Showing papers in "Transactions of the American Mathematical Society in 1988"

Hausdorff dimension in graph directed constructions

Abstract: We introduce the notion of geometric constructions in Rm governed by a directed graph G and by similarity ratios which are labelled with the edges of this graph. For each such construction, we calculate a number a which is the Hausdorff dimension of the object constructed from a realization of the construction. The measure of the object with respect to fl\"* is always positive and

Tauberian theorems and stability of one-parameter semigroups

Abstract: The main result is the following stability theorem: Let T = (T(t))tzo be a bounded Co-semigroup on a reflexive space X. Denote by A the generator of T and by a(A) the spectrum of A. If a(A) n iR is countable and no eigenvalue of A lies on the imaginary axis, then limt BOO T(t)x = O for all x E X.

Bounds on the ² spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality

Abstract: We prove a general version of Cheeger's inequality for discrete-time Markov chains and continuous-time Markovian jump processes, both reversible and nonreversible, with general state space. We also prove a version of Cheeger's inequality for Markov chains and processes with killing. As an application, we prove L 2 exponential convergence to equilibrium for random walk with inward drift on a class of countable rooted graphs

Interpolation of Besov-Spaces

Abstract: We investigate Besov spaces and their connection with dyadic spline approximation in Lp(Q), 0 < p < oo. Our main results are: the determination of the interpolation spaces between a pair of Besov spaces; an atomic decomposition for functions in a Besov space; the characterization of the class of functions which have certain prescribed degree of approximation by dyadic splines.

New combinatorial interpretations of Ramanujan’s partition congruences mod 5,7 and 11

Abstract: Soit p(n) le nombre de partitions non restreintes de n. On donne de nouvelles interpretations combinatoires des congruences p(5n+4), p(7n+5) et p(11n+6)≡0 (mod 5,7 et 11, respectivement)

A characterization of two weight norm inequalities for fractional and Poisson integrals

Abstract: Caracterisation de deux inegalites de normes ponderees pour les integrales fractionnaires et de Poisson. Applications aux operateurs differentiels elliptiques degeneres

Finite order solutions of second order linear differential equations

Abstract: On considere l'equation differentielle f''+A(z)f'+B(z) f=0 ou A(z) et B(z) sont des fonctions entieres. On trouve des conditions sur A(z) et B(z) qui garantissent que chaque solution f¬=0 de l'equation a un ordre fini

Uniformly fat sets

Abstract: In this paper we study closed sets E which are "locally uniformly fat" with respect to a certain nonlinear Riesz capacity. We show that E is actually "locally uniformly fat" with respect to a weaker Riesz capacity. Two applications of this result are given. The first application is concerned with proving Sobolev-type inequalities in domains whose complements are uniformly fat. The second application is concerned with the Fekete points of E. Introduction. Let x = (X ,X2, ... , x) be a point in Euclidean n space, Rn, with jxl the norm of x. For a > 0, let I,>(x) = IXI(a-n) denote the Riesz kernel of order a and put I, * f(x) = f Ix _ y(&f-n)f(y) dy, x E R , Rn IC, * ,u(x) = fR _ -yj(c-n) d,u(y), x E R , Rn when f is a Lebesgue integrable function and ,u a Borel measure on Rn. Here dy denotes Lebesgue measure on Rn. If E C Rn, 0 1, define the (a, p) outer Riesz capacity of E by Ra,P(E) = inf{jfllfP: Ia, * f > 1 on El, where If lIp is the Lebesgue p norm of f. We note that a = 1, p = 2, is the classical Newtonian capacity. If ap = n, we let Rc,p(E) = inf{jfjjf|: J, * f > 1 on El, where Jc, is the truncated Riesz kernel defined by Je(x) = IXI(a-n) _ (100)(a-n), |x ? 100, = 0, lxl > 100. We shall say that a property holds (a, p) quasi everywhere (abbreviated (a, p) q.e.) on a set E if it holds on E except perhaps for a set of Rop capacity zero. Next we list some properties of Riesz capacities. To simplify the writing we state these properties only for 0 * u)l/(P-l) > 1, (a,p) q.e., on E and P = 1, (a,p) q.e., on supp,u C E. Moreover, _ _(E) = jjlI * pI,ujj = Rop(E) Received by the editors March 11, 1987. 1980 Mathematic3 Subject Cla33ification (1985 Revi3ion). Primary 31C15, 31B15. (?1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page

