Nonlinear equations and weighted norm inequalities
N.
J.
Kalton;
I.
E.
Verbitsky
3441-3497
Abstract: We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for the solvability of the Dirichlet problem \begin{equation*}\begin{split} -& \Delta u = v \, u^{q} + w, \quad u \ge 0 \quad \text{on} \quad \Omega , &u = 0 \quad \text{on} \quad \partial \Omega , \end{split}\end{equation*} on a regular domain $\Omega$ in $\mathbf{R}^{n}$ in the ``superlinear case'' $q > 1$. The coefficients $v, w$ are arbitrary positive measurable functions (or measures) on $\Omega$. We also consider more general nonlinear differential and integral equations, and study the spaces of coefficients and solutions naturally associated with these problems, as well as the corresponding capacities. Our characterizations of the existence of positive solutions take into account the interplay between $v$, $w$, and the corresponding Green's kernel. They are not only sufficient, but also necessary, and are established without any a priori regularity assumptions on $v$ and $w$; we also obtain sharp two-sided estimates of solutions up to the boundary. Some of our results are new even if $v \equiv 1$ and $\Omega$ is a ball or half-space. The corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so-called $3 G$-inequality by an elementary ``integration by parts'' argument. This also gives a new unified proof for some classical inequalities including the Carleson measure theorem for Poisson integrals and trace inequalities for Riesz potentials and Green potentials.
On minimal parabolic functions and time-homogeneous parabolic $h$-transforms
Krzysztof
Burdzy;
Thomas
S.
Salisbury
3499-3531
Abstract: Does a minimal harmonic function $h$ remain minimal when it is viewed as a parabolic function? The question is answered for a class of long thin semi-infinite tubes $D\subset \mathbb{R}^{d}$ of variable width and minimal harmonic functions $h$ corresponding to the boundary point of $D$ ``at infinity.'' Suppose $f(u)$ is the width of the tube $u$ units away from its endpoint and $f$ is a Lipschitz function. The answer to the question is affirmative if and only if $\int ^{\infty }f^{3}(u)du = \infty$. If the test fails, there exist parabolic $h$-transforms of space-time Brownian motion in $D$ with infinite lifetime which are not time-homogenous.
Spectral gap estimates on compact manifolds
Kevin
Oden;
Chiung-Jue
Sung;
Jiaping
Wang
3533-3548
Abstract: For a compact Riemannian manifold with boundary, its mass gap is the difference between the first and second smallest Dirichlet eigenvalues. In this paper, taking a variational approach, we obtain an explicit lower bound estimate of the mass gap for any compact manifold in terms of geometric quantities.
Convex functions on Alexandrov surfaces
Yukihiro
Mashiko
3549-3567
Abstract: We investigate the topological structure of Alexandrov surfaces of curvature bounded below which possess convex functions. We do not assume the continuities of these functions. Nevertheless, if the convex functions satisfy a condition of local nonconstancy, then the topological structures of Alexandrov surfaces and the level sets configurations of these functions in question are determined.
A symplectic jeu de taquin bijection between the tableaux of King and of De Concini
Jeffrey
T.
Sheats
3569-3607
Abstract: The definitions, methods, and results are entirely combinatorial. The symplectic jeu de taquin algorithm developed here is an extension of Schützenberger's original jeu de taquin and acts on a skew form of De Concini's symplectic standard tableaux. This algorithm is used to construct a weight preserving bijection between the two most widely known sets of symplectic tableaux. Anticipated applications to Knuth relations and to decomposing symplectic tensor products are indicated.
Weight distributions of geometric Goppa codes
Iwan
M.
Duursma
3609-3639
Abstract: The in general hard problem of computing weight distributions of linear codes is considered for the special class of algebraic-geometric codes, defined by Goppa in the early eighties. Known results restrict to codes from elliptic curves. We obtain results for curves of higher genus by expressing the weight distributions in terms of $L$-series. The results include general properties of weight distributions, a method to describe and compute weight distributions, and worked out examples for curves of genus two and three.
Dense Egyptian fractions
Greg
Martin
3641-3657
Abstract: Every positive rational number has representations as Egyptian fractions (sums of reciprocals of distinct positive integers) with arbitrarily many terms and with arbitrarily large denominators. However, such representations normally use a very sparse subset of the positive integers up to the largest denominator. We show that for every positive rational there exist representations as Egyptian fractions whose largest denominator is at most $N$ and whose denominators form a positive proportion of the integers up to $N$, for sufficiently large $N$; furthermore, the proportion is within a small factor of best possible.
