Showing papers in "Transactions of the American Mathematical Society in 1972"

Weighted norm inequalities for the Hardy maximal function

Abstract: The principal problem considered is the determination of all nonnegative functions, U(x), for which there is a constant, C, such that | [f*(x)rUix)dx g CJ \\f(x)\\'U(x) dx, where l

Equational bases and nonmodular lattice varieties

Abstract: This paper is focused on equational theories and equationally defined varieties of lattices which are not assumed to be modular. It contains both an elementary introduction to the subject and a survey of open problems and recent work. The concept of a \"splitting\" of the lattice of lattice theories is defined here for the first time in print. These splittings are shown to correspond bi-uniquely with certain finite lattices, called \"splitting lattices\". The problems of recognizing whether a given finite lattice is a splitting lattice, whether it can be embedded into a free lattice, and whether a given interval in a free lattice is atomic are shown to be closely related and algorithmically solvable. Finitely generated projective lattices are characterized as being those finitely generated lattices that can be embedded into a free lattice. Introduction. What we do in this paper is fairly described by the phrase \"equational model theory of lattices.\" We take this in a broad sense, as the study of lattices and their equational theories, with particular emphasis on finding connections relating algebraic properties of lattices to various properties of their theories. Since many interesting properties of lattice theories can be defined abstractly, purely by reference to the lattice of all such theories, we are naturally led to investigate the structure of S the lattice of all equational theories of lattices. Recent studies in this largely unexplored field have been greatly stimulated by a paper of Bjarni Jónsson [6]. The central result of that paper (L6, Corollary 3.2], stated below as Lemma 1.4) has already found a number of applications, in the original paper as well as in [l], [7], L9J and the present paper. Broadly speaking, Jónsson's result tells us that the cords binding lattices to their theories are much more tightly drawn than one would expect on the basis of experience with other equationally defined classes of algebras, such as groups and rings. The happy consequences of this fact, both for the algebraic and the equational study of lattices, will hardly be exhausted by the present paper; and we expect other researchers to take up the challenge. To that end, we have given Received by the editors March 5, 1969. AMS (MOS) subject classifications (1970). Primary 08A15, 06A20, 02G05, 02G15.

Entropy-expansive maps

Abstract: Let /: X -> X be a uniformly continuous map of a metric space. / is called /(-expansive if there is an oO so that the set $>c(x) = {y : d(Jn(x),fn(y))^e for all naO) has zero topological entropy for each xe X. For Xcompact, the topo- logical entropy of such an /is equal to its estimate using e: h(f) = h(f, e). If X is com- pact finite dimensional and y. an invariant Borel measure, then /V"(/) = A"(/, A) for any finite measurable partition A of X into sets of diameter at most e. A number of examples are given. No diffeomorphism of a compact manifold is known to be not A-expansive.

A decomposition for combinatorial geometries

Abstract: A construction based on work by Tutte and Grothendieck is applied to a decomposition on combinatorial pregeometries in order to study an important class of invariants. The properties of this Tutte decomposition of a pregeometry into a subgeometry G\e and contraction G/e is explored in a categorically integrated view using factored strong maps. After showing that direct sum decomposition distributes over the Tutte decomposition we construct a universal pair (R, t) where R is a free commutative ring with two generators corresponding to a loop and an isthmus; and t, the Tutte polynomial assigns a ring element to each pregeometry. Evaluations of t(G) give the Mobius function, characteristic polynomial, Crapo invariant, and numbers of subsets, bases, spanning and independent sets of G and its Whitney dual. For geometries a similar decomposition gives the same information as the chromatic polynomial throwing new light on the critical problem. A basis is found for all linear identities involving Tutte polynomial coefficients. In certain cases including Hartmanis partitions one can recover all the Whitney numbers of the associated geomnetric lattice L(G) from t(G) and conversely. Examples and counterexamples show that duals, minors, connected pregeometries, series-parallel networks, free geometries (on which many invariants achieve their upper bounds), and lower distributive pregeometries are all characterized by their polynomials. However, inequivalence, Whitney numbers, and representability are not always invariant. Applying the decomposition to chain groups we generalize the classical twocolor theorem for graphs to show when a geometry can be imbedded in binary affine space. The decomposition proves useful also for graphical pregeometries and for unimodular (orientable) pregeometries in the counting of cycles and coboundaries.