A construction of pseudo-Anosov homeomorphisms

Abstract: We describe a generalization of Thurston's original construction of pseudo-Anosov maps on a surface F of negative Euler characteristic. In fact, we construct whole semigroups of pseudo-Anosov maps by taking appropriate compositions of Dehn twists along certain families of curves; our arguments furthermore apply to give examples of pseudo-Anosov maps on nonorientable surfaces. For each self-map f: F -* F arising from our recipe, we construct an invariant "bigon track" (a slight generalization of train track) whose incidence matrix is Perron-Frobenius. Standard arguments produce a projective measured foliation invariant by f. To finally prove that f is pseudo-Anosov, we directly produce a transverse invariant projective measured foliation using tangential measures on bigon tracks. As a consequence of our argument, we derive a simple criterion for a surface automorphism to be pseudo-Anosov. Introduction. A homeomorphism p of a surface F is said to be pseudo-Anoosov if no iterate of p fixes any essential nonboundaryor puncture-parallel free homotopy class of simple curves in F. Examples of these homeomorphisms date back to the work of Nielsen (see [N and Gi]), but a systematic study of these maps was not undertaken until the work of Thurston [T1]. Anosov [A] studied maps of the torus which preserve two foliations of the torus by lines of irrational slope, and pseudo-Anosov maps on F similarly preserve a pair of foliations (with singularities). Pseudo-Anosov maps are by no means special; indeed, the monodromy of any nontorus fibred knot which is not a satellite is pseudo-Anosov [T4]. (Note that being pseudo-Anosov is a conjugacy invariant.) Moreover, these maps play an important role in the geometrization of three-manifolds; indeed, a mapping torus has hyperbolic structure if and only if the monodromy is pseudo-Anosov [T4]. In the original preprint [T1], there is described a construction of pseudo-Anosov maps which we will recall later. In this paper, we generalize Thurston's construction and give a recipe for constructing whole semigroups of pseudo-Anosov maps, many of which do not arise from Thurston's construction. Our recipe is also applicable to nonorientable surfaces, and we give examples of pseudo-Anosov maps in this setting. (In [T3], Thurston proved the existence of such, and [AY] gave the first explicit examples.) This paper is organized as follows. In ?1, we review the basic terminology and results on train tracks in surfaces and indicate the connection between measured train tracks and measured foliations. ?2 is devoted to tangential measure on bigon Received by the editors March 7, 1986 and, in revised form, June 17, 1987. 1980 Mathematics Subject (Cassification (1985 Revision). Primary 57N06, 57N50.

First- and second-order epi-differentiability in nonlinear programming

Abstract: Problems are considered in which an objective function expressible as a max of finitely many C2 functions, or more generally as the composition of a piecewise linear-quadratic function with a C2 mapping, is minimized subject to finitely many C2 constraints. The essential objective function in such a problem, which is the sum of the given objective and the indicator of the constraints, is shown to be twice epi-differentiable at any point where the active constraints (if any) satisfy the Mangasarian-Fromovitz qualification. The epi-derivatives are defined by taking epigraphical limits of classical firstand second-order difference quotients instead of pointwise limits, and they reveal properties of local geometric approximation that have not previously been observed.

