. If H is a topological vector space, G a subspace and A closed in G for the induced topology, under what conditions on A in G is it true that the closure of A is preserved in H, i.e., A is closed in H? In this paper a fundamental lifting proposition is proved. 'Preservation of closure' will prove to be a fruitful technique in obtaining some interesting results in the theory of locally convex spaces. Using this technique, we will first show when closure is equivalent to completeness. Then we will prove a generalization to locally convex spaces of the classical Heine-Borel Theorem for Euclidean n-space. Generalizing a result of Petunin, we will also give some necessary and sufficient conditions on semireflexivity. Finally, we will give a necessary and sufficient condition for the sum of two closed subspaces to be closed.
Deformation theory and the tame fundamental group
David
Harbater
399-415
Abstract: Let U be a curve of genus g with $n\, + \,1$ points deleted, defined over an algebraically closed field of characteristic $p\, \geqslant \,0$. Then there exists a bijection between the Galois finite étale covers of U having degree prime to p, and the finite $ p'$-groups on $n\, + \,2g$ generators. This fact has been proven using analytic considerations; here we construct such a bijection algebraically. We do this by algebraizing an analytic construction of covers which uses Hurwitz families. The process of algebraization relies on a deformation theorem, which we prove using Artin's Algebraization Theorem, and which allows the patching of local families into global families. That our construction provides the desired bijection is afterwards verified analytically.
The product of two countably compact topological groups
Eric K.
van Douwen
417-427
Abstract: We use MA ( = Martin's Axiom) to construct two countably compact topological groups whose product is not countably compact. To this end we first use MA to construct an infinite countably compact topological group which has no nontrivial convergent sequences.
Nonexistence of nontrivial $cm''$-harmonic 1-forms on a complete foliated Riemannian manifold
Haruo
Kitahara
429-435
Abstract: We study the nonexistence of nontrivial $\square''$-harmonic 1-forms on a complete foliated riemannian manifold with positive definite Ricci curvature. It is well known that the harmonic 1-form on a compact and orientable riemannian manifold with positive definite Ricci curvature is trivial. Our main theorem is an extension of this fact in the complete foliated riemannian case.
The category of $D$-completely regular spaces is simple
N. C.
Heldermann
437-446
Abstract: In a recent paper H. Brandenburg characterized the objects of the epireflective hull of all developable spaces-that are those spaces which are homeomorphic to a subspace of a product of developable spaces-by intrinsic properties. It is shown here that these spaces, called D-completely regular, can be generated from a single second countable developable space D which has the same cardinality as the reals. As an application of this result we obtain a new characterization of D-normal spaces analogous to Urysohn's lemma and a new (external) characterization of perfect spaces (meaning every closed set is a ${G_\delta }$).
Composition factors of the principal series representations of the group ${\rm Sp}(n,\,1)$
M. W.
Baldoni Silva;
H.
Kraljević
447-471
Abstract: Using Vogan's algorithm the composition factors of any principal series representation of the group $Sp(n,\,1)$ are determined.
Regularity of certain small subharmonic functions
P. C.
Fenton
473-486
Abstract: Suppose that u is subharmonic in the plane and that $ {\underline {\lim } _{r \to \infty }}B(r)/{(\log \,r)^{2\,}}\, = \,\sigma \, < \,\infty$. It is known that, given $\varepsilon > 0$, there are arbitrarily large values of r such that $ A(r)\, > \,B(r)\, - (\sigma \, + \,\varepsilon){\pi ^2}$. The following result is proved. Let u be subharmonic and let $ \sigma$ be any positive number. Then either $A(r)\, > \,B(r)\, - {\pi ^2}\sigma$ for certain arbitrarily large values of r or, if this is false, then $\displaystyle \mathop {\lim }\limits_{r \to \infty } \left( {B\left( r \right)\, - \,\sigma {{\left( {\log \,r} \right)}^2}} \right)/\log \,r$ exists and is either $+ \,\infty$ or finite.
