Some P.V.-equivalences and a classification of $2$-simple prehomogeneous vector spaces of type ${\rm II}$
Tatsuo
Kimura;
Shin-ichi
Kasai;
Masanobu
Taguchi;
Masaaki
Inuzuka
433-494
Abstract: A classification of $ 2$-simple prehomogeneous vector spaces is completed by using some P.V.-equivalences together with [3]. Some part is very different from the previous classification of the irreducible or simple cases [1, 2], and some new method is necessary. This result shows the difficult point of a classification problem of reductive prehomogeneous vector spaces.
Hardy spaces of vector-valued functions: duality
Oscar
Blasco
495-507
Abstract: We prove here that the Hardy space of $B$-valued functions ${H^1}(B)$ defined by using the conjugate function and the one defined in terms of $B$-valued atoms do not coincide for a general Banach space. The condition for them to coincide is the UMD property on $B$. We also characterize the dual space of both spaces, the first one by using $ {B^{\ast}}$-valued distributions and the second one in terms of a new space of vector-valued measures, denoted $ \mathcal{B}\mathcal{M}\mathcal{O}({B^{\ast}})$, which coincides with the classical $\operatorname{BMO} ({B^{\ast}})$ of functions when ${B^{\ast}}$ has the RNP.
The number of solutions to linear Diophantine equations and multivariate splines
Wolfgang
Dahmen;
Charles A.
Micchelli
509-532
Abstract: In this paper we study how the number of nonnegative integer solutions of $s$ integer linear equations in $n \geqslant s$ unknowns varies as a function of the inhomogeneous terms. Aside from deriving various recurrence relations for this function, we establish some of its detailed structural properties. In particular, we show that on certain subsets of lattice points it is a polynomial. The univariate case ($s = 1$) yields E. T. Bell's description of Sylvester's denumerants. Our approach to this problem relies upon the use of polyhedral splines. As an example of this method we obtain results of R. Stanley on the problem of counting the number of magic squares.
A characterization of two weight norm inequalities for fractional and Poisson integrals
Eric T.
Sawyer
533-545
Abstract: For $1 < p \leqslant q < \infty$ and $ w(x)$, $v(x)$ nonnegative functions on $ {{\mathbf{R}}^n}$, we show that the weighted inequality $\displaystyle {\left( {\int {\vert Tf{\vert^q}w} } \right)^{1/q}} \leqslant C{\left( {\int {{f^p}v} } \right)^{1/p}}$ holds for all $f \geqslant 0$ if and only if both $\displaystyle \int {{{[T({\chi _Q}{v^{1 - p'}})]}^q}w \leqslant {C_1}{{\left( {\int_Q {{v^{1 - p'}}} } \right)}^{q/p}} < \infty }$ and $\displaystyle {\int {{{[T({\chi _Q}w)]}^{p'}}{v^{1 - p'}} \leqslant {C_2}\left( {\int_Q w } \right)} ^{p'/q'}} < \infty $ hold for all dyadic cubes $Q$. Here $T$ denotes a fractional integral or, more generally, a convolution operator whose kernel $K$ is a positive lower semicontinuous radial function decreasing in $\vert x\vert$ and satisfying $K(x) \leqslant CK(2x)$, $x \in {{\mathbf{R}}^n}$. Applications to degenerate elliptic differential operators are indicated. In addition, a corresponding characterization of those weights $v$ on $ {{\mathbf{R}}^n}$ and $ w$ on ${\mathbf{R}}_ + ^{n + 1}$ for which the Poisson operator is bounded from ${L^p}(v)$ to ${L^q}(w)$ is given.
Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators
K. F.
Andersen;
E. T.
Sawyer
547-558
Abstract: The weight functions $u(x)$ for which $ {R_\alpha }$, the Riemann-Liouville fractional integral operator of order $\alpha > 0$, is bounded from ${L^p}({u^p}\,dx)$ to $ {L^q}({u^q}\,dx)$, $1 < p < 1/\alpha$, $1/q = 1/p - \alpha$, are characterized. Further, given $p$,$q$ with $ 1/q \geqslant 1/p - \alpha$, the weight functions $u > 0$ a.e. (resp. $ v < \infty$ a.e.) for which there is $v < \infty $ a.e. (resp. $ u > 0$ a.e.) so that ${R_\alpha }$ is bounded from ${L^p}({v^p}\,dx)$ to ${L^q}({u^q}\,dx)$ are characterized. Analogous results are obtained for the Weyl fractional integral. The method involves the use of complex interpolation of analytic families of operators to obtain similar results for fractional "one-sided" maximal function operators which are of independent interest.
Orthogonal polynomials on several intervals via a polynomial mapping
J. S.
Geronimo;
W.