Spinor bundles on quadrics

Abstract: We define some stable vector bundles on the complex quadric hypersurface Qn of dimension n as the natural generalization of the universal bundle and the dual of the quotient bundle on Q4 ~ Gr(l,3). We call them spinor bundles. When n = 2fc — 1 there is one spinor bundle of rank 2k~1. When n = 2k there are two spinor bundles of rank 2k~1. Their behavior is slightly different according as n = 0 (mod 4) or n = 2 (mod 4). As an application, we describe some moduli spaces of rank 3 vector bundles on Q5 and Qe- Introduction. Let Qn be the smooth quadric hypersurface of the complex pro- jective space Pn+1. In this paper we define in a geometrical way some vector bundles on the quadric Qn as the natural generalization of the universal bundle and the dual of the quotient bundle on Q4 ~ Gr(l,3). We call them spinor bundles. On Q4 this definition is equivalent to the usual one. Spinor bundles are homogeneous and stable (according to the definition of Mum- ford-Takemoto). We study their first properties using the geometrical description given and some standard techniques available in (OSS). We also use a theorem of Ramanan (see (Um)) about the stability of homoge- neous bundles induced by irreducible representations. When n is odd there is only one spinor bundle, while when n is even there are two nonisomorphic spinor bun- dles. When n is even the behavior of spinor bundles is slightly different according as n = 0 (mod4) or n = 2 (mod4). In (Ot2) we have given a cohomological splitting criterion for vector bundles on quadrics involving spinor bundles. Qn ~ Spin(n + 2)/P(cty) (St) is a homogeneous manifold, and the semisimple part of the Lie algebra of -P(ai) is o(n). At the level of Lie algebras, spinor bundles are defined from the spin and half-spin representations of o(n). The paper is divided into three sections. In §1 we give some preliminary results and we define the spinor bundles. In §2 we study the first properties of spinor bundles. In §3, as an application, we describe some moduli spaces of rank 3 vector bundles on Q5 and Qq.

The automorphism group of a shift of finite type

Abstract: Let (XT,AT) be a shift of finite type, and G = aut(vT) denote the group of homeomorphisms of XT commuting with ¢T. We investigate the algebraic properties of the countable group G and the dynamics of its action on XT and associated spaces. Using "marker" constructions, we show G contains many groups, such as the free group on two generators. However, G is residually finite, so does not contain divisible groups or the infinite symmetric group. The doubly exponential growth rate of the number of automorphisms depending on n coordinates leads to a new and nontrivial topological invariant of CRT whose exact value is not known. We prove that, modulo a few points of low period, G acts transitively on the set of points with least aT-period n. Using padic analysis, we generalize to most finite type shifts a result of Boyle and Krieger that the gyration function of a full shift has infinite order. The action of G on the dimension group of aT iS investigated. We show there are no proper infinite compact G-invariant sets. We give a complete characterization of the G-orbit closure of a continuous probability measure, and deduce that the only continuous G-invariant measure is that of maximal entropy. Examples, questions, and problems complement our analysis, and we conclude with a brief survey of some remaining open problems.

Orthogonal polynomials on several intervals via a polynomial mapping

Abstract: Starting from a sequence {pn{x; no)} of orthogonal polynomials with an orthogonality measure yurj supported on Eo C (—1,1), we construct a new sequence {p"(x;fi)} of orthogonal polynomials on£ = T~1(Eq) (T is a polynomial of degree TV) with an orthogonality measure (i that is related to no- If Eo = (—1,1), then E = T_1((-l,l)) will in general consist of TV intervals. We give explicit formulas relating {pn(x;fi)} and {pn(x;fio)} 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.

Homology of smooth splines: generic triangulations and a conjecture of Strang

Abstract: For A a triangulated d-dimensional region in Rd, let Sr (A) denote the vector space of all cr functions F on A that, restricted to any simplex in A, are given by polynomials of degree at most m. We consider the problem of computing the dimension of such spaces. We develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds A in the plane, getting lower bounds on the dimension of Sr (A) for all r. For r = 1, we prove a conjecture of Strang concerning the generic dimension of the space of Cl splines over a triangulated manifold in R2. Finally, we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region.

