Showing papers in "Annals of Mathematics in 1976"

Factorization theorems for Hardy spaces in several variables

Abstract: The purpose of this paper is to extend to Hardy spaces in several variables certain well known factorization theorems on the unit disk. The extensions will be carried out for various spaces of holomorphic functions on the unit ball of C" as well as for Hardy spaces defined by the Riesz systems on R". These results together with their proofs yield new characterizations of the space BMO (Bounded Mean Oscillation) and show a close relationship between BMO functions and certain linear operators on various LI and H2 spaces. The main tools are the result of Fefferman and Stein [8] on the duality between HI and BMO and a new characterization of BMO in terms of boundedness on L2 of the commutator of a singular integral operator with a multiplication operator. We begin by illustrating these ideas in the one dimensional case: Let F be holomorphic in {I z I < 1} and satisfy sup, 5 F(rete) I dO ? 1 (i.e., F is in H'(dO)). It is well known that F = GG2 with G1, G2 holomorphic and sup, I G,(rel0) 1' ! 1 (i.e., G, e H2(dO)). Write F = f + if, G, = gj + ig withf, g1, g, real. Thenf = Im(GG2) = sg1 1 + gi. Thusafunction f is an imaginary (or real) part of an HI function if and only if it can be represented as glg2 + g192 for L2 functions g, and g2. Furthermore,

Representations of reductive groups over finite fields

Abstract: JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Annals of Mathematics.

The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces

Abstract: In this paper we provide a complete classification of the local structure of singularities in a wide class of two-dimensional surfaces in R3 collected under the adjective (M, i, a) minimal by Almgren [A3] (see I(8)). The results, Theorems II. 4, IV. 5, IV. 8, are that the singular set of an (M, i, a) minimal set consists of H6lder continuously differentiable curves along which three sheets of the surface meet (Holder continuously) at equal (120?) angles, together with isolated points at which four such curves meet bringing together six sheets of the surface (H6lder continuously) at equal anglesin fact, in a neighborhood of each singular point, the surface is H6lder continuously diffeomorphic to either the surface Y of Figure 1 or the surface T of Figure 2 (both of which are defined in I(11)). The results apply to (idealizations of) many actual surfaces which are governed by surface tension, such as soap films as in Figure 4 and compound soap bubbles as in Figure 3 (and therefore to aggregates of some kinds of biological and metallurgical cells) (Corollary IV. 9 (i), (ii)), and thus are a proof of a result deduced experimentally by Plateau over 100 years ago [P]. They also apply to surfaces which minimize integrals which equal the area integral times some Holder continuous function on R3. A necessary first step in classifying singularities is to determine all possible area minimizing cones (Proposition II. 3). (In 1864 Lamarle claimed to make such a determination but his analysis of the technically most difficult case Figure 12 (p. 503)-was wrong.) Also included is a proof that the surface T of Figure 2 is in fact area minimizing (Theorem IV. 6); it seems to require the full force of Theorem IV.5 and I have never seen it proved elsewhere. The methods of this paper are

The completely positive lifting problem for C*-algebras

Abstract: (1.1)~~~~~~~~~~~~~~~~~~~1 where A, B are C*-algebras (resp., unital C*-algebras), J is a closed twosided ideal in B, 7 is the quotient map, and A, * are contractive (resp., unital) completely positive maps. The lifting problem for q is to determine whether or not one can find * so that the diagram commutes. In this paper, we will show that this is the case if A is separable, and either A, B/J, or B

Harmonic Analysis on Real Reductive Groups III. The Maass-Selberg Relations and the Plancherel Formula

Abstract: Part I. The c-, jand p-functions 2. Some elementary results on integrals 120 3. A lemma of Arthur 125 4. Induced representations 127 5. Intertwining operators 129 6. The mapping T-+XT ..131 131 7. The relation between induced representations and Eisenstein integrals 132 8. Some simple properties of E(P: f: v) 134 9. Proof of Theorem 7.1 136 10. An application of Theorem 7.1 138 11. Some properties of the j-functions 139 12. The p-function in a special case 141 13. The p-function in the general case and irreducibility of representations 142

