Duality theorems in deformation theory
Hubert
Goldschmidt
1-49
Abstract: We give a unified treatment of the construction of the Calabi sequence, which is a resolution of the sheaf of Killing vector fields on a Riemannian manifold of constant curvature, and of the resolution of the sheaf of conformal Killing vector fields on a conformally flat Riemannian manifold of dimension $ \geqslant 3$ introduced in [7]. We also explain why the latter resolution is selfadjoint and associate to certain geometric structures selfadjoint resolutions of their infinitesimal automorphisms.
$(n-1)$-axial ${\rm SO}(n)$ and ${\rm SU}(n)$ actions on homotopy spheres
R. D.
Ball
51-79
Abstract: Let $G(n) = O(n)$ or $U(n)$ and $ SG(n) = SO(n)$ or $ SU(n)$. For each integer $m \geqslant 1$ a family $\{ {S_{\gamma ,\sigma }}:\gamma \in H,\sigma \in K\}$ of $(n - 1)$-axial $SG(n)$ homotopy spheres ${S_{\gamma ,\sigma }}$ is constructed. Each ${S_{\gamma ,\sigma }}$ has fixed point set of dimension $ (m - 1) \geqslant 0$ and orbit space of dimension $r = \tfrac{1} {2}n(n - 1) + (m - 1)$ (resp. $r = {(n - 1)^2} + m - 1$) if $SG(n) = SO(n)$ (resp. $SU(n)$). $H$ is ${\pi _{r - 1}}(SG(n)/G(n - 1))$. $K$ is trivial if $SG(n) = SO(n)$ and is a homotopy theoretically defined subgroup of sections of an ${S^2}$ bundle depending only on $m$ and $n$ if $SG(n) = SU(n)$. Assume that $ m$ and $n$ satisfy the mild restriction $ \S5$, (1). It is shown that the above family is universal for $(n - 1)$-axial $SG(n)$ homotopy spheres and provides a classification analogous to the classification of fibre bundles: for each $(n - 1)$-axial $SG(n)$ homotopy sphere $\Sigma$ there is a ${S_{\gamma ,\sigma }}$ and a unique equivariant stratified map $\Sigma \to {S_{\gamma ,\sigma }}$. $ \Sigma$ is equivariantly diffeomorphic to the pullback of ${S_{\gamma ,\sigma }}$ via the map $ B(\Sigma ) \to B({S_{\gamma ,\sigma }})$ of orbit spaces. If $SG(n) = SO(n)$ then $\gamma$ is unique (and $ \sigma = 1$). If $SG(n) = SU(n)$ then $\gamma$ is unique modulo the image of $\displaystyle {\pi _{r - 1}}S(U(n - 2) \times U(2))/U(k - 1) \times U(1)\quad {\text{in}}\;H.$ An example is given showing that the differentiable structure of the underlying smooth manifold of $ {S_{\gamma ,\sigma }}$ may be exotic.
${\rm Aut}(F)\to{\rm Aut}(F/F'')$ is surjective for free group $F$ of rank $\geq 4$
Seymour
Bachmuth;
Horace Y.
Mochizuki
81-101
Abstract: In this article, it is shown that the group of automorphisms of the free metabelian group $\Phi (n)$ of rank $ n \geqslant 4$ is not only finitely generated but in fact every automorphism of $ \Phi (n)$ is induced by an automorphism of the free group of the same rank $ n$. This contrasts sharply with the authors' earlier result [4] that any set of generators of the group of automorphisms of the free metabelian group $\Phi (3)$ of rank $3$ contains infinitely many automorphisms which are not induced by an automorphism of the free group of rank $3$.
A commutator theorem and weighted BMO
Steven
Bloom
103-122
Abstract: The main result of this paper is a commutator theorem: If $\mu$ and $\lambda$ are ${A_p}$ weights, then the commutator $H$, ${M_b}$ is a bounded operator from ${L^p}(\mu )$ into $ {L^p}(\lambda )$ if and only if $b \in {\operatorname{BMO} _{{{(\mu {\lambda ^{ - 1}})}^{1/p}}}}$. The proof relies heavily on a weighted sharp function theorem. Along the way, several other applications of this theorem are derived, including a doubly-weighted $ {L^p}$ estimate for BMO. Finally, the commutator theorem is used to obtain vector-valued weighted norm inequalities for the Hilbert transform.
