Degree theory for equivariant maps. I
J.
Ize;
I.
Massabò;
A.
Vignoli
433-510
Abstract: A degree theory for equivariant maps is constructed in a simple geometrical way. This degree has all the basic properties of the usual degree theories and takes its values in the equivariant homotopy groups of spheres. For the case of a semifree ${S^1}$-action, a complete computation of these groups is given, the range of the equivariant degree is determined, and the general ${S^1}$-action is reduced to that special case. Among the applications one recovers and unifies both the degree for autonomous differential equations defined by Fuller [F] and the ${S^1}$-degree for gradient maps introduced by Dancer [Da]. Also, a simple but very useful formula of Nirenberg [N] is generalized (see Theorem 4.4(ii)).
Geometric quantization and the universal enveloping algebra of a nilpotent Lie group
Niels Vigand
Pedersen
511-563
Abstract: We study geometric quantization in connection with connected nilpotent Lie groups. First it is proved that the quantization map associated with a (real) polarized coadjoint orbit establishes an isomorphism between the space of polynomial quantizable functions and the space of polynomial quantized operators. Our methods allow noninductive proofs of certain basic facts from Kirillov theory. It is then shown how the quantization map connects with the universal enveloping algebra. This is the main result of the paper. Finally we show how one can explicitly compute global canonical coordinates on coadjoint orbits, and that this can be done simultaneously on all orbits contained in a given stratum of what we call "the fine $ \mathcal{F}$-stratification of the dual of the Lie algebra". This is a generalization of a result of M. Vergne about simultaneous canonical coodinates for orbits in general position.
Area integral estimates for caloric functions
Russell M.
Brown
565-589
Abstract: We study the relationship between the area integral and the parabolic maximal function of solutions to the heat equation in domains whose boundary satisfies a $ \left({\frac{1}{2},1}\right)$ mixed Lipschitz condition. Our main result states that the area integral and the parabolic maximal function are equivalent in $ {L^p}(\mu)$, $0 < p < \infty$. The measure $\mu$ must satisfy Muckenhoupt's ${A_\infty }$-condition with respect to caloric measure. We also give a Fatou theorem which shows that the existence of parabolic limits is a.e. (with respect to caloric measure) equivalent to the finiteness of the area integral.
The universal von Staudt theorems
Francis
Clarke
591-603
Abstract: We prove general forms of von Staudt's theorems on the Bernoulli numbers. As a consequence we are able to deduce strong versions of a number of congruences involving various generalisations of the Bernoulli numbers. For example we obtain an improved form of a congruence due to Hurwitz involving the Laurent series coefficients of the Weierstrass elliptic function associated with a square lattice.
Value functions on central simple algebras
Patrick J.
Morandi
605-622
Abstract: In this paper we study noncommutative valuation rings as defined by Dubrovin. While there is in general no valuation associated to a Dubrovin valuation ring, we show that there is a value function associated to any Dubrovin valuation ring integral over its center. By using value functions we obtain information on Dubrovin valuation rings in a tensor product, both generalizing and giving a much simpler proof of a result about valued division algebras. By being able to work directly with central simple algebras we gain new information about division algebras over Henselian fields.
Integral Dubrovin valuation rings
Patrick J.
Morandi;
Adrian R.
Wadsworth
623-640
Abstract: In the preceding paper, Dubrovin valuation rings integral over their centers in central simple algebras were characterized by value functions. Here, these value functions are used to give a method for extending integral Dubrovin valuation rings in generalized crossed product algebras. Several applications of this extension theorem are given, including new and more natural proofs of some theorems on valued division algebras over Henselian fields.
A Mandelbrot set whose boundary is piecewise smooth
M. F.
Barnsley;
D. P.
Hardin
641-659
Abstract: It is proved that the Mandelbrot set associated with the pair of maps $ {w_{1,2}}:{\mathbf{C}} \to {\mathbf{C}}, {w_1}(z) = sz + 1, {w_2}(z) = {s^\ast }z - 1$, with parameter $s \in {\mathbf{C}}$, is connected and has piecewise smooth boundary.