Coexistence theorems of steady states for predator-prey interacting systems

Abstract: On donne des conditions necessaires et suffisantes pour l'existence des solutions positives des etats stationnaires pour des systemes predateur-proie sous des conditions aux limites de Dirichlet sur Ω∈R n

Conditional gauge and potential theory for the Schrödinger operator

Abstract: On etend le theoreme de jauge conditionnelle a des operateurs plus generaux et a des domaines moins reguliers. On obtient des resultats de theorie du potentiel pour l'equation de Schrodinger

Fast algorithms for multiple evaluations of the riemann zeta function

Abstract: The best previously know algorithm for evaluating the Riemann zeta function, ζ(σ+it), with σ bounded and t large to moderate accuracy was based on the Rienmann-Siegel formula and required on the order of t 1/2 operations for each value that was computed. New algorithms are presented in this paper which enable one to compute any single value of ζ(σ+it) with σ fixed and T≤t≤T+T 1/2 in O(te) operations on numbers of O(log t) bits for any e>0 for example. These algorithms lead to methods for numerically verifying the Riemann hypothesis for the first n zeros in what is expected to be O(n 1+ e) operations

Ideals associated to deformations of singular plane curves

Abstract: We consider in this paper the geometry of certain loci in deformation spaces of plane curve singularities. These loci are the equisingular locus ES which parametrizes equisingular or topologically trivial deformations, the equigeneric locus EG which parametrizes deformations of constant geometric genus, and the equiclassical locus EC which parametrizes deformations of constant geometric genus and class. (The class of a reduced plane curve is the degree of its dual.) It was previously known that the tangent space to ES corresponds to an ideal called the equisingular ideal and that the support of the tangent cone to EG corresponds to the conductor ideal. We show that the support of the tangent cone to EC corresponds to an ideal which we call the equiclassical ideal. By studying these ideals we are able to obtain information about the geometry and dimensions of ES, EC, and EG. This allows us to prove some theorems about the dimensions of families of plane curves with certain specified singularities.

Crossed products of continuous-trace *-algebras by smooth actions

Abstract: We study in detail the structure of C*-crossed products of the form A> G, where A is a continuous-trace algebra and a is an action of a locally compact abelian group G on A, especially in the case where the action of G on A has a Hausdorff quotient and only one orbit type. Under mild conditions, the crossed product has continuous trace, and we are often able to compute its spectrum and Dixmier-Douady class. The formulae for these are remarkably interesting even when G is the real line. In recent years, considerable progress has been made in understanding the structure of a transformation group C *-algebra C*(G, X) (sometimes written C0(X) X G) when the orbit space X/G is reasonable. For example, a well-known theorem by Green [10] asserts that if G acts freely and properly on X, then C*(G, X) is isomorphic to C0(X/G, ./((L2(G))) (for a good discussion of this, and some generalizations, see [29]), and for abelian G a description of the topology on the spectrum of C*(G, X) has been given by Williams [38]. The structure of crossed products AA a G with A noncommutative is much more complicated, due to an extra step in the description of their representation theory by the "Mackey machine" (e.g., [34 and 11]): if 77 E A, there is an obstruction in H2(G, T) to extending so to a covariant representation of (A, G), where G,, is the stabilizer of v in G, and even if this obstruction vanishes, it may not be possible to find a canonical extension of s. Here we shall study the crossed product A X a G, when A is a continuous-trace C *-algebra, G is abelian, and the action of G on the spectrum X of A satisfies various local triviality hypotheses. Somewhat surprisingly, our point of view yields new information even about transformation group algebras. For example, it turns out that for a certain action of R on S3, the associated transformation group algebra has nonzero Dixmier-Douady invariant (Example 4.6 below). There are two extreme cases which have already been investigated to some extent. First of all, locally unitary automorphism groups a: G -Aut A [27] are group actions which act trivially on the spectrum and for which Mackey obstructions do not arise, and they include all such actions if G and A are separable and G is Received by the editors February 23, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 46L05, 46L40, 46L55, 46M20; Secondary 46L80, 19K99, 22D25, 22E27. This project was made possible by support from the Australian Research Grants Scheme and by NSF Grant 8120790. Partially supported by the Alfred P. Sloan Foundation and by NSF Grant DMS-8400900. ?)1988 American Mathematical Society 0002-9947/88 $1.00 + S.25 per page This content downloaded from 207.46.13.113 on Thu, 06 Oct 2016 04:24:49 UTC All use subject to http://about.jstor.org/terms 2 IAIN RAEBURN AND JONATHAN ROSENBERG compactly generated and abelian [32, ?2]. For such a, the spectrum of the crossed product A A, a G is in a natural way a principal G-bundle over A [27, Theorem 2.2], and in fact A A a G is isomorphic to the balanced tensor product Co((A A a G)A) ?co(x)A [28, Proposition 1.5]. Secondly, some things are known about diagonal actions on the pull-backs of C *-algebras. If B is a C *-algebra with spectrum T and p: Q -> T is a principal G-bundle, then the pull-back p*B is by definition Co(Q) ?CO(T) B (so the previously quoted result asserts A >A a G _p*A for p: (A >A a G) ^A when a is locally unitary). A diagonal action on p*B, denoted p*/B, is one inherited from a tensor product action y ? /3 of G on Co(Q) ? B, where y is translation on Q and /3 is an action of G on B which commutes with the action of Cb(T) Z(M(B)). The spectrum of p*B is canonically homeomorphic to Q, and p*,B induces the original action of G on Q, so these give nontrivial examples of actions which induce principal bundle structures on the spectrum. In [28] it is shown that p*B A p* G is often, but not always, Morita equivalent to B. (It is if ,B is implemented by a unitary group, or if the bundle p: Q -> T is trivial.) We begin where [28] leaves off: we consider actions of G on A which make A into a locally trivial principal G-bundle over A/G. Our first main result (Theorem 1.1) says that, up to stable isomorphism, the pull-back actions studied in [28] are the only examples of such automorphism groups. The crossed product is again a continuoustrace C *-algebra, and we give a formula for its Dixmier-Douady class