On Vassiliev knot invariants induced from finite type 3-manifold invariants
Matt
Greenwood;
Xiao-Song
Lin
3659-3672
Abstract: We prove that the knot invariant induced by a $\mathbb{Z}$-homology 3-sphere invariant of order $\leq k$ in Ohtsuki's sense, where $k\geq 4$, is of order $\leq k-2$. The method developed in our computation shows that there is no $\mathbb{Z}$-homology 3-sphere invariant of order 5.
Connectedness properties of limit sets
B.
H.
Bowditch
3673-3686
Abstract: We study convergence group actions on continua, and give a criterion which ensures that every global cut point is a parabolic fixed point. We apply this result to the case of boundaries of relatively hyperbolic groups, and consider implications for connectedness properties of such spaces.
Second-order subgradients of convex integral functionals
Mohammed
Moussaoui;
Alberto
Seeger
3687-3711
Abstract: The purpose of this work is twofold: on the one hand, we study the second-order behaviour of a nonsmooth convex function $F$ defined over a reflexive Banach space $X$. We establish several equivalent characterizations of the set $\partial^2F(\overline x,\overline y)$, known as the second-order subdifferential of $F$ at $\overline x$ relative to $\overline y\in \partial F(\overline x)$. On the other hand, we examine the case in which $F=I_f$ is the functional integral associated to a normal convex integrand $f$. We extend a result of Chi Ngoc Do from the space $X=L_{\mathbb R^d}^p$ $(1<p<+\infty)$ to a possible nonreflexive Banach space $X=L_E^p$ $(1\le p<+\infty)$. We also establish a formula for computing the second-order subdifferential $\partial ^2I_f(\overline x,\overline y)$.
A global condition for periodic Duffing-like equations
Piero
Montecchiari;
Margherita
Nolasco;
Susanna
Terracini
3713-3724
Abstract: We study Duffing-like equations of the type $\ddot q= q - \alpha (t)W'(q)$,with $\alpha \in C({\mathbb{R}},{\mathbb{R}})$ periodic. We prove that if the stable and unstable manifolds to the origin do not coincide, then the system exhibits positive topological entropy.
Exact Hausdorff measure and intervals of maximum density for Cantor sets
Elizabeth
Ayer;
Robert
S.
Strichartz
3725-3741
Abstract: Consider a linear Cantor set $K$, which is the attractor of a linear iterated function system (i.f.s.) $S_{j}x = \rho _{j}x+b_{j}$, $j = 1,\ldots ,m$, on the line satisfying the open set condition (where the open set is an interval). It is known that $K$ has Hausdorff dimension $\alpha$ given by the equation $\sum ^{m}_{j=1} \rho ^{\alpha }_{j} = 1$, and that $\mathcal{H}_{\alpha }(K)$ is finite and positive, where $\mathcal{H}_{\alpha }$ denotes Hausdorff measure of dimension $\alpha$. We give an algorithm for computing $\mathcal{H}_{\alpha }(K)$ exactly as the maximum of a finite set of elementary functions of the parameters of the i.f.s. When $\rho _{1} = \rho _{m}$ (or more generally, if $\log \rho _{1}$ and $\log \rho _{m}$ are commensurable), the algorithm also gives an interval $I$ that maximizes the density $d(I) = \mathcal{H}_{\alpha }(K \cap I)/|I|^{\alpha }$. The Hausdorff measure $\mathcal{H}_{\alpha }(K)$ is not a continuous function of the i.f.s. parameters. We also show that given the contraction parameters $\rho _{j}$, it is possible to choose the translation parameters $b_{j}$ in such a way that $\mathcal{H}_{\alpha }(K) = |K|^{\alpha }$, so the maximum density is one. Most of the results presented here were discovered through computer experiments, but we give traditional mathematical proofs.
On the $L^2\rightarrow L^\infty$ norms of spectral multipliers of ``quasi-homogeneous'' operators on homogeneous groups
Adam
Sikora
3743-3755
Abstract: We study the $L^2 \to L^{\infty}$ norms of spectral projectors and spectral multipliers of left-invariant elliptic and subelliptic second-order differential operators on homogeneous Lie groups. We obtain a precise description of the $L^2 \to L^{\infty}$ norms of spectral multipliers for some class of operators which we call quasi-homogeneous. As an application we prove a stronger version of Alexopoulos' spectral multiplier theorem for this class of operators.
