On functions in the little Bloch space and inner functions
S.
Rohde
2519-2531
Abstract: We prove that analytic functions in the little Bloch space assume every value as a radial limit on a set of Hausdorff dimension one, unless they have radial limits on a set of positive measure. The analogue for inner functions in the little Bloch space is also proven, and characterizations of various classes of Bloch functions in terms of their level sets are given.
Propagation of Gevrey Regularity for a Class of Hypoelliptic Equations
Antonio
Bove;
David
S.
Tartakoff
2533-2575
Abstract: We prove results on the propagation of Gevrey and analytic wave front sets for a class of $C^\infty$ hypoelliptic equations with double characteristics.
Global (and local) analyticity for second order operators constructed from rigid vector fields on products of tori
David
S.
Tartakoff
2577-2583
Abstract: We prove global analytic hypoellipticity on a product of tori for partial differential operators which are constructed as rigid (variable coefficient) quadratic polynomials in real vector fields satisfying the Hörmander condition and where $P$ satisfies a ``maximal'' estimate. We also prove an analyticity result that is local in some variables and global in others for operators whose prototype is \begin{displaymath}P=\left ( \frac \partial {\partial x_1}\right ) ^2+\left ( \frac \partial { \partial x_2}\right ) ^2+\left ( a(x_1,x_2)\frac \partial {\partial t}\right )^2 \end{displaymath} (with analytic $a(x),a(0)=0$, naturally, but not identically zero). The results, because of the flexibility of the methods, generalize recent work of Cordaro and Himonas in [4] and Himonas in [8] which showed that certain operators known not to be locally analytic hypoelliptic (those of Baouendi and Goulaouic [1], Hanges and Himonas [6], and Christ [3]) were globally analytic hypoelliptic on products of tori.
Immersed $n$-manifolds in $\mathbf{R}^{2n}$ and the double points of their generic projections into $\mathbf{R}^{2n-1}$
Osamu
Saeki;
Kazuhiro
Sakuma
2585-2606
Abstract: We give two congruence formulas concerning the number of non-trivial double point circles and arcs of a smooth map with generic singularities --- the Whitney umbrellas --- of an $n$-manifold into $\text {\bf R}^{2n-1}$, which generalize the formulas by Szücs for an immersion with normal crossings. Then they are applied to give a new geometric proof of the congruence formula due to Mahowald and Lannes concerning the normal Euler number of an immersed $n$-manifold in $\text {\bf R}^{2n}$. We also study generic projections of an embedded $n$-manifold in $\text {\bf R}^{2n}$ into $\text {\bf R}^{2n-1}$ and prove an elimination theorem of Whitney umbrella points of opposite signs, which is a direct generalization of a recent result of Carter and Saito concerning embedded surfaces in $\text {\bf R}^{4}$. The problem of lifting a map into $\text {\bf R}^{2n-1}$ to an embedding into $\text {\bf R}^{2n}$ is also studied.
Simplifying stable mappings into the plane from a global viewpoint
Mahito
Kobayashi;
Osamu
Saeki
2607-2636
Abstract: Let $f : M \to \text {\bf R}^{2}$ be a $C^{\infty }$ stable map of an $n$-dimensional manifold into the plane. The main purpose of this paper is to define a global surgery operation on $f$ which simplifies the configuration of the critical value set and which does not change the diffeomorphism type of the source manifold $M$. For this purpose, we also study the quotient space $W_{f}$ of $f$, which is the space of the connected components of the fibers of $f$, and we completely determine its local structure for arbitrary dimension $n$ of the source manifold $M$. This is a completion of the result of Kushner, Levine and Porto for dimension 3 and that of Furuya for orientable manifolds of dimension 4. We also pay special attention to dimension 4 and obtain a simplification theorem for stable maps whose regular fiber is a torus or a 2-sphere, which is a refinement of a result of Kobayashi.
Rotation Vectors and Fixed Points of Area Preserving Surface Diffeomorphisms
John
Franks
2637-2662
Abstract: We consider the (homological) rotation vectors for area preserving diffeomorphisms of compact surfaces which are homotopic to the identity. There are two main results. The first is that if $0$ is in the interior of the convex hull of the rotation vectors for such a diffeomorphism then $f$ has a fixed point of positive index. The second result asserts that if $f$ has a vanishing mean rotation vector then $f$ has a fixed point of positive index. There are several applications including a new proof of the Arnold conjecture for area preserving diffeomorphisms of compact surfaces.