Van Assche
559-581
Abstract: Starting from a sequence $ \{ {p_n}(x;\,{\mu _0})\}$ of orthogonal polynomials with an orthogonality measure ${\mu _0}$ supported on ${E_0} \subset [ - 1,\,1]$, we construct a new sequence $ \{ {p_n}(x;\,\mu )\}$ of orthogonal polynomials on $E = {T^{ - 1}}({E_0})$ ($T$ is a polynomial of degree $N$) with an orthogonality measure $ \mu$ that is related to $ {\mu _0}$. If ${E_0} = [ - 1,\,1]$, then $E = {T^{ - 1}}([ - 1,\,1])$ will in general consist of $N$ intervals. We give explicit formulas relating $\{ {p_n}(x;\,\mu )\}$ and $\{ {p_n}(x;\,{\mu _0})\}$ and show how the recurrence coefficients in the three-term recurrence formulas for these orthogonal polynomials are related. If one chooses $T$ to be a Chebyshev polynomial of the first kind, then one gets sieved orthogonal polynomials.
Deficient values and angular distribution of entire functions
Lo
Yang
583-601
Abstract: Let $f(z)$ be an entire function of positive and finite order $\mu$. If $f(z)$ has a finite number of Borel directions of order $\geqslant \mu $, then the sum of numbers of finite nonzero deficient values of $f(z)$ and all its primitives does not exceed $2\mu$. The proof is based on several lemmas and application of harmonic measure.
The central limit theorem for empirical processes under local conditions: the case of Radon infinitely divisible limits without Gaussian component
Niels T.
Andersen;
Evarist
Giné;
Joel
Zinn
603-635
Abstract: Weak convergence results are obtained for empirical processes indexed by classes $ \mathcal{F}$ of functions in the case of infinitely divisible purely Poisson (in particular, stable) Radon limits, under conditions on the local modulus of the processes $\{ f(X):\,f \in \mathcal{F}\}$ ("bracketing" conditions). They extend (and slightly improve upon) a central limit theorem of Marcus and Pisier (1984) for Lipschitzian processes. The law of the iterated logarithm is also considered. The examples include Marcinkiewicz type laws of large numbers for weighted empirical processes and for the dual-bounded-Lipschitz distance between a probability in $ {\mathbf{R}}$ and its associated empirical measures.
Eigensharp graphs: decomposition into complete bipartite subgraphs
Thomas
Kratzke;
Bruce
Reznick;
Douglas
West
637-653
Abstract: Let $\tau (G)$ be the minimum number of complete bipartite subgraphs needed to partition the edges of $G$, and let $r(G)$ be the larger of the number of positive and number of negative eigenvalues of $ G$. It is known that $\tau (G) \geqslant r(G)$; graphs with $\tau (G) = r(G)$ are called eigensharp. Eigensharp graphs include graphs, trees, cycles ${C_n}$ with $n = 4$ or $n \ne 4k$, prisms ${C_n}\square {K_2}$ with $n \ne 3k$, "twisted prisms" (also called "Möbius ladders") ${M_n}$ with $n = 3$ or $n \ne 3k$, and some Cartesian products of cycles. Under some conditions, the weak (Kronecker) product of eigensharp graphs is eigensharp. For example, the class of eigensharp graphs with the same number of positive and negative eigenvalues is closed under weak products. If each graph in a finite weak product is eigensharp, has no zero eigenvalues, and has a decomposition into $\tau (G)$ stars, then the product is eigensharp. The hypotheses in this last result can be weakened. Finally, not all weak products of eigensharp graphs are eigensharp.
Differentiation theorem for Gaussian measures on Hilbert space
Jaroslav
Tišer
655-666
Abstract: It is shown that the differentiation theorem is valid in infinitely dimensional Hilbert space with certain Gaussian measures. The proof uses result from harmonic analysis concerning the behavior of Hardy-Littlewood maximal operator in highly dimensional space.
Explicit formula for weighted scalar nonlinear hyperbolic conservation laws
Philippe
LeFloch;
Jean-Claude
Nédélec
667-683
Abstract: We prove a uniqueness and existence theorem for the entropy weak solution of nonlinear hyperbolic conservation laws of the form $\displaystyle \frac{\partial } {{\partial t}}(ru) + \frac{\partial } {{\partial x}}(rf(u)) = 0,$ with initial data and boundary condition. The scalar function $u = u(x,\,t)$, $x > 0$, $t > 0$, is the unknown, the function $f = f(u)$ is assumed to be strictly convex with inf $ f( \cdot ) = 0$ and the weight function $r = r(x)$, $x > 0$, to be positive (for example, $r(x) = {x^\alpha }$, with an arbitrary real $ \alpha$). We give an explicit formula, which generalizes a result of P. D. Lax. In particular, a free boundary problem for the flux $ r( \cdot )f(u( \cdot , \cdot ))$ at the boundary is solved by introducing a variational inequality. The uniqueness result is obtained by extending a semigroup property due to B. L. Keyfitz.