Classes of singular integrals along curves and surfaces
Andreas
Seeger;
Stephen
Wainger;
James
Wright;
Sarah
Ziesler
3757-3769
Abstract: This paper is concerned with singular convolution operators in $\mathbb{R}^{d}$, $d\ge 2$, with convolution kernels supported on radial surfaces $y_{d}=\Gamma (|y'|)$. We show that if $\Gamma (s)=\log s$, then $L^{p}$ boundedness holds if and only if $p=2$. This statement can be reduced to a similar statement about the multiplier $m(\tau ,\eta )=|\tau |^{-i\eta }$ in $\mathbb{R}^{2}$. We also construct smooth $\Gamma$ for which the corresponding operators are bounded for $p_{0}<p\le 2$ but unbounded for $p\le p_{0}$, for given $p_{0}\in [1,2)$. Finally we discuss some examples of singular integrals along convex curves in the plane, with odd extensions.
Causal compactification and Hardy spaces
G.
Ólafsson;
B.
Ørsted
3771-3792
Abstract: Let $\mathcal{M}=G/H$ be a irreducible symmetric space of Cayley type. Then $\mathcal{M}$ is diffeomorphic to an open and dense $G$-orbit in the Shilov boundary of $G/K\times G/K$. This compactification of $\mathcal{M}$ is causal and can be used to give answers to questions in harmonic analysis on $\mathcal{M}$. In particular we relate the Hardy space of $\mathcal{M}$ to the classical Hardy space on the bounded symmetric domain $G/K\times G/K$. This gives a new formula for the Cauchy-Szegö kernel for $\mathcal{M}$.
On the degree of groups of polynomial subgroup growth
Aner
Shalev
3793-3822
Abstract: Let $G$ be a finitely generated residually finite group and let $a_n(G)$ denote the number of index $n$ subgroups of $G$. If $a_n(G) \le n^{\alpha}$ for some $\alpha$ and for all $n$, then $G$ is said to have polynomial subgroup growth (PSG, for short). The degree of $G$ is then defined by ${\mathrm{deg}}(G) = \limsup {{\log a_n(G)} \over {\log n}}$. Very little seems to be known about the relation between ${\mathrm{deg}}(G)$ and the algebraic structure of $G$. We derive a formula for computing the degree of certain metabelian groups, which serves as a main tool in this paper. Addressing a problem posed by Lubotzky, we also show that if $H \le G$ is a finite index subgroup, then ${\mathrm{deg}}(G) \le {\mathrm{deg}}(H)+1$. A large part of the paper is devoted to the structure of groups of small degree. We show that $a_n(G)$ is bounded above by a linear function of $n$ if and only if $G$ is virtually cyclic. We then determine all groups of degree less than $3/2$, and reveal some connections with plane crystallographic groups. It follows from our results that the degree of a finitely generated group cannot lie in the open interval $(1, 3/2)$. Our methods are largely number-theoretic, and density theorems à la Chebotarev play essential role in the proofs. Most of the results also rely implicitly on the Classification of Finite Simple Groups.
The trace space and Kauffman's knot invariants
Keqin
Liu
3823-3842
Abstract: The traces in the construction of Kauffman's knot invariants are studied. The trace space is determined for a semisimple finite-dimensional quantum Hopf algebra and the best lower bound of the dimension of the trace space is given for a unimodular finite-dimensional quantum Hopf algebra.
Direct sum decompositions of infinitely generated modules
D.
J.
Benson;
Wayne
W.
Wheeler
3843-3855
Abstract: Almost all of the basic theorems in the representation theory of finite groups have proofs that depend upon the Krull-Schmidt Theorem. Because this theorem holds only for finite-dimensional modules, however, the recent interest in infinitely generated modules raises the question of which results may hold more generally. In this paper we present an example showing that Green's Indecomposability Theorem fails for infinitely generated modules. By developing and applying some general properties of idempotent modules, we are also able to construct explicit examples of modules for which the cancellation property fails.
On the number of terms in the middle of almost split sequences over tame algebras
J.
A.
de la Peña;
M.
Takane
3857-3868
Abstract: Let $A$ be a finite dimensional tame algebra over an algebraically closed field $k$. It has been conjectured that any almost split sequence $0 \to X \to \oplus _{i=1} ^n Y_i \to Z \to 0$ with $Y_i$ indecomposable modules has $n \le 5$ and in case $n=5$, then exactly one of the $Y_i$ is a projective-injective module. In this work we show this conjecture in case all the $Y_i$ are directing modules, that is, there are no cycles of non-zero, non-iso maps $Y_i =M_1 \to M_2 \to \cdots \to M_s=Y_i$ between indecomposable $A$-modules. In case, $Y_1$ and $Y_2$ are isomorphic, we show that $n \le 3$ and give precise information on the structure of $A$.