On the decomposition numbers of the finite general linear groups. II
Richard
Dipper
123-133
Abstract: Let $q$ be a prime power, $G = {\operatorname{GL} _n}(q)$ and let $ r$ be a prime not dividing $ q$. Using representations of Hecke algebras associated with symmetric groups over arbitrary fields, the $r$-modular irreducible $G$-modules are classified. The decomposition matrix $D$ of $G$ (with respect to $r$) is partly described in terms of decomposition matrices of Hecke algebras, and it is shown that $ D$ is lower unitriangular, provided the irreducible characters and irreducible Brauer characters of $G$ are suitably ordered.
Singular integrals and approximate identities on spaces of homogeneous type
Hugo
Aimar
135-153
Abstract: In this paper we give conditions for the ${L^2}$-boundedness of singular integrals and the weak type $(1,1)$ of approximate identities on spaces of homogeneous type. Our main tools are Cotlar's lemma and an extension of a theorem of Zó.
Group-graded rings and duality
Declan
Quinn
155-167
Abstract: We give an alternative construction of the duality between finite group actions and group gradings on rings which was shown by Cohen and Montgomery in [1]. This duality is then used to extend known results on skew group rings to corresponding results for large classes of group-graded rings. Finally we modify the construction slightly to handle infinite groups.
Quasiconformal and bi-Lipschitz homeomorphisms, uniform domains and the quasihyperbolic metric
Gaven J.
Martin
169-191
Abstract: Let $D$ be a proper subdomain of ${R^n}$ and ${k_D}$ the quasihyperbolic metric defined by the conformal metric tensor $d{\overline s ^2} = \operatorname{dist} {(x,\partial D)^{ - 2}}d{s^2}$. The geodesics for this and related metrics are shown, by purely geometric methods, to exist and have Lipschitz continuous first derivatives. This is sharp for ${k_D}$; we also obtain sharp estimates for the euclidean curvature of such geodesics. We then use these results to prove a general decomposition theorem for uniform domains in ${R^n}$, in terms of embeddings of bi-Lipschitz balls. We also construct a counterexample to the higher dimensional analogue of the decomposition theorem of Gehring and Osgood.
Classification of semisimple algebraic monoids
Lex E.
Renner
193-223
Abstract: Let $X$ be a semisimple algebraic monoid with unit group $G$. Associated with $E$ is its polyhedral root system $(X,\Phi ,C)$, where $X = X(T)$ is the character group of the maximal torus $T \subseteq G$, $\Phi \subseteq X(T)$ is the set of roots, and $C = X(\overline T )$ is the character monoid of $ \overline T \subseteq E$ (Zariski closure). The correspondence $E \to (X,\Phi ,C)$ is a complete and discriminating invariant of the semisimple monoid $E$, extending the well-known classification of semisimple groups. In establishing this result, monoids with preassigned root data are first constructed from linear representations of $G$. That done, we then show that any other semisimple monoid must be isomorphic to one of those constructed. To do this we devise an extension principle based on a monoid analogue of the big cell construction of algebraic group theory. This, ultimately, yields the desired conclusions.
Brownian motion with polar drift
R. J.
Williams
225-246
Abstract: Consider a strong Markov process ${X^0}$ that has continuous sample paths in ${R^d}(d \geqslant 2)$ and the following two properties. (1) Away from the origin $ {X^0}$ behaves like Brownian motion with a polar drift given in spherical polar coordinates by $\mu (\theta )/2r$. Here $\mu$ is a bounded Borel measurable function on the unit sphere in ${R^d}$, with average value $\overline \mu$. (2) ${X^0}$ is absorbed at the origin. It is shown that $ {X^0}$ reaches the origin with probability zero or one as $\overline \mu \geqslant 2 - d$ or $< 2 - d$. Indeed, ${X^0}$ is transient to $+ \infty$ if $\overline \mu > 2 - d$ and null recurrent if $\bar \mu = 2 - d$. Furthermore, if $\bar \mu < 2 - d$ (i.e., ${X^0}$ reaches the origin), then $ {X^0}$ does not approach the origin in any particular direction. Indeed, there is a single Martin boundary point for ${X^0}$ at the origin. The question of the existence and uniqueness of a strong Markov process with continuous sample paths in ${R^d}$ that behaves like ${X^0}$ away from the origin, but spends zero time there (in the sense of Lebesgue measure), is also resolved here.
Algebraic and etale $K$-theory
William G.
Dwyer;
Eric M.