Pseudo-orbit shadowing in the family of tent maps

Abstract: On montre que les applications en tente ont la propriete d'ombrage pour presque tous les parametres s, mais elles n'ont pas cette propriete pour un ensemble dense non denombrable des parametres. On montre que pour une application tente, chaque pseudo-orbite peut etre approchee par une orbite actuelle d'une application en tente avec une boucle peut etre legerement plus grande

Generalized quotients in Coxeter groups

Abstract: For (W, S) a Coxeter group, we study sets of the form W/V = {w E W I l(wv) = 1(w) + I(v) for all v E V}, where V C W. Such sets W/V, here called generalized quotients, are shown to have much of the rich combinatorial structure under Bruhat order that has previously been known only for the case when V C S (i.e., for minimal coset representatives modulo a parabolic subgroup). We show that Bruhat intervals in W/V, for general V C W, are lexicographically shellable. The Mobius function on W/V under Bruhat order takes values in {-1, 0, +1}. For finite groups W, generalized quotients are the same thing as lower intervals in the weak order. This is, however, in general not true. Connections with the weak order are explored and it is shown that W/V is always a complete meet-semilattice and a convex order ideal as a subset of W under weak order. Descent classes DI = {w E W Il1(ws) < 1(w) Xt s e I, for all s E S}, I C S, are also analyzed using generalized quotients. It is shown that each descent class, as a poset under Bruhat order or weak order, is isomorphic to a generalized quotient under the corresponding ordering. The latter half of the paper is devoted to the symmetric group and to the study of some specific examples of generalized quotients which arise in combinatorics. For instance, the set of standard Young tableaux of a fixed shape or the set of linear extensions of a rooted forest, suitably interpreted, form generalized quotients. We prove a factorization result for the quotients that come from rooted forests, which shows that algebraically these quotients behave as a system of minimal "coset" representatives of a subset which is in general not a subgroup. We also study the rank generating function for certain quotients in the symmetric group.