Complex interpolation of normed and quasinormed spaces in several dimensions. I
Zbigniew
Slodkowski
685-711
Abstract: A variety of complex interpolation methods for families of normed or quasi-normed spaces, parametrized by points of domains in complex homogeneous spaces, parametrized by points of domains in complex homogeneous spaces, is developed. Results on existence, continuity, uniqueness, reiteration and duality for interpolation are proved, as well as on interpolation of operators. A minimum principle for plurisubharmonic functions is obtained and used as a tool for the duality theorem.
Construction of an inner function in the little Bloch space
Kenneth
Stephenson
713-720
Abstract: An explicit construction using Riemann surfaces and Brownian motion is given for an inner function in the unit disc which is not a finite Blaschke product yet belongs to the little Bloch space $ {\mathcal{B}_0}$. In addition to showing how an inner function can meet the geometric conditions for $ {\mathcal{B}_0}$, this example settles an open question concerning the finite ranges of inner functions: the values which it takes only finitely often are dense in the disc.
Equivalence and strong equivalence of actions on handlebodies
John
Kalliongis;
Andy
Miller
721-745
Abstract: An algebraic characterization is given for the equivalence and strong equivalence classes of finite group actions on $ 3$-dimensional handlebodies. As one application it is shown that each handlebody whose genus is bigger than one admits only finitely many finite group actions up to equivalence. In another direction, the algebraic characterization is used as a basis for deriving an explicit combinatorial description of the equivalence and strong equivalence classes of the cyclic group actions of prime order on handlebodies with genus larger than one. This combinatorial description is used to give a complete closed-formula enumeration of the prime order cyclic group actions on such handlebodies.
${\rm SO}(2)$-equivariant vector fields on $3$-manifolds: moduli of stability and genericity
Genesio Lima
dos Reis;
Geovan Tavares
dos Santos
747-763
Abstract: An open and dense class of vector fields on $3$-dimensional compact manifolds equivariant under the action of $ \operatorname{SO} (2)$ is defined. Each such vector field has finite moduli of stability. We also exhibit an open and dense subset of the $ \operatorname{SO} (2)$-equivariant gradient vector fields which are structurally stable.
Templates and train tracks
George
Frank
765-784
Abstract: Within the context of Smale flows on compact manifolds, this article deals with a relationship between abstract templates, branched $1$-manifolds (train tracks), and laminations representing unstable separatrices of basic sets. We show that an abstract template, the richest in information of the above three entities, determines a member of each of the remaining two groups, and partial determinations in other directions are developed. As a result of this relationship, an obstruction to the realization of certain abstract templates in nonsingular Smale flows on homology $3$-spheres is raised.
Normal derivative for bounded domains with general boundary
Guang Lu
Gong;
Min Ping
Qian;
Martin L.
Silverstein
785-809
Abstract: Let $D$ be a general bounded domain in the Euclidean space ${R^n}$. A Brownian motion which enters from and returns to the boundary symmetrically is used to define the normal derivative as a functional for $ f$ with $f$, $\nabla f$ and $\Delta f$ all in ${L^2}$ on $D$. The corresponding Neumann condition (normal derivative $= 0$) is an honest boundary condition for the $ {L^2}$ generator of reflected Brownian notion on $D$. A conditioning argument shows that for $ D$ and $f$ sufficiently smooth this general definition of the normal derivative agrees with the usual one.
Wiener's criterion for parabolic equations with variable coefficients and its consequences
Nicola
Garofalo;
Ermanno
Lanconelli
811-836
Abstract: In a bounded set in ${{\mathbf{R}}^{n + 1}}$ we study the problem of the regularity of boundary points for the Dirichlet problem for a parabolic operator with smooth coefficients. We give a geometric characterization, modelled on Wiener's criterion for Laplace's equation, of those boundary points that are regular. We also present some important consequences. Here is the main one: a point is regular for a variable coefficient operator if and only if it is regular for the constant coefficient operator obtained by freezing the coefficients at that point.
Rigidity theorems for right angled reflection groups
Ennis
Rosas
837-848
Abstract: Let $\Gamma$ be a right angled reflection group. Let $M$ and $M'$ be Coxeter manifolds. Then any $ \Gamma$-map
Engulfing and subgroup separability for hyperbolic groups
D. D.
Long
849-859
Abstract: If a group is subgroup separable, otherwise known as locally extended residually finite or LERF, one can pass from immersions to embeddings in some finite covering space. We show that a certain 'engulfing' property gives subgroup separability for a large and useful class of subgroups of hyperbolic $3$-manifold groups.