Operators on $C(K)$ spaces preserving copies of Schreier spaces
Ioannis
Gasparis
1-30
Abstract: It is proved that an operator \(T \colon C(K) \to X\), \(K\)compact metrizable, \(X\) a separable Banach space, for which the \(\epsilon\)-Szlenk index of \(T^*(B_{X^*})\) is greater than or equal to \(\omega^\xi\), \(\xi < \omega_1\), is an isomorphism on a subspace of \(C(K)\) isomorphic to \(X_\xi\), the Schreier space of order \(\xi\). As a corollary, one obtains that a complemented subspace of \(C(K)\) with Szlenk index equal to \(\omega^{\xi + 1}\) contains a subspace isomorphic to \(X_\xi\).
Symmetrically approximately continuous functions, consistent density theorems, and Fubini type inequalities
P.
D.
Humke;
M.
Laczkovich
31-44
Abstract: Using the continuum hypothesis, Sierpinski constructed a nonmeasurable function $f$ such that $\{ h: f(x+h)\ne f(x-h)\}$ is countable for every $x.$ Clearly, such a function is symmetrically approximately continuous everywhere. Here we to show that Sierpinski's example cannot be constructed in ZFC. Moreover, we show it is consistent with ZFC that if a function is symmetrically approximately continuous almost everywhere, then it is measurable.
The $\alpha$-invariant on certain surfaces with symmetry groups
Jian
Song
45-57
Abstract: The global holomorphic $\alpha$-invariant introduced by Tian is closely related to the existence of Kähler-Einstein metrics. We apply the result of Tian, Yau and Zelditch on polarized Kähler metrics to approximate plurisubharmonic functions and compute the $\alpha$-invariant on $CP^2\char93 n\overline{CP^2}$ for $n=1,2,3$.
Local zeta function for curves, non-degeneracy conditions and Newton polygons
M.
J.
Saia;
W.
A.
Zuniga-Galindo
59-88
Abstract: This paper is dedicated to a description of the poles of the Igusa local zeta function $Z(s,f,v)$ when $f(x,y)$ satisfies a new non-degeneracy condition called arithmetic non-degeneracy. More precisely, we attach to each polynomial $f(x,y)$ a collection of convex sets $\Gamma ^{A}(f)=\left\{ \Gamma _{f,1},\dots ,\Gamma _{f,l_{0}}\right\}$called the arithmetic Newton polygon of $f(x,y)$, and introduce the notion of arithmetic non-degeneracy with respect to $\Gamma ^{A}(f)$. If $L_{v}$ is a $p$-adic field, and $f(x,y)\in L_{v}\left[ x,y \right]$ is arithmetically non-degenerate, then the poles of $Z(s,f,v)$ can be described explicitly in terms of the equations of the straight segments that form the boundaries of the convex sets $\Gamma _{f,1},\dots , \Gamma _{f,l_{0}}$. Moreover, the proof of the main result gives an effective procedure for computing $Z(s,f,v)$.
Some logical metatheorems with applications in functional analysis
Ulrich
Kohlenbach
89-128
Abstract: In previous papers we have developed proof-theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis. Here `uniform' means independence from parameters in compact spaces. A recent case study in fixed point theory systematically yielded uniformity even w.r.t. parameters in metrically bounded (but noncompact) subsets which had been known before only in special cases. In the present paper we prove general logical metatheorems which cover these applications to fixed point theory as special cases but are not restricted to this area at all. Our theorems guarantee under general logical conditions such strong uniform versions of non-uniform existence statements. Moreover, they provide algorithms for actually extracting effective uniform bounds and transforming the original proof into one for the stronger uniformity result. Our metatheorems deal with general classes of spaces like metric spaces, hyperbolic spaces, CAT(0)-spaces, normed linear spaces, uniformly convex spaces, as well as inner product spaces.
A Lyndon-Hochschild-Serre spectral sequence for certain homotopy fixed point spectra
Ethan
S.