Bounded continuous functions on a completely regular space

Abstract: Three locally convex topologies on C(X) are introduced and developed, and in particular shown to coincide with the strict topology on locally compact A1 and yield dual spaces consisting of tight, -r-additive and x where p. is a bounded regular Borel measure on X. This yields a very satisfactory relationship between the topology on X, the space C(X), a natural class of linear functionals on it, and those measures on Jfthat measure at least the sets determined by the topology on X in the usual way, the Borel sets. This kind of representation was subsequently extended to locally compact spaces : first to functionals on the space C0(X) of continuous functions vanishing at infinity, and then further, to the bounded continuous functions on X. The last result, due to R. C. Buck, demanded the use of a locally convex topology, the strict topology, rather than a norm topology. In both extensions the same satisfactory relationship between measure and topology was obtained. In this paper we begin the development of locally convex topologies for C(X) which extend this kind of representation to its last reasonable setting, completely regular Hausdorff spaces. This setting appears to be ultimate in the sense of Hewitt's example of a regular space upon which the only continuous functions are constants. 1. Definitions and preliminaries. The actual work of integral representation of linear forms has been done by other authors, going back to Aleksandrov (1) and, following his work, by Varadarajan (39), and later Knowles (21) and more recently Kirk (20) and Moran ((24), (25)). Our work relies heavily on theirs and will not extend the representations they have obtained but will relate these works to earlier

Integrability of expected increments of point processes and a related random change of scale

Abstract: Given a stationary point process with finite intensity on the real line R, denote by N(Q) (Q Borel set in R) the random number of points that the process throws in Q and by ^ (t s R) the c-field of events that happen in ( — co, t). The main results are the following. If for each partition A = {¿»=f0-0 (the a.s. convergence requires a judicious choice of versions). If the random transformation / » W(

Ultrafilters and independent sets

Abstract: Independent families of sets and of functions are used to prove some theorems about ultrafilters. All of our results are well known to be provable from some form of the generalized continuum hypothesis, but had remained open without such an assumption. Independent sets are used to show that the Rudin-Keisler ordering on ultrafilters is nonlinear. Independent functions are used to prove the existence of good ultrafilters.

Characteristic classes of stable bundles of rank 2 over an algebraic curve

Abstract: Let X be a complete nonsingular algebraic curve over C and L a line bundle of degree 1 over X. It is well known that the isomorphism classes of stable bundles of rank 2 and determinant L over X form a nonsingular projective variety S(X). The Betti numbers of S(X) are also known. In this paper we define certain distinguished cohomology classes of S(X) and prove that these classes generate the rational cohomology ring. We also obtain expressions for the Chern character and Pontrjagin classes of S(X) in terms of these generators. Introduction. Let X be a complete nonsingular algebraic curve of genus g> 2 over the complex numbers and let L be a line bundle of degree 1 over X. The isomorphism classes of stable bundles of rank 2 and determinant L over X form a nonsingular projective variety S(X) (see [2], [6]), whose Betti numbers were calculated in [3]. The main object of this paper is to show that certain naturally occurring elements generate the rational cohomology ring of S(X) (Theorem 1). We shall also obtain expressions for the Chern character and Pontrjagin classes of S(X) in terms of these generators. My thanks are due to S. Ramanan for informing me of his work on this topic in advance of publication (see [5]). He has obtained generators and relations for the rational cohomology ring of S(X) in the case g = 3. He has also obtained some information for the spaces of stable bundles of rank n (in particular a generalisation of Corollary 1 to Theorem 2) by methods similar to those used in the proof of Proposition 2.2. Unless the contrary is indicated, all cohomology groups in this paper will have integral coefficients. Also, if E is any bundle over V x W (or W x V) and v E V, we shall denote by Ev the bundle over W obtained by restricting E to vx W (or W x lv). 1. Statement of the main theorem. We recall [1, ? 1] that there exists an algebraic vector bundle U over S(X) x X with the property that, for all s e SX, Received by the editors September 9, 1971. AMS 1970 subject classifications. Primary 14D20, 14F05, 14F25; Secondary 55F40, 57D20.