On infinite root systems
R. V.
Moody;
A.
Pianzola
661-696
Abstract: We define in an axiomatic fashion the concept of a set of root data that generalizes the usual concept of root system of a Kac-Moody Lie algebra. We study these objects from a purely formal and geometrical point of view as well as in relation to their associated Lie algebras. This leads to a coherent theory of root systems, bases, subroot systems, Lie algebras defined by root data, and subalgebras.
Pixley-Roy hyperspaces of $\omega$-graphs
J. D.
Mashburn
697-709
Abstract: The techniques developed by Wage and Norden are used to show that the Pixley-Roy hyperspaces of any two $\omega$-graphs are homeomorphic. The Pixley-Roy hyperspaces of several subsets of ${{\mathbf{R}}^n}$ are also shown to be homeomorphic.
Isolated singularities of the Schr\"odinger equation with a good potential
Juan Luis
Vázquez;
Cecilia
Yarur
711-720
Abstract: We study the behaviour near an isolated singularity, say 0, of nonnegative solutions of the Schrödinger equation $- \Delta u + Vu = 0$ defined in a punctured ball $0 < \vert x\vert < R$. We prove that whenever the potential $V$ belongs to the Kato class ${K_n}$ the following alternative, well known in the case of harmonic functions, holds: either $\vert x{\vert^{n - 2}}u(x)$ has a positive limit as $\vert x\vert \to 0$ or $u$ is continuous at 0. In the first case $ u$ solves the equation $- \Delta u + Vu = a\delta$ in $\{ \vert x\vert < R\} $. We discuss the optimality of the class ${K_n}$ and extend the result to solutions $ u \ngeq 0$ of $- \Delta u + Vu = f$.
Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities
A. C.
Lazer;
P. J.
McKenna
721-739
Abstract: We give conditions for the existence, uniqueness, and asymptotic stability of periodic solutions of a second-order differential equation with piecewise linear restoring and $2\pi$-periodic forcing where the range of the derivative of the restoring term possibly contains the square of an integer. With suitable restrictions on the restoring and forcing in the undamped case, we give a necessary and sufficient condition.
Deviations of trajectory averages and the defect in Pesin's formula for Anosov diffeomorphisms
Steven
Orey;
Stephan
Pelikan
741-753
Abstract: A large deviation theorem at the Donsker Varadhan level three is obtained for the convergence of trajectory averages of Anosov diffeomorphisms. It is possible to provide an explicit description of the rate function.
Hyperconvexity and nonexpansive multifunctions
Robert
Sine
755-767
Abstract: It is shown that a ball intersection valued nonexpansive multifunction on a hyperconvex space admits a nonexpansive point valued selection. This implies fixed point theorems for such multifunctions and to certain point valued nonexpansive maps. The result is used to study best approximation and to show the space of all nonexpansive maps of a bounded hyperconvex space is hyperconvex.
Relations between $H\sp p\sb u$ and $L\sp p\sb u$ in a product space
Jan-Olov
Strömberg;
Richard L.
Wheeden
769-797
Abstract: Relations between $ L_u^p$ and $H_u^p$ are studied for the product space ${{\mathbf{R}}^1} \times {{\mathbf{R}}^1}$ in the case $1 < p < \infty$ and $u({x_1},{x_2}) = \vert{Q_1}({x_1}){\vert^p}\vert{Q_2}({x_2}){\vert^p}w({x_1},{x_2})$, where ${Q_1}$ and ${Q_2}$ are polynomials and $w$ satisfies the ${A_p}$ condition for rectangles. A description of the distributions in $ H_u^p$ is given. Questions about boundary values and about the existence of dense subsets of smooth functions satisfying appropriate moment conditions are also considered.