On the first Betti number of a constant negatively curved manifold

Abstract: constant negatively curved n-dimensional manifolds with arbitrarily large first Betti number. In fact we show that any constant negatively curved manifold whose fundamental group is an arithmetic group commensurable with the group of units of a quadratic form admits a finite covering with first Betti number not equal to zero; in particular, the examples given by Borel at the end of [4] all admit such coverings. Previous to this a few examples were known in low dimensions (cf. Vinberg [14]) with examples up to dimension 5 in the compact case. However, his construction uses hyperbolic Coxeter groups which exist only in low dimensions. Rather than attack the problem algebraically by computing the abelianized fundamental group by group theory an approach that appears hopeless except for the above low dimensional examples, we take a geometric approach and construct explicit nonbounding codimension 1 cycles and then appeal to Poincare duality. Although these examples are of obvious geometric interest their main significance is group theoretic. The vanishing theorem of Kajdan [6] has as a consequence that the first Betti number of all compact locally irreducible, locally symmetric spaces of rank greater than 2 vanishes. This was extended by S. P. Wang [15] and Kostant [8] who show that Kajdan's criterion applies to all compact locally irreducible locally symmetric spaces except those associated with SO(n, 1) and SU(n, 1). The results of this paper show that the vanishing theorem does not hold for SO(n, 1). Whether or not it holds for SU(n, 1) remains unsolved. Our results are of interest in connection with the congruence subgroup problem. Bass, Milnor, Serre [1] show that if an arithmetic group F satisfies the congruence subgroup property that every subgroup of finite index contains a congruence subgroup, then:

Existence of codimension-one foliations

Abstract: A codimension-k foliation of a manifold Mn is a geometric structure which is formally defined by an atlas {qf: U. - Mn}, with U c Rn-k x R , such that the transition functions have the form 9pj(x, y) = (f(x, y), g(y)), [x e Rnk, y e Rk]. Intuitively, a foliation is a pattern of (n - k)-dimensional stripes-i.e., submanifolds-on M", called the leaves of the foliation, which are locally well-behaved. See the survey article of Lawson [11], for basic examples and better explanations of the definitions. The tangent space to the leaves of a foliation If forms a vector bundle over Mz, denoted TR. The complementary bundle vf = TMn/TJY is the normal bundle of WF. We define a codimension-k Haefliger structure, SC, to be a k-dimensional Rn-bundle v(XJ) over Mn, together with a foliation Y(ThC) transverse to the fibers of >(UC). A foliation If has a Haefliger structure SCT naturally associated to it, with normal bundle >(ACE) = Iff). The foliation IF(UXW) is constructed via the exponential map, exp: vfy) - Mn, which is transverse to If in a neighborhood of the zero section so that it induces a foliation 2(XYQ) in some neighborhood isomorphic to the entire bundle. 7Cy has the special property

Sous-espaces de dimension finie des espaces de Banach reticules

Abstract: Tous les espaces vectoriels consideres ici sont sur le corps des reels. Un espace norme C-reticule (C reel 2 1) est un espace vectoriel L muni d'une relation d'ordre fermee compatible avec la structure d'espace vectoriel (x ? y x + z ! y + z et Xx ? Xy pour x, y, z e L, X e R+) dans lequel deux elements quelconques x, y ont une borne superieure x U y, une borne inferieure x n y, et tel que si x, y e L, et I x I 0 de L; x, y e L seront dits etrangers si I x I n I Y 1 = 0. Un espace de Banach C-reticule est, par definition, un espace norme C-reticule qui est complet. Lorsque C = 1, L est appele espace norme reticule (resp. espace de Banach reticule). Un espace de Banach muni d'une base inconditionnelle (X%)%eN (telle que 11 = XX II :H C II In:l I Xx I xn II) est un espace C-reticule, pour la relation d'ordre dans laquelle =lXnxg ! 0 Xo, Xi, ..., Xk 2 0. E, F etant des espaces normes, on dit que F est finiment representable dans E si, pour tout s>0 et tout sous-espace F' de F de dimension finie, il existe un sous-espace E' de E et une bijection lineaire T: E' _* F' telle que

Continued fractions of algebraic power series in characteristic 2

Abstract: the field with two elements. There is a continued fraction theory for K, analogous to that for real numbers, with polynomials in x playing the role of the integers. We show that the continued fraction of the unique f e K satisfying the irreducible equation (over F(x)) Xf3 +f + X = 0

Some new four-manifolds

Abstract: The quotient space Q = X'/Z, of the involution is a smooth manifold of the (simple) homotopy type of real projective 4-space P', but not diffeomorphic or even piecewise linear (PL) homeomorphic to P'. As a corollary of the theorem and the vanishing of L,(Z2, -), there are precisely two s-cobordism classes of homotopy 4-dimensional real projective spaces. The manifold Q is obtained from Pi by removing the complement of a tube around P2 c P', a non-orientable 3-disk bundle over S', and replacing it by a certain bundle over S' with fibre the complement of an open cell in the 3-torus T3. In [6] we gave a classification theory of 4-manifolds, modulo connected sum with copies of S2 x S2, extending the results of Wall for the simplyconnected case. In particular, the connected sum of Q with many copies of S2 x SI is diffeomorphic to the manifold constructed in [6, 2.4], and the connected sum of Q with arbitrarily many copies of S2 x S2 is not PL homeomorphic or even h-cobordant to the connected sum of Pi with arbitrarily many copies of S2 x S2. Let V be a manifold with dim V > 1, and dim V > 2 if Vhas non-empty boundary. Then one can show, using topological surgery and the open h-cobordism theorem, that Pi x V and Q x V are topologically homeomorphic. It is not known if any of the manifolds Q constructed below are actually homeomorphic to P'. It is also not known if their universal covering spaces are diffeomorphic, PL homeomorphic, or even just homeomorphic to the 4-sphere S. 1 Partially supported by NSF.