Geodesic flow in certain manifolds without conjugate points

Abstract: A complete simply connected Riemannian manifold H without conju- gate points satisfies the uniform Visibility axiom if the angle subtended at a point p by any geodesic y of H tends uniformly to zero as the distance from p to y tends uniformly to infinity. A complete manifold Mis a uniform Visibility manifold if it has no conju- gate points and if the simply connected covering H satisfies the uniform Visibility axiom. We derive criteria for the existence of uniform Visibility manifolds. Let M be a uniform Visibility manifold, SM the unit tangent bundle of M and Tt the geodesic flow on SM. We prove that if every point of SM is nonwandering with respect to Tt then Tt is topologically transitive on SM. We also prove that if M' is a normal covering of Mthen Tt is topologically transitive on SM' if Tt is topologically transitive on SM. Introduction. Much research, past and present, has been devoted to proving the topological transitivity of the geodesic flow in manifolds with curvature K^O that satisfy various conditions, usually including compactness. An essential part of all the proofs is the fact that if Af is a complete manifold with K^O and simply connected covering space H then any two distinct points of H are joined by a unique geodesic. Manifolds Af without conjugate points are characterized by this property in the simply connected covering space H and it is natural to ask what further conditions on Af are sufficient to imply that the geodesic flow is topo- logically transitive. Results have been obtained (mostly in the compact case) by Green, Hedlund, Klingenberg, Morse and others under the assumption that the manifold Af admits another (closely related) metric g* with curvature K= — 1 or #gc<0. In this paper we extend and unify many of these results by requiring that Af be a uniform Visibility manifold (defined above and more precisely in §1). To comple- ment the main results stated above we obtain the following criteria for the existence of uniform Visibility manifolds.

Groups with finite dimensional irreducible representations

Abstract: It will be shown that a locally compact group has a finite bound for the dimensions of its irreducible unitary representations if and only if it has a closed abelian subgroup of finite index. It will further be shown that a locally compact group has all of its irreducible representations of finite dimension if and only if it is a projective limit of Lie groups with the same property, and finally that a Lie group has this property if and only if it has a closed subgroup H of finite index such that //"modulo its center is compact. 1. Let G be a locally compact group; a (unitary) representation of G is by definition a strongly continuous homomorphism n of G into the group of unitary operators on some Hubert space H(tt) [11]. One says that 77 is irreducible if the only closed subspaces of 7/(77) invariant under all the operators 77(g), ge G, are (0) and H (it). We shall denote by G the set of equivalence classes under unitary equivalence of irreducible unitary representations [11], and we shall use 77 to denote both a representation and its equivalence class. Associated with any representation 7T we have a cardinal number d(n), the degree of 77, which is by definition the cardinality of an orthonormal basis of the Hubert space H(w). It is not assumed here that the topology of G satisfies the second axiom of countability nor that d(tr) is restricted to be ^ X0. Although, in general, examples show that representations with d(n) finite are rather rare, the purpose of this paper is to investigate two closely related hypotheses involving finiteness conditions on d(-rr). To be precise, we say that G satisfies (1) if there is an integer M

Normed convex processes

Abstract: : The paper shows that several well-known results about continuous linear operators on Banach spaces can be generalized to the wider class of convex processes, as defined by Rockafellar. In particular, the open mapping theorem and the standard bound for the norm of the inverse of a perturbed linear operator can be extended to convex processes. In the last part of the paper, these theorems are exploited to prove results about the stability of solution sets of certain operator inequalities and equations in Banach spaces. These results yield quantitative bounds for the displacement of the solution sets under perturbations in the operators and/or in the right-hand sides. They generalize the standard results on stability of unique solutions of linear operator equations.