Mappings of trees and the fixed point property
M. M.
Marsh
799-810
Abstract: We investigate weakly confluent, universal, and related mappings of trees and their relationships to the fixed point property for tree-like continua. This investigation leads to some new results, to generalizations of some known results, and to a partial solution of a question of H. Cook.
On a maximal function on compact Lie groups
Michael
Cowling;
Christopher
Meaney
811-822
Abstract: Suppose that $ G$ is a compact Lie group with finite centre. For each positive number $ s$ we consider the $ \operatorname{Ad}(G)$-invariant probability measure ${\mu _s}$ carried on the conjugacy class of $\exp (s{H_\rho })$ in $G$. This one-parameter family of measures is used to define a maximal function $\mathcal{M}\,f$, for each continuous function $ f$ on $G$. Our theorem states that there is an index ${p_0}$ in $(1,2)$, depending on $G$, such that the maximal operator $\mathcal{M}$ is bounded on ${L^p}(G)$ when $p$ is greater than ${p_0}$. When the rank of $G$ is greater than one, this provides an example of a controllable maximal operator coming from averages over a family of submanifolds, each of codimension greater than one.
Scattering for the Yang-Mills equations
John C.
Baez
823-832
Abstract: We construct wave and scattering operators for the Yang-Mills equations on Minkowski space, ${{\mathbf{M}}_0} \cong {{\mathbf{R}}^4}$. Sufficiently regular solutions of the Yang-Mills equations on $ {{\mathbf{M}}_0}$ are known to extend uniquely to solutions of the corresponding equations on the universal cover of its conformal compactification, $ \tilde{\mathbf{M}} \cong {\mathbf{R}} \times {S^3}$. Moreover, the boundary of $ {{\mathbf{M}}_0}$ as embedded in $ \tilde{\mathbf{M}}$ is the union of "lightcones at future and past infinity", $ {C_ \pm }$. We construct wave operators ${W_ \pm }$ as continuous maps from a space ${\mathbf{X}}$ of time-zero Cauchy data for the Yang-Mills equations to Hilbert spaces ${\mathbf{H}}({C_ \pm })$ of Goursat data on $ {C_ \pm }$. The scattering operator is then a homeomorphism $ S:{\mathbf{X}} \to {\mathbf{X}}$ such that $ U{W_ + } = {W_ - }S$, where $U:{\mathbf{H}}({C_ + }) \to {\mathbf{H}}({C_ - })$ is the linear isomorphism arising from a conformal transformation of $ \tilde{\mathbf{M}}$ mapping ${C_ - }$ onto ${C_ + }$. The maps ${W_ \pm }$ and $S$ are shown to be smooth in a certain sense.
Regularity of the metric entropy for expanding maps
Marek
Rychlik
833-847
Abstract: The main result of the current paper is an estimate of the radius of the nonperipheral part of the spectrum of the Perron-Frobenius operator for expanding mappings. As a consequence, we are able to show that the metric entropy of an expanding map has modulus of continuity $x\log (1/x)$ on the space of ${C^2}$-expandings. We also give an explicit estimate of the rate of mixing for ${C^1}$-functions in terms of natural constants. It seems that the method we present can be generalized to other classes of dynamical systems, which have a distinguished invariant measure, like $ \operatorname{Axiom}$ A diffeomorphisms. It also can be adopted to show that the entropy of the quadratic family ${f_\mu }(x) = 1 - \mu {x^2}$ computed with respect to the absolutely continuous invariant measure found in Jakobson's Theorem varies continuously (the last result is going to appear somewhere else).
Infima of convex functions
Gerald
Beer
849-859
Abstract: Let $\Gamma (X)$ be the lower semicontinuous, proper, convex functions on a real normed linear space $ X$. We produce a simple description of what is, essentially, the weakest topology on $\Gamma (X)$ such that the value functional $f \to \inf f$ is continuous on $\Gamma (X)$. When $X$ is reflexive, convergence of a sequence in this topology is equivalent to Mosco plus pointwise convergence of the corresponding sequence of conjugate convex functions.