On the singularities of Gegenbauer (ultraspherical) expansions
Ahmed I.
Zayed
487-503
Abstract: The results of Gilbert on the location of the singular points of an analytic function $f(z)$ given by Gegenbauer (ultraspherical) series expansion $f(z)\, = \,\Sigma _{n\, = \,0}^\infty \,{a_n}\,C_n^\mu (z)$ are extended to the case where the series converges to a distribution. On the other hand, this generalizes Walter's results on distributions given by Legendre series: $ f(z)\, = \,\Sigma _{n\, = \,0}^\infty \,{a_n}\,C_n^{1/2}(z)$. The singularities of the analytic representation of $ f(z)$ are compared to those of the associated power series $ g(z)\, = \,\Sigma _{n\, = \,0}^\infty \,{a_n}{z^n}$. The notion of value of a distribution at a point is used to study the boundary behavior of the associated power series. A sufficient condition for Abel summability of Gegenbauer series is also obtained in terms of the distribution to which the series converges.
A new result on the convergence of nonhomogeneous stochastic chains
Arunava
Mukherjea
505-520
Abstract: Nonhomogeneous stochastic chains with a finite number of states are considered in this paper. Convergence of such chains is established here in terms of strong ergodicity of certain related chains of smaller size. These results are shown to be best possible and extend earlier results of Maksimov. Nonnegative idempotent matrices are also considered.
$L\sp{p}$ behavior of certain second order partial differential operators
Carlos E.
Kenig;
Peter A.
Tomas
521-531
Abstract: We give examples of bounded inverses of polynomials in ${{\textbf{R}}^n}$, $n\, > \,1$, which are not Fourier multipliers of $ {L^p}\,({{\textbf{R}}^n})$ for any $p\, \ne \,2$. Our main tool is the Kakeya set construction of C. Fefferman. Using these results, we relate the invertibility on $ {L^p}$ of a linear second order constant coefficient differential operator D on $ {{\textbf{R}}^n}$ to the geometric structure of quadratic surfaces associated to its symbol d. This work was motivated by multiplier conjectures of N. Rivière and R. Strichartz.
On locally and globally conformal K\"ahler manifolds
Izu
Vaisman
533-542
Abstract: Some relations between the locally conformal Kähler (l.c.K.) and the globally conformal Kähler (g.c.K.) properties are established. Compact l.c.K. manifolds which are not g.c.K. do not have Kähler metrics. l.c.K. manifolds which are not g.c.K. are analytically irreducible. Various curvature restrictions on l.c.K. manifolds imply the g.c.K. property. Total spaces of induced Hopf fibrations are l.c.K. and not g.c.K. manifolds. Conjecture. A compact l.c.K. manifold which is not g.c.K. has at least one odd odd-dimensional Betti number.
Linear spaces with an $H\sp{\ast} $-algebra-valued inner product
Parfeny P.
Saworotnow
543-549
Abstract: The paper deals with a particular class of VH-spaces of Loynes [5] whose inner product assumes its values in a trace-algebra associated with an $ H^{\ast}$-algebra. It is shown that these spaces admit a structure of a ``nonassociative module", and this structure could be used to characterize such spaces. Also we characterize other related spaces.
The asymptotic behavior of gas in an $n$-dimensional porous medium
Avner
Friedman;
Shoshana
Kamin
551-563
Abstract: Consider the flow of gas in an n-dimensional porous medium with initial density ${u_0}(x)\, \geqslant \,0$. The density $ u(x,\,t)$ then satisfies the nonlinear degenerate parabolic equation ${u_t}\, = \,\Delta {u^m}$ where $ m\, > \,1$ is a physical constant. Assuming that $ I\, \equiv \,\int {\,{u_0}(x)} dx\, < \,\infty$ it is proved that $ u(x,\,t)$ behaves asymptotically, as $ t\, \to \,\infty$, like the special (explicitly given) solution $V(\vert x\vert,\,t)$ which is invariant by similarity transformations and which takes the initial values $\delta (x)I\,(\delta (x)\, = \,$ the Dirac measure) in the distribution sense.