Nonresidually finite one-relator groups

Abstract: The study of one-relator groups includes the connections between group properties and the form of the relator. In this paper we discuss conditions on the form uiv uvm which force the corresponding one-relator groups to be nonresidually finite, i.e. the intersection of the normal subgroups of finite index to be nontrivial. Moreover we show that these forms can be detected amongst the words of a free group.

Approximation in the mean by analytic functions

Abstract: Let E be a compact set in the plane, let LP(E) have its usual meaning, and let LP(E) be the subspace of functions analytic in the interior of E. The problem studied in this paper is whether or not rational functions with poles off E are dense in LP(E) (or in L\"(E) in the case when E has no interior). For 1 ?p 2 which improve earlier results by Sinanjan. The results are given in terms of capacities.

On the degrees and rationality of certain characters of finite Chevalley groups

Abstract: Let a'' be a system of finite groups with (5, N)-pairs, with Coxeter system (W, R) and set of characteristic powers {q} (see [4]). Let A be the generic algebra of the system, over the polynomial ring o—Q[u]. Let K be ß(w), K an algebraic closure of K, and o* the integral closure of o in K. For the specialization /: «->- Q be a fixed extension of/. For each irreducible character x of the algebra As, there exists an irreducible character £,,,. of the group G{q) in the system corresponding to q, such that (£»,/•, lf0, and x -*■ ix.r is a bijective correspondence between the irreducible characters of A* and the irreducible constituents of lg$. Assume almost all primes occur among the characteristic powers {q}. The first main result is that, for each x, there exists a polynomial dx(t) e Q [/] such that, for each specialization /: u^-q, the degree {,,/•(!) is given by dx{q). The second result is that, with two possible exceptions in type E7, the characters £/t/. are afforded by rational representations of G(q).

Characteristic classes of real manifolds immersed in complex manifolds

Abstract: Let M be a compact, orientable, k-dimensional real differentia- able manifold and N an n-dimensional complex manifold, where k > n. Given an immersion t: M -- N, a point x E M is called an RC-singular point of the immersion if the tangent space to t(M) at L(x) contains a complex subspace of dimension > k - n. This paper is devoted to the study of the cohomological properties of the set of RC-singular points of an immersion. When k = 2n - 2, the following formula is obtained:

Existence theorems for infinite particle systems

Abstract: Sufficient conditions are given for a countable sum of bounded generators of semigroups of contractions on a Banach space to be a generator. This result is then applied to obtain existence theorems for two classes of models of infinite particle systems. The first is a model of a dynamic lattice gas, while the second describes a lattice spin system.

The equivariant Plateau problem and interior regularity

Abstract: Let M C R\" be a compact submanifold of Euclidean space which is invariant by a compact group G C SO (zz). When dim (M) = zz 2, it is shown that there always exists a solution to the Plateau problem for M which is invariant by G and, furthermore, that uniqueness of this solution among G-invariant currents implies uniqueness in general. This result motivates the subsequent study of the Plateau problem for M within the class of G-invariant integral currents. It is shown that this equivariant problem reduces to the study of a corresponding Plateau problem in the orbit space R/G where, for \"big\" groups, questions of uniqueness and regularity are simplified. The method is then applied to prove that for a constellation of explicit manifolds M, the cone C(M) = \\tx; x € M and 0 < t < l! is the unique solution to the Plateau problem for M, (Thus, there is no hope for general interior regularity of solutions in codimension one.) These manifolds include the original examples of type S\" x S\" C R n+ , zz > 3, due to Bombieri, DeGiorgi, Giusti and Simons. They also include a new example in R and examples in R\" for zz > 10 with any prescribed Betti number nonzero.