The number of solutions to linear diophantine equations and multivariate splines

Abstract: Etude de la variation du nombre des solutions entieres non negatives de s equations diophantiennes lineaires en fonction des termes non homogenes

Bifurcation phenomena associated to the p-Laplace operator

Abstract: On determine la structure de l'ensemble des solutions u de −(|u x | p−2 u x ) x +f(u)=λ|u| p−2 u sur (0,1) telles que u(0)=u(1)=0, ou p>1 et λ∈R. On demontre que les solutions avec k zeros sont uniques quand 1 2

Paracommutators—boundedness and Schatten-von Neumann properties

Abstract: Definition d'une classe d'operateurs, denommes paracommutateurs, operant sur les fonctions de L 2 (R d ), par analogie avec la para-multiplication de Bony et les commutateurs des operateurs de Calderon-Zygmund. Etablissement de la borne et des proprietes de Schatten-von Neumann

The existence of least area surfaces in 3-manifolds

Abstract: This paper presents a new and unified approach to the existence theorems for least area surfaces in 3-manifolds. Introduction. A surface F smoothly embedded or immersed in a Riemannian manifold M is minimal if it has mean curvature zero at all points. It is a least area surface in a class of surfaces if it has finite area which realizes the infimum of all possible areas for surfaces in this class. The connection between these two ideas is that a surface which is of least area in a reasonable class of surfaces must be minimal. The converse is false; minimal surfaces are in general only critical points for the area function. There are close analogies between these two concepts and the theory of geodesics in a Riemannian manifold. Minimal surfaces correspond to geodesics, and least area surfaces correspond to geodesic arcs or closed geodesics which have shortest length in some class of paths. Any geodesic A in a Riemannian manifold M has the property that it is locally shortest, i.e., if P and Q are nearby points on A, then the subarc of A which joins P and Q is the shortest path in M from P to Q. It can also be proved that minimal surfaces are locally of least area, but the proof is difficult and involves substantial knowledge of the theory of partial differential equations. There are now a large number of theorems asserting the existence of surfaces of least area in various classes. Surfaces of this type have become an important tool in 3-dimensional topology. In this paper we present a new approach to the proofs of these existence theorems. This yields a simplified and unified method for the proof of the existence of minimal surfaces in 3-dimensional Riemannian manifolds. In 1930 Douglas [Do] and Rado [Ra] independently showed that a simple closed curve in RI which bounds a disk of finite area bounds a disk of least possible area. This result was extended by Morrey [MoI] in 1948 to a general class of Riemannian manifolds, the homogeneously regular manifolds, a class which includes all closed manifolds. Work of Osserman [0] and Gulliver [G] later showed that least area disks in 3-manifolds were immersed in their interiors. More recently there has been a series of new existence results for closed surfaces of least area in manifolds of any dimension. It follows from work of Sacks and Uhlenbeck [S-UI] that if M is closed and 7r2(M) is nonzero then there is an essential map of the 2-sphere into M which has least area among all essential maps. Sacks and Uhlenbeck [S-UII] and Schoen and Yau [S-Y] independently showed that if f: F -* M is a map of a closed orientable surface F, not the 2-sphere, into a closed Riemannian manifold M, such Received by the editors June 23, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 53A10; Secondary 57M35. The first author was supported by NSF Postdoctoral Fellowship DMS84-14097. ?)1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page