Devinatz
129-150
Abstract: Let $H$ and $K$ be closed subgroups of the extended Morava stabilizer group $G_n$ and suppose that $H$ is normal in $K$. We construct a strongly convergent spectral sequence \begin{displaymath}H^\ast_c(K/H, (E^{hH}_n)^\ast X) \Rightarrow (E^{hK}_n)^\ast X, \end{displaymath} where $E^{hH}_n$ and $E^{hK}_n$ are the continuous homotopy fixed point spectra of Devinatz and Hopkins. This spectral sequence turns out to be an Adams spectral sequence in the category of $K(n)_\ast$-local $E^{hK}_n$-modules.
Boundary Hölder and $L^p$ estimates for local solutions of the tangential Cauchy-Riemann equation
Christine
Laurent-Thiébaut;
Mei-Chi
Shaw
151-177
Abstract: We study the local solvability of the tangential Cauchy-Riemann equation on an open neighborhood $\omega$ of a point $z_0\in M$ when $M$ is a generic $q$-concave $CR$ manifold of real codimension $k$ in $\mathbb{C} ^n$, where $1\le k\le n-1$. Our method is to first derive a homotopy formula for $\overline\partial_b$ in $\omega$ when $\omega$ is the intersection of $M$ with a strongly pseudoconvex domain. The homotopy formula gives a local solution operator for any $\overline\partial_b$-closed form on $\omega$ without shrinking. We obtain Hölder and $L^p$ estimates up to the boundary for the solution operator. RÉSUMÉ. Nous étudions la résolubilité locale de l'opérateur de Cauchy- Riemann tangentiel sur un voisinage $\omega$ d'un point $z_0$d'une sous-variété $CR$ générique $q$-concave $M$ de codimension quelconque de $\mathbb C^n$. Nous construisons une formule d'homotopie pour le $\overline\partial_b$ sur $\omega$, lorsque $\omega$ est l'intersection de $M$ et d'un domaine strictement pseudoconvexe. Nous obtenons ainsi un opérateur de résolution pour toute forme $\overline\partial_b$-fermée sur $\omega$. Nous en déduisons des estimations $L^p$ et des estimations hölderiennes jusqu'au bord pour la solution de l'équation de Cauchy-Riemann tangentielle sur $\omega$.
On a refinement of the generalized Catalan numbers for Weyl groups
Christos
A.
Athanasiadis
179-196
Abstract: Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$, coroot lattice $\check{Q}$ and Coxeter number $h$, spanning a Euclidean space $V$, and let $m$ be a positive integer. It is known that the set of regions into which the fundamental chamber of $W$ is dissected by the hyperplanes in $V$ of the form $(\alpha, x) = k$ for $\alpha \in \Phi$ and $k = 1, 2,\dots,m$ is equinumerous to the set of orbits of the action of $W$ on the quotient $\check{Q} / \, (mh+1) \, \check{Q}$. A bijection between these two sets, as well as a bijection to the set of certain chains of order ideals in the root poset of $\Phi$, are described and are shown to preserve certain natural statistics on these sets. The number of elements of these sets and their corresponding refinements generalize the classical Catalan and Narayana numbers, which occur in the special case $m=1$ and $\Phi = A_{n-1}$.
One-dimensional dynamical systems and Benford's law
Arno
Berger;
Leonid
A.
Bunimovich;
Theodore
P.
Hill
197-219
Abstract: Near a stable fixed point at 0 or $\infty$, many real-valued dynamical systems follow Benford's law: under iteration of a map $T$ the proportion of values in $\{x, T(x), T^2(x),\dots, T^n(x)\}$ with mantissa (base $b$) less than $t$ tends to $\log_bt$ for all $t$ in $[1,b)$ as $n\to\infty$, for all integer bases $b>1$. In particular, the orbits under most power, exponential, and rational functions (or any successive combination thereof), follow Benford's law for almost all sufficiently large initial values. For linearly-dominated systems, convergence to Benford's distribution occurs for every $x$, but for essentially nonlinear systems, exceptional sets may exist. Extensions to nonautonomous dynamical systems are given, and the results are applied to show that many differential equations such as $\dot x=F(x)$, where $F$ is $C^2$ with
Elements of specified order in simple algebraic groups
R.