Extremal functions for Moser's inequality
Kai-Ching
Lin
2663-2671
Abstract: Let $\Omega$ be a bounded smooth domain in $R^{n}$, and $u(x)$ a $C^{1}$ function with compact support in $\Omega$. Moser's inequality states that there is a constant $c_{o}$, depending only on the dimension $n$, such that \begin{equation*}% \frac {1}{|\Omega |} \int _{\Omega } e^{n \omega _{n-1}^{\frac {1}{n-1}} u^{\frac {n}{n-1}}}\, dx \leq c_{o}% , \end{equation*} where $|\Omega |$ is the Lebesgue measure of $\Omega$, and $\omega _{n-1}$ the surface area of the unit ball in $R^{n}$. We prove in this paper that there are extremal functions for this inequality. In other words, we show that the \begin{equation*}% \sup \{\frac {1}{|\Omega |} \int _{\Omega } e^{n \omega _{n-1}^{\frac {1}{n-1}} u^{\frac {n}{n-1}}}\, dx: u \in W_{o}^{1,n}, \|\nabla u\|_{n} \leq 1 \} % \end{equation*} is attained. Earlier results include Carleson-Chang (1986, $\Omega$ is a ball in any dimension) and Flucher (1992, $\Omega$ is any domain in 2-dimensions).
Spectral Convergence for Degenerating Sequences of Three Dimensional Hyperbolic Manifolds
Lizhen
Ji
2673-2688
Abstract: For degenerating sequences of three dimensional hyperbolic manifolds of finite volume, we prove convergence of their eigenfunctions, heat kernel and spectral measure.
Reciprocity Laws in the Verlinde Formulae for the Classical Groups
W.
M.
Oxbury;
S.
M. J.
Wilson
2689-2710
Abstract: The Verlinde formula is computed for each of the simply-connected classical Lie groups, and it is shown that the resulting formula obeys certain reciprocity laws with respect to the exchange of the rank and the level. Some corresponding dualities between spaces of sections of theta line bundles over moduli spaces of $G$-bundles on curves are conjectured but not proved.
Low codimensional submanifolds of Euclidean space with nonnegative isotropic curvature
Francesco
Mercuri;
Maria
Helena
Noronha
2711-2724
Abstract: In this paper we study compact submanifolds of Euclidean space with nonnegative isotropic curvature and low codimension. We determine their homology completely in the case of hypersurfaces and for some low codimensional conformally flat immersions.
Divisors on Generic Complete Intersections in Projective Space
Geng
Xu
2725-2736
Abstract: Let $V$ be a generic complete intersection of hypersurfaces of degree $d_{1}, d_{2}, \cdots , d_{m}$ in $n$-dimensional projective space. We study the question when a divisor on $V$ is nonrational or of general type, and give an alternative proof of a result of Ein. We also give some improvement of Ein's result in the case $d_{1}+d_{2}+\cdots + d_{m}=n+2$.
On complete nonorientable minimal surfaces with low total curvature
Francisco
J.
Lopez
2737-2758
Abstract: We classify complete nonorientable minimal surfaces in $\mathbb R^3$ with total curvature $-8\pi$.
On Jacobian Ideals Invariant by a Reducible $s\ell(2,\mathbf{C})$ Action
Yung
Yu
2759-2791
Abstract: This paper deals with a reducible $s\ell (2, \mathbf {C})$ action on the formal power series ring. The purpose of this paper is to confirm a special case of the Yau Conjecture: suppose that $s\ell (2, \mathbf {C})$ acts on the formal power series ring via $(0.1)$. Then $I(f)=(\ell _{i_{1}})\oplus (\ell _{i_{2}})\oplus \cdots \oplus (\ell _{i_{s}})$ modulo some one dimensional $s\ell (2, \mathbf {C})$ representations where $(\ell _{i})$ is an irreducible $s\ell (2, \mathbf {C})$ representation of dimension $\ell _{i}$ or empty set and $\{\ell _{i_{1}},\ell _{i_{2}},\ldots ,\ell _{i_{s}}\}\subseteq \{\ell _{1},\ell _{2},\ldots ,\ell _{r}\}$. Unlike classical invariant theory which deals only with irreducible action and 1--dimensional representations, we treat the reducible action and higher dimensional representations succesively.