Correction to "Monge-Ampere Equations, the Bergman Kernel, and Geometry of Pseudoconvex Domains"

Abstract: When one takes account of the sign change, the definite integral 7 on p. 414 vanishes identically, thus invalidating the arguments on pp. 414 and 415. With aD, substituted for aDO, the arguments of Section V apply with obvious small changes, and the definite integral analogous to -f is non-zero. Starting with the defining function r = z1 + 1 Z22 + i(z1 1)zl2, one obtains the following formulas, analogous to those in Section V of [1]: 2i = (112r12 + 48ir4)-2 288yr4 208r14 1568yr20

There exist inequivalent knots with the same complement

Abstract: The study and classification of knots has been based upon invariants of the knot complement. In this paper we give examples of inequivalent smooth spherical knots with diffeomorphic complements. These examples will also not be reflections, inversions or reflected inversions of each other. The knots we consider can be described as the class of all fibred knots with fibre a punctured n-torus, n > 3, and with the monodromy map on first homology having no negative eigenvalues. With easy modifications the arguments show that the constructed examples are not even piecewise linearly or topologically equivalent. A smooth n-knot is a smooth submanifold K of the (n + 2)-sphere S"+2, homeomorphic to So, and if K is diffeomorphic to S", it is said that the

Measure of degenerate relative equilibria. I

Abstract: Relative equilibria of three positive masses always correspond to the well known central configurations found by Euler and Lagrange-the collinear and equilateral triangle solutions of the three body problem. The name "relative equilibrium" suggests the fact that each of the masses appears at rest in a rotating barycentric coordinate system. By identifying any two configurations provided that one can be made congruent to the other by a rotation followed by a scalar multiplication, we define a relative equilibria class. Wintner [7] proves that there are only five relative equillibria classes in this case. The three Euler classes and the two Lagrange classes are always nondegenerate [3]. In [2] we showed how a degeneracy arises in the four body problem. In the plane E2 we place three unit masses at the vertices of an equilateral triangle with center of mass at the origin. We place at the origin an arbitrary fourth positive mass, mi. It follows easily for all values of m4 that this configuration is a relative equilibrium. We find that the relative equilibria class to which this configuration belongs is degenerate when m4 equals a unique positive mass ma* < 1.

A non-commutative real Nullstellensatz and Hilbert's 17th problem

Abstract: 1. Generalities Our object of study will be algebras with involution, or *-algebras as we will often call them. All our rings will be algebras over a field F of characteristic # 2; F is endowed with an automorphism a of period one or two with respect to which the involution * will be assumed to be semi-linear. In Section I we identify the central simple algebras which can support an involution of "positive" type. In Section IV we prove a Nullstellensatz for representations of prime algebras into such central simple algebras. This is used in Section V to classify the elements of a central simple algebra which are positive in all orderings. Section V follows closely the ideas of Artin-Schreier theory developed in proving that positive-valued rational functions are sums of squares. Let R be a *-algebra. A *-ideal I of R is an ideal which is stable under * R will be called *-simple if it has no non-trivial *-ideals. The *-center of R will be those elements of the center of R which are fixed by *. The *-center of a *-simple algebra is easily seen to be a field. Let R be *-simple with *-center G, G algebraically closed, and R finite dimensional over G. Then by [4, Chapter 0], R is one of the following types: [InJ R G, (n x n matrices over G) with the usual transpose involution; [II,] R -G2 with the usual symplectic involution; [III"J R G, ED Go (Go = the opposite ring of G,) with the exchange involution.

Solving Diophantine Problems Over All Residue Class Fields of a Number Field and All Finite Fields

Abstract: 0. Introduction 203 A. History and explanation of the problem 203 B. Background from logic and elimination theory 208 1. Generalizing the quantifier elimination problem . 210 A. Notations and terminology 210 B. The Frobenius symbol 212 C. Galois stratification and generalization of the diophantine problem .213 2. The intersection-union process 215 A. The intersection-union process over a finite field 215 B. The intersection-union process over a perfect field 217 3. A generalization of the theorems of Bertini and Noether . 219 4. Diophantine problems over all residue class fields of a number field . . .225 5. Diophantine problems over finite fields . 230 A. Over all extensions of a fixed finite field 230 B. Over all finite fields 231