Lawther
221-245
Abstract: In this paper we let $G$ be a simple algebraic group and $r$ be a natural number, and consider the codimension in $G$ of the variety of elements $g\in G$ satisfying $g^r=1$. We shall obtain a lower bound for this codimension which is independent of characteristic, and show that it is attained if $G$ is of adjoint type.
Well-posedness of the Dirichlet problem for the non-linear diffusion equation in non-smooth domains
Ugur
G.
Abdulla
247-265
Abstract: We investigate the Dirichlet problem for the parablic equation \begin{displaymath}u_t = \Delta u^m, m > 0, \end{displaymath} in a non-smooth domain $\Omega \subset \mathbb{R}^{N+1}, N \geq 2$. In a recent paper [U.G. Abdulla, J. Math. Anal. Appl., 260, 2 (2001), 384-403] existence and boundary regularity results were established. In this paper we present uniqueness and comparison theorems and results on the continuous dependence of the solution on the initial-boundary data. In particular, we prove $L_1$-contraction estimation in general non-smooth domains.
Dupin indicatrices and families of curve congruences
J.
W.
Bruce;
F.
Tari
267-285
Abstract: We study a number of natural families of binary differential equations (BDE's) on a smooth surface $M$ in ${\mathbb{R}}^3$. One, introduced by G. J. Fletcher in 1996, interpolates between the asymptotic and principal BDE's, another between the characteristic and principal BDE's. The locus of singular points of the members of these families determine curves on the surface. In these two cases they are the tangency points of the discriminant sets (given by a fixed ratio of principle curvatures) with the characteristic (resp. asymptotic) BDE. More generally, we consider a natural class of BDE's on such a surface $M$, and show how the pencil of BDE's joining certain pairs are related to a third BDE of the given class, the so-called polar BDE. This explains, in particular, why the principal, asymptotic and characteristic BDE's are intimately related.
Stability of transonic shock fronts in two-dimensional Euler systems
Shuxing
Chen
287-308
Abstract: We study the stability of stationary transonic shock fronts under two-dimensional perturbation in gas dynamics. The motion of the gas is described by the full Euler system. The system is hyperbolic ahead of the shock front, and is a hyperbolic-elliptic composed system behind the shock front. The stability of the shock front and the downstream flow under two-dimensional perturbation of the upstream flow can be reduced to a free boundary value problem of the hyperbolic-elliptic composed system. We develop a method to deal with boundary value problems for such systems. The crucial point is to decompose the system to a canonical form, in which the hyperbolic part and the elliptic part are only weakly coupled in their coefficients. By several sophisticated iterative processes we establish the existence and uniqueness of the solution to the described free boundary value problem. Our result indicates the stability of the transonic shock front and the flow field behind the shock.
Glauberman-Watanabe corresponding $p$-blocks of finite groups with normal defect groups are Morita equivalent
Morton
E.
Harris
309-335
Abstract: Let $G$ be a finite group and let $A$ be a solvable finite group that acts on $G$ such that the orders of $G$ and $A$are relatively prime. Let $b$ be a $p$-block of $G$ with normal defect group $D$ such that $A$ stabilizes $b$ and $D\leq C_{G}(A)$. Then there is a Morita equivalence between the block $b$ and its Watanabe correspondent block $W(b)$ of $C_{G}(A)$ given by a bimodule $M$ with vertex $\Delta D$ and trivial source that on the character level induces the Glauberman correspondence (and which is an isotypy by a theorem of Watanabe).
Estimations $L^p$ des fonctions du Laplacien sur les variétés cuspidales
Hong-Quan
Li
337-354
Abstract: Le but de cet article est d'étudier la continuité $L^p$ des fonctions du Laplacien sur les variétés cuspidales.