Finite-dimensional lattice-subspaces of $C(\Omega)$ and curves of $\mathbb{R}^n$
Ioannis
A.
Polyrakis
2793-2810
Abstract: Let $x_1,\dotsc ,x_n$ be linearly independent positive functions in $C(\Omega )$, let $X$ be the vector subspace generated by the $x_i$ and let $\beta$ denote the curve of $\mathbb R^n$ determined by the function $\beta (t)=\frac {1}{z(t)} (x_1(t),x_2(t),\dotsc ,x_n(t))$, where $z(t)=x_1(t)+x_2(t)+\dotsb +x_n(t)$. We establish that $X$ is a vector lattice under the induced ordering from $C(\Omega )$ if and only if there exists a convex polygon of $\mathbb R^n$ with $n$ vertices containing the curve $\beta$ and having its vertices in the closure of the range of $\beta$. We also present an algorithm which determines whether or not $X$ is a vector lattice and in case $X$ is a vector lattice it constructs a positive basis of $X$. The results are also shown to be valid for general normed vector lattices.
On a parabolic equation with a singular lower order term
Qi
Zhang
2811-2844
Abstract: We obtain the existence of the weak Green's functions of parabolic equations with lower order coefficients in the so called parabolic Kato class which is being proposed as a natural generalization of the Kato class in the study of elliptic equations. As a consequence we are able to prove the existence of solutions of some initial boundary value problems. Moreover, based on a lower and an upper bound of the Green's function, we prove a Harnack inequality for the non-negative weak solutions.
Strong laws for $L$- and $u$-statistics
J.
Aaronson;
R.
Burton;
H.
Dehling;
D.
Gilat;
T.
Hill;
B.
Weiss
2845-2866
Abstract: Strong laws of large numbers are given for $L$-statistics (linear combinations of order statistics) and for $U$-statistics (averages of kernels of random samples) for ergodic stationary processes, extending classical theorems of Hoeffding and of Helmers for iid sequences. Examples are given to show that strong and even weak convergence may fail if the given sufficient conditions are not satisfied, and an application is given to estimation of correlation dimension of invariant measures.
On Gelfand-Kirillov Transcendence Degree
James
J.
Zhang
2867-2899
Abstract: We study some basic properties of the Gelfand-Kirillov transcendence degree and compute the transcendence degree of various infinite-dimensional division algebras including quotient division algebras of quantized algebras related to quantum groups, 3-dimensional Artin-Schelter regular algebras and the 4-dimensional Sklyanin algebra.
Properties of extremal functions for some nonlinear functionals on Dirichlet spaces
Alec
Matheson;
Alexander
R.
Pruss
2901-2930
Abstract: Let $\mathfrak {B}$ be the set of holomorphic functions $f$ on the unit disc $D$ with $f(0)=0$ and Dirichlet integral $(1/\pi ) \iint _{D} |f'|^{2}$ not exceeding one, and let $\mathfrak {b}$ be the set of complex-valued harmonic functions $f$ on the unit disc with $f(0)=0$ and Dirichlet integral $(1/2)(1/\pi ) \iint _{D} |\nabla f|^{2}$ not exceeding one. For a (semi)continuous function $\Phi :[0,\infty ) \to \mathbb {R}$, define the nonlinear functional $\Lambda _{\Phi }$ on $\mathfrak {B}$ or $\mathfrak {b}$ by $\Lambda _{\Phi }(f)={\frac {1}{2\pi }} \int _{0}^{2\pi }\Phi (|f(e^{i\theta })|)\,d\theta$. We study the existence and regularity of extremal functions for these functionals, as well as the weak semicontinuity properties of the functionals. We also state a number of open problems.
Representations of monoids arising from finite groups of Lie type
A.
Salwa
2931-2945
Abstract: A class of finite monoids $M$ constructed from a group $G$ of Lie type is considered. We describe the irreducible complex representations and prove the complete reducibility of the representations of $M$. The sandwich matrix of $M$ is decomposed into a product of matrices corresponding to maximal parabolic subgroups of $G$.