Strictly regular elements in Freudenthal triple systems

Abstract: Strictly regular elements play a role in the structure theory of Freudenthal triple systems analogous to that played by idempotents in nonassociative algebras with identity. In this paper we study the coordinatization of reduced triple systems relative to a connected pair of strictly regular elements and use the explicit form of strictly regular elements in terms of the coordinatization to prove uniqueness of the coordinatizing Jordan algebra, as well as several generalizations of known results regarding groups of transformations related to triple systems. Finally, we classify forms of a particularly important triple system (the representation module for the Lie algebra E7) over finite, p-adic or real fields. Following Freudenthal [71, Meyberg [91 and Brown [11 have introduced an axiomatic approach to the study of the ternary algebraic structure of the minimal dimensional module for the lie algebra E7. In particular, this module is one of a class of ternary algebras we refer to as Freudenthal Triple Systems (FTS). In this paper we analyze further the internal structure of such algebras using as a basic tool the Peirce decomposition relative to a pair of supplementary strictly regular elements. Using this decomposition we obtain, in a manner similar to [1 ], a coordinatization theorem-every simple, reduced FTS is isomorphic to VQ), ?S a member of a small class of Jordan algebras specified in ?1. In ?6 we analyze the forms of strictly regular elements in K2?) and use the results to show I) lN(S) if and only if ?and A are norm equivalent. In ?7 the action of the group of q-similarities on the strictly regular elements is studied, yielding conjugacy theorems generalizing a result of Brown [1], as well as information regarding ratios of similitudes generalizing results of Faulkner [41 and Seligman (unpublished). Results of this section are useful also in the classification problem for Lie algebras of type E7. In the final section we classify completely all forms of lQ(?), 3 exceptional central simple, for finite, p-adic, or real fields. In [51, symplectic algebras (an asymmetric version of FTS's) are studied, yielding different proofs of several results obtained in this paper. Received by the editors December 22, 1971. AMS (MOS) subject classifications (1970). Primary 17A30; Secondary 17B60.

Embedding theorems and quasi-linear elliptic boundary value problems for unbounded domains

Abstract: The Sobolev-Kondrachov embedding and compactness theorems are extended to cover general unbounded domains, by introducing appropriate weighted Lp norms. These results are then applied to the Dirichlet problem for quasi-linear elliptic partial differential equations and isoperimetric variational problems defined on general unbounded domains in R\". The general study of boundary value problems for quasi-linear elliptic partial differential equations has been generally limited to bounded domains. Perhaps one reason for this fact is that such compactness theorems as that of SobolevKondrachov and its extensions are no longer valid for general unbounded domains in R . Consequently the degree theory of Leray-Schauder and the critical point theory based on such compactness properties as Condition (C) of Palais-Smale are not applicable in the study of quasi-linear elliptic problems defined on such general domains. In this article, we extend the Sobolev-Kondrachov compactness and embedding theorems to general unbounded domains and apply these results to quasi-linear Dirichlet problems and to nonquadratic isoperimetric variational problems. For quadratic isoperimetric problems some special embedding and compactness theorems of the type discussed here have been obtained recently ([l], [2], and [3]) in conjunction with the study of discrete spectra of linear elliptic partial differential operators of order 2tt2 defined on R' . Our embedding theorems also extend some research of Glusko and Kreïn [4]. Some of the results presented here were announced by us in [O]. The present article is organized as follows: In §1, we mention the types of elliptic boundary value problems to be discussed. In §2, we state the L embedding and compactness theorems that extend the results of Sobolev-Kondrachov. Applications of these theorems to quasi-linear elliptic boundary value problems are given in §3. Finally, in §4, we prove the embedding theorems of §2. 1. Quasi-linear elliptic problems on unbounded domains. Let ß be an open set in R\" with boundary ail. In this section we mention some problems that arise Received by the editors June 16, 1971 and, in revised form, January 5, 1972. AMS 1970 subject classifications. Primary 35J60, 58E15.