Maximal holonomy of infra-nilmanifolds with $2$-dimensional quaternionic Heisenberg geometry
Ku
Yong
Ha;
Jong
Bum
Lee;
Kyung
Bai
Lee
355-383
Abstract: Let $\mathbf{H}_{4n-1}(\mathbb{H} )$ be the quaternionic Heisenberg group of real dimension $4n-1$ and let $I_{n}$ denote the maximal order of the holonomy groups of all infra-nilmanifolds with $\mathbf{H}_{4n-1}(\mathbb{H} )$-geometry. We prove that $I_2=48$. As an application, by applying Kim and Parker's result, we obtain that the minimum volume of a $2$-dimensional quaternionic hyperbolic manifold with $k$ cusps is at least $\frac{\sqrt{2}k}{720}.$
Lack of natural weighted estimates for some singular integral operators
José
María
Martell;
Carlos
Pérez;
Rodrigo
Trujillo-González
385-396
Abstract: We show that the classical Hörmander condition, or analogously the $L^r$-Hörmander condition, for singular integral operators $T$ is not sufficient to derive Coifman's inequality \begin{displaymath}\int_{\mathbb{R} ^n} \vert Tf(x)\vert^p\, w(x)\, dx \le C\,\int_{\mathbb{R} ^n} M f(x)^p\, w(x)\,dx, \end{displaymath} where $0<p<\infty$, $M$ is the Hardy-Littlewood maximal operator, $w$ is any $A_{\infty}$ weight and $C$ is a constant depending upon $p$ and the $A_{\infty}$ constant of $w$. This estimate is well known to hold when $T$ is a Calderón-Zygmund operator. As a consequence we deduce that the following estimate does not hold: \begin{displaymath}\int_{\mathbb{R} ^n} \vert Tf(x)\vert^p\, w(x)\, dx \le C\,\int_{\mathbb{R} ^n} Mf(x)^p\, Mw(x)\,dx, \end{displaymath} where $0<p\le 1$ and where $w$ is an arbitrary weight. However, by a recent result due to A. Lerner, this inequality is satisfied whenever $T$ is a Calderón-Zygmund operator. One of the main ingredients of the proof is a very general extrapolation theorem for $A_\infty$ weights.
Convergence of double Fourier series and $W$-classes
M.
I.
Dyachenko;
D.
Waterman
397-407
Abstract: The double Fourier series of functions of the generalized bounded variation class $\{n/\ln (n+1)\}^{\ast }BV$ are shown to be Pringsheim convergent everywhere. In a certain sense, this result cannot be improved. In general, functions of class $\Lambda ^{\ast }BV,$ defined here, have quadrant limits at every point and, for $f\in \Lambda ^{\ast }BV,$ there exist at most countable sets $P$ and $Q$ such that, for $x\notin P$ and $y\notin Q,$ $f$is continuous at $(x,y)$. It is shown that the previously studied class $\Lambda BV$ contains essentially discontinuous functions unless the sequence $\Lambda$ satisfies a strong condition.
A novel dual approach to nonlinear semigroups of Lipschitz operators
Jigen
Peng;
Zongben
Xu
409-424
Abstract: Lipschitzian semigroup refers to a one-parameter semigroup of Lipschitz operators that is strongly continuous in the parameter. It contains $C_{0}$-semigroup, nonlinear semigroup of contractions and uniformly $k$-Lipschitzian semigroup as special cases. In this paper, through developing a series of Lipschitz dual notions, we establish an analysis approach to Lipschitzian semigroup. It is mainly proved that a (nonlinear) Lipschitzian semigroup can be isometrically embedded into a certain $C_{0}$-semigroup. As application results, two representation formulas of Lipschitzian semigroup are established, and many asymptotic properties of $C_{0}$-semigroup are generalized to Lipschitzian semigroup.