Weak subordination and stable classes of meromorphic functions
Kenneth
Stephenson
565-577
Abstract: This paper introduces the notion of weak subordination: If F and G are meromorphic in the unit disc $\mathcal{u}$, then F is weakly subordinate to G, written $F\, < \,G$, provided there exist analytic functions $\phi$ and $\omega :\,\mathcal{u}\, \to \,\mathcal{u}$, with $ \phi$ an inner function, so that $F\, \circ \,\phi \, = \,G\, \circ \,\omega$. A class $\mathcal{X}$ of meromorphic functions is termed stable if $F\,\mathop w\limits_ < \,G$ and $G\, \in \,\mathcal{X}\, \Rightarrow \,F\, \in \,\mathcal{X}$. The motivation is recent work of Burkholder which relates the growth of a function with its range and boundary values. Assume F and G are meromorphic and G has nontangential limits, a.e. Assume further that $ F(\mathcal{u})\, \cap \,G(\mathcal{u})\, \ne \,\emptyset $ and $ G({e^{i\theta }})\, \notin \,F(\mathcal{u})$, a.e. This is denoted by $ F\, < \,G$. Burkholder proved for several classes $ \mathcal{X}$ that $\displaystyle F\, < \,G \qquad{\text{and}}\quad G\, \in \,\mathcal{X}\, \Rightarrow \,F\, \in \,\mathcal{X}.$ ($ (\ast)$) The main result of this paper is the Theorem: $ F\, < \,G\, \Rightarrow \,F\,{ \prec ^w}\,G$. In particular, implication (*) holds for all stable classes $ \mathcal{X}$. The paper goes on to study various stable classes, which include BMOA, ${H^p}$, $0\, < \,p\, \leqslant \,\infty$, ${N_{\ast}}$, the space of functions of bounded characteristic, and the ${M^\Phi }$ spaces introduced by Burkholder. VMOA and the Bloch functions are examples of classes which are not stable.
A maximal function characterization of $H\sp{p}$ on the space of homogeneous type
Akihito
Uchiyama
579-592
Abstract: Let ${\psi _0}(x)\, \in \,{\mathcal{S(}}{R^n}{\text{)}}$ and let $\int_{{R^n}} {{\psi _0}(y)\,dy\, \ne \,0}$.For $x\, \in \,{R^n}$ and $M\, \geqslant \,0$, let $\displaystyle {f^ + }(x)\, = \,\mathop {\sup }\limits_{t\, > \,0} \,\left\vert {f\,{\ast}\,{\psi _{0t}}(x)} \right\vert$ and let $ {f^{{\ast}M}}(x)\, = \,\sup \{ \left\vert {f\,{\ast}\,{\psi _t}(x)} \right\vert:\,t\, > \,0$, $\psi (y)\, \in \,{\mathcal{S(}}{R^n})$, $\operatorname{supp} \,\psi \, \subset \,\{ y\, \in \,{R^n}:\,\left\vert y \right\vert\, < \,1\}$, ${\left\Vert {{D^\alpha }\psi } \right\Vert _{{L^\infty }}}\, \leqslant \,1$ for any multi-index $\alpha \, = \,({\alpha _1},\, \ldots ,\,{\alpha _n})$ such that $\Sigma _{i = 1}^n\,{\alpha _i}\, \leqslant \,M\}$ where ${\psi _t}(y)\, = \,{t^{ - n}}\psi (y/t)$. Fefferman-Stein [11] showed Theorem A. Let $ p\, > \,0$. Then there exists $M(p,\,n)$, depending only on p and n, such that if $ M\, \geqslant \,M(p,\,n)$, then $\displaystyle c\left\Vert {{f^ + }} \right\Vert{L^p}\, \leqslant \,\left\Vert {... ...right\Vert{L^p}\, \leqslant \,{\textbf{C}}\left\Vert {{f^ + }} \right\Vert{L^p}$ for any