Friedlander
247-280
Abstract: We define etale $ K$-theory, interpret various conjectures of Quillen and Lichtenbaum in terms of a map from algebraic $K$-theory to etale $K$-theory, and then prove that this map is surjective in many cases of interest.
The extended module of indecomposables for mod $2$ finite $H$-spaces
R. M.
Kane
281-304
Abstract: We construct a generalization of the usual module of indecomposables for the $\bmod 2$ cohomology of a finite $H$-space. Structure theorems are obtained regarding how the Steenrod algebra acts on this module.
On the solution of certain systems of partial difference equations and linear dependence of translates of box splines
Wolfgang
Dahmen;
Charles A.
Micchelli
305-320
Abstract: This paper is concerned with solving systems of partial difference equations which arise when studying linear dependence of translates of box splines.
On the restriction of the Fourier transform to a conical surface
Bartolome Barcelo
Taberner
321-333
Abstract: Let $\Gamma$ be the surface of a circular cone in $ {{\mathbf{R}}^3}$. We show that if $1 \leqslant p < 4/3$, $1/q = 3(1 - 1/p)$ and $f \in {L^p}({{\mathbf{R}}^3})$, then the Fourier transform of $f$ belongs to ${L^q}(\Gamma ,d\sigma )$ for a certain natural measure $\sigma$ on $\Gamma$. Following P. Tomas we also establish bounds for restrictions of Fourier transforms to conic annuli at the endpoint $p = 4/3$, with logarithmic growth of the bound as the thickness of the annulus tends to zero.
Multiparameter maximal functions along dilation-invariant hypersurfaces
Hasse
Carlsson;
Peter
Sjögren;
Jan-Olov
Strömberg
335-343
Abstract: Consider the hypersurface ${x_{n + 1}} = \Pi _1^nx_i^{{\alpha _i}}$ in ${{\mathbf{R}}^{n + 1}}$. The associated maximal function operator is defined as the supremum of means taken over those parts of the surface lying above the rectangles $\{ 0 \leqslant {x_i} \leqslant {h_i},\;i = 1, \ldots ,n\}$. We prove that this operator is bounded on ${L^p}$ for $p > 1$. An analogous result is proved for a quadratic surface in $ {{\mathbf{R}}^3}$.
Saturation of the closed unbounded filter on the set of regular cardinals
Thomas J.
Jech;
W. Hugh
Woodin
345-356
Abstract: For any $\alpha < {\kappa ^ + }$, the following are equiconsistent: (a) $\kappa$ is measurable of order $\alpha$, (b) $\kappa$ is $\alpha$-Mahlo and the filter $ {\mathbf{C}}[\operatorname{Reg} ]$ is saturated.
$L\sp p$ estimates for Schr\"odinger evolution equations
M.
Balabane;
H. A.
Emamirad
357-373
Abstract: We prove that for Cauchy data in $ {L^1}({{\mathbf{R}}^n})$, the solution of a Schrödinger evolution equation with constant coefficients of order $ 2m$ is uniformly bounded for $t \ne 0$, with bound $(1 + \vert t{\vert^{ - c}})$, where $ c$ is an integer, $c > n/2m - 1$. Moreover it belongs to ${L^q}({{\mathbf{R}}^n})$ if $q > q(m,n)$, with its ${L^p}$ norm bounded by $ ({L^p},{L^q})$ estimates. On the other hand, we prove that for Cauchy data in $ {L^p}({{\mathbf{R}}^n})$, such a Cauchy problem is well posed as a distribution in the $t$-variable with values in $ {L^p}({{\mathbf{R}}^n})$, and we compute the order of the distribution. We apply these two results to the study of Schrödinger equations with potential in $ {L^p}({{\mathbf{R}}^n})$. We give an estimate of the resolvent operator in that case, and prove an asymptotic boundedness for the solution when the Cauchy data belongs to a subspace of $ {L^p}({{\mathbf{R}}^n})$.
Existence in the large for Riemann problems for systems of conservation laws
Michael
Sever
375-381
Abstract: An existence theorem in the large is obtained for the Riemann problem for nonlinear systems of conservation laws. Our principal assumptions are strict hyperbolicity, genuine nonlinearity in the strong sense, and the existence of a convex entropy function. The entropy inequality is used to obtain an a priori estimate of the strengths of the shocks and refraction waves forming a solution; existence of such a solution then follows by an application of finite-dimensional degree theory. The case of a single degenerate field is also included, with an additional assumption on the existence of Riemann invariants.