Isometry groups of Riemannian solvmanifolds

Abstract: A simply connected solvable Lie group R together with a leftinvariant Riemannian metric g is called a (simply connected) Riemannian solvmanifold. Two Riemannian solvmanifolds (R,g) and (R',g') may be isometric even when R and R' are not isomorphic. This article addresses the problems of (i) finding the \"nicest\" realization (R, g) of a given solvmanifold, (ii) describing the embedding of R in the full isometry group I(R,g), and (iii) testing whether two given solvmanifolds are isometric. The paper also classifies all connected transitive groups of isometries of symmetric spaces of noncompact type and partially describes the transitive subgroups of I(R, g) for arbitrary solvmanifolds (R,g). Introduction. A Riemannian solvmanifold is a connected Riemannian manifold At which admits a transitive solvable group of isometries. It is well known (see §1) that every such manifold actually admits an almost simply transitive solvable group R of isometries, simply transitive if At is simply connected. We will assume for simplicity in our introductory remarks that At is simply connected, although this assumption is dropped throughout much of the paper. The simply transitive group R is in general not unique even up to isomorphism. Given any choice of R, At is isometric to R equipped with a left-invariant metric ( , ). We address the following problems: (i) Given the data (R, ( , )) for At, find the \"nicest\" simply transitive solvable group 5 of isometries of At, i.e. find the nicest realization (S, ( , )') of AI. (ii) Given (R, ( , )), describe the full isometry group of At. (iii) Develop a test to determine whether any two given Riemannian solvmanifolds are isometric. In §1, we construct a single conjugacy class of simply transitive solvable subgroups of the full isometry group I(M) which we call the subgroups in \"standard position\". They are defined entirely in terms of their embeddings in /(At). These play the role of the \"nicest\" groups in (i). §2 develops the abstract theory of modifications of solvable Lie algebras needed for §3. In §3, beginning with the data (R,{ , )) and no other knowledge of /(AI), we algorithmically modify (R,( , )) to obtain (S, ( , )') where S is a simply transitive subgroup of /(AI) in standard position. We also compute the normalizer of S in /(Ai). This together with the results of §1 gives considerable information about /(At). In many cases, such as when R is unimodular (see §4), S is normal in /(At) and hence a complete solution to problem (ii) is obtained. Received by the editors March 25, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C20, 53C30; Secondary 53C35. The first author was partially supported by the National Science Foundation grant 8502084. ©1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page

Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators

Abstract: Caracterisation des fonctions-poids pour lesquelles l'operateur integral fractionnaire de Riemann-Liouville d'ordre α>0 est borne

Iterating the basic construction

Abstract: Let N C M be a pair of type II1 factors with finite Jones' index and N cM CM1 C M2 C C Mn C C M2n+l be the associated tower of type II1 factors obtained by iterating Jones' basic construction. We give an explicit formula of a projection in M2n+l which implements the conditional expectation of Mn onto N, thus showing that M2n+1 comes naturally from the basic construction associated to the pair N c Mn. From this we deduce several properties of the relative commutant N' n Mn. Introduction. Let N C M be a pair of finite factors. Jones defined in [1] the index [M: N] of N in M to be the coupling constant of N in its representation on L2 (M). If this index is finite, then the trace preserving conditional expectation of M onto N, regarded as an operator on L2 (M), generates together with M a finite factor Ml. This factor is called in Jones' terminology the extension of M by N and the construction of Ml from M and N, the basic construction. The pair M C M1 has the remarkable property that [Ml: M] = [M: N], so this procedure may be iterated to get an increasing sequence of finite factors N c M c M1 C M2 C ... and together with it a sequence of projections ei E Mi+i, i > 0, implementing the conditional expectations at consecutive steps. We prove in this paper that in this sequence of factors the basic construction arises periodically from n to n steps, for any n. In fact we give a formula for a projection fn in M2n+l that implements the conditional expectation of Mn onto N: fn is a scalar multiple of the word of maximal length in {e }o<-<2n, namely fn = [M: N]n(n+l)/2(enen.1 ... eo)(en+len el) ... (e2n ... en) We mention that this result was independently obtained by A. Ocneanu [2]. We apply this theorem to show that if the logarithm of the index [M : N] equals the relative entropy H(MIN) considered in [3], then one also has H(MnIN) = ln[Mn : N] for every n. Since this equality characterizes an extremal case for an inclusion of factors, from the analysis of a similar situation in [3] we deduce several properties of the inclusion N c Mn and of the relative commutant N' n Mn. 1. Preliminaries. Throughout this paper M will be a finite factor with normalized trace T, T(l) = 1. We denote by 11x112 = T(X*X)1/2, x E M, the Hilbert norm given by r and by L2 (M, T) the Hilbert space completion of M in this norm. The canonical conjugation of L2(M, T) is denoted by J. It acts on M C L2(M, r) Received by the editors February 23, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 46L35; Secondary 46L10. (?)1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page