Ascent, descent, and commuting perturbations

Abstract: In the present paper we investigate the stability of the ascent and descent of a linear operator T when T is subjected to a perturbation by a linear operator C which commutes with T. The domains and ranges of T and C lie in some linear space X. The results are used to characterize the Browder essential spectrum of T. We conclude with a number of remarks concerning the notion of commutativity used in the present paper. Introduction. To discuss ascent and descent one must consider iterates of operators. Some sort of commutativity of T and C is necessary in order to meaningfully compare operators such as T and (T + C) and to \"factor\" operator products (cf. Lemma 1.4). We shall say a linear operator C commutes with T if (i) the domain of C, JJ(C), contains the domain of T, (ii) Cx e 3/(T) whenever x £ J)(T), and (iii) TCx = CTx for x e J)(T ) This definition coincides with the usual one when T and C ate defined on all of X. Note that T commutes with itself if and only if T maps i)(T) into J'(T). In $T we collect together a number of preliminary lemmas about operators T and C such that C commutes with T. In §2 we show by purely algebraic methods that the finiteness of the ascent or descent of T is retained by the operator T + C when C commutes with T and a certain power of C has finitedimensional range. In general this does not hold if C is a compact operator, but similar results may be obtained when some restrictions are placed on T. This is shown in §3, where we consider perturbations by compact operators, Riesz operators and T -compact operators. In §4 we use the results of §3 to characterize the Browder essential spectrum. The main results of this section have been announced earlier by the second author (see [9]). In the final section we discuss the commutativity condition used here. Among other things we show that there exists a closed operator T such that the only bounded operators commuting with T ate scalar multiples of the identity operator. Received by the editors September 8, 1971. AMS 1970 subject classifications. Primary 47A10, 47A55, 47B05; Secondary 47B30.

Solvable fundamental groups of compact 3-manifolds

Abstract: A classification is given for groups which can occur as the fundamental group of some compact 3-manifold. In most cases we are able to determine the topological structure of a compact 3-manifold whose fundamental group is known to be solvable. Using the results obtained for solvable groups, we are able to extend some known results concerning nilpotent groups of closed 3-manifolds to the more general class of compact 3-manifolds. In the final section it is shown that each nonfinitely generated abelian group which occurs as a subgroup of the fundamental group of a 3-manifold is a subgroup of the additive group of rationals. (1) Introduction. This paper is primarily concerned with the classification of those solvable groups which can occur as the fundamental group of a compact 3-manifold. We also consider the problem of determining the structure of a compact 3-manifold whose fundamental group is known to be solvable. Our results are complete except in the category of almost sufficiently large 3-manifolds and the category of 3-manifolds whose nontrivial second homotopy group is generated by projective plane boundary components. If M is a compact, sufficiently large 3-manifold with trivial second homotopy group, and if ir^M) is solvable, then ^(M) appears in the following list of groups: (1) Z,Z©Z, or Jf, the fundamental group of the Klein bottle, (2) an extension 1 -> A -> iti(M) -> Z -> 1 where A is either Z © Z or ^T, (3) a free product of two copies of JT amalgamated along certain subgroups isomorphic with Z © Z. These groups may be presented by (a, b, x, y \\ bab~1 = a~1, yxy~1 = x~1, a = xpy2q, b2 = xTy2s) where//, q, r, s are integers such that//s — rq= ± 1. Further, the above list is complete. That is, for each group G listed above, there is a compact sufficiently large 3-manifold M with n2(M) = 0, and tt1(M)~G. If the restrictions that M be sufficiently large and that ir2(M) = 0 are dropped, then further groups must be added to the list. Such groups are discussed in detail in §§3, 4, 5, and 6. C. Thomas [16] has listed those nilpotent groups which can act as the fundamental group of a closed 3-manifold. Making use of the information we gain Received by the editors February 2, 1971. AMS 1970 subject classifications. Primary 57A10, 55A05; Secondary 20E15, 20K45.

Some asymptotic fixed point theorems

Abstract: By an asymptotic fixed point theorem we mean a theorem in functional analysis in which the existence of fixed points of a map f is established with the aid of assumptions on the iterates f" of f. We prove below some new theorems of this type, and we obtain as corollaries results of F. E. Browder, G. Darbo, R. L. Frum-Ketkov, W. A. Horn and others. We also state a number of conjectures about fixed point theorems at the end of the paper. Our interest in the results here is two-fold. First, asymptotic fixed point theorems have proved useful in the theory of ordinary and functional differential equations (see [17], [18], [19] and [34]), and in fact we hope to indicate in a future paper some applications of our results to functional differential equations of neutral type (see [11] or [15]). Second, and perhaps more relevant to our immediate line of development, asymptotic fixed point theorems provide a framework for unifying and generalizing many of the known fixed point theorems of functional anal-