Showing papers in "Acta Mathematica in 1973"

Random difference equations and Renewal theory for products of random matrices

Abstract: where Mn and Qn are random d • d matrices respectively d-vectors and Yn also is a d-vector. Throughout we take the sequence of pairs (Mn, Q~), n >/1, independently and identically distributed. The equation (1.1) arises in various contexts. We first met a special case in a paper by Solomon, [20] sect. 4, which studies random walks in random environments. Closely related is the fact tha t if Yn(i) is the expected number of particles of type i in the nth generation of a d-type branching process in a random environment with immigration, then Yn = (Yn(1) ..... Yn(d)) satisfies (1.1) (Qn represents the immigrants in the nth generation). (1.1) has been used for the amount of radioactive material in a compar tment ([17]) and in control theory [9 a]. Moreover, it is the principal feacture in a model for evolution and cultural inheritance by Cavalli-Sforza and Feldman [2]. Notice also tha t the dth order linear difference equation

The Lp-integrability of the partial derivatives of A quasiconformal mapping

Abstract: Jf(x) = lim sup m(f(B(x, r)))/m(B(x, r)), r->0 where B(x, r) denotes the open ^-dimensional ball of radius r about x and m denotes Lebesgue measure in R. We call Lf(x) and Jf(x\ respectively, the maximum stretching and generalized Jacobian for the homeomorphism ƒ at the point x. These functions are nonnegative and measurable in D, and Lebesgue's theorem implies that Jf is locally LMntegrable there. Suppose in addition that ƒ is X-quasiconformal in D. Then Lf ^ KJf a.e. in D, and thus Lf is locally L -integrable in D. Bojarski has shown in [1] that a little more is true in the case where n = 2, namely that Lf is locally L-integrable in D for p e [2, 2 + c), where c is a positive constant which depends only on K. His proof consists of applying the CalderonZygmund inequality [2] to the Hubert transform which relates the complex derivatives of a normalized plane quasiconformal mapping. Unfortunately this elegant two-dimensional argument does not suggest what the situation is when n > 2. The purpose of this note is to announce the following n-dimensional version of Bojarski's theorem. THEOREM. Suppose that D is a domain in R and that f\D^Risa K-quasiconformal mapping. Then Lf is locally L -integrable in D for p e [1, n + c\ where c is a positive constant which depends only on K and n.

Duality for crossed products and the structure of von Neumann algebras of type III

Abstract: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Preliminary 251 Construction of crossed products . . . . . . . . . . . . . . . . . . . . . . . 253 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Dual weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Bi-dual weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 Subgroups and subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . 284 The structure of a v o n Neumann algebra of type I I I . . . . . . . . . . . . . . 286 Algebraic invariants S ( ~ ) and T ( ~ ) of A. Connes . . . . . . . . . . . . . . . 294 Induced action and crossed products . . . . . . . . . . . . . . . . . . . . . 297 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

The Radon-Nikodym theorem for von neumann algebras

Abstract: Let ϕ be a faithful normal semi-finite weight on a von Neumann algebraM. For each normal semi-finite weight ϕ onM, invariant under the modular automorphism group Σ of ϕ, there is a unique self-adjoint positive operatorh, affiliated with the sub-algebra of fixed-points for Σ, such that ϕ=ϕ(h·). Conversely, each suchh determines a Σ-invariant normal semi-finite weight. An easy application of this non-commutative Radon-Nikodym theorem yields the result thatM is semi-finite if and only if Σ consists of inner automorphisms.

Nevanlinna theory and holomorphic mappings between algebraic varieties

Abstract: 0. NOTATIONS AND TERMINOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . 151 (a) D i v i s o r s a n d l ine b u n d l e s . . . . . . . . . . . . . . . . . . . . . . . . . 151 (b) T h e c a n o n i c a l b u n d l e a n d v o l u m e f o r m s . . . . . . . . . . . . . . . . . . . 154 (c) D i f f e r e n t i a l f o r m s a n d c u r r e n t s ( t e r m i n o l o g y ) . . . . . . . . . . . . . . . . 155

C*-algebras of almost periodic pseudo-differential operators

Abstract: Our basic goal is to develop an index theory for almost periodic pseudo-differential operators on R ~. The prototype of this theory is [5] which has direct application to the almost periodic Toeplitz operators. Here, we s tudy index theory for a C*-algebra of operators on R ~ which contains most almost periodic pseudo-differential operators such as those arising in the s tudy of elliptic boundary value problems for constant coefficient elliptic operators on a half space with almost periodic boundary conditions. Our program is as follows: We begin with a discussion of a C*-algebra with symbol which contains all of the classical pseudo-differential operators on R ~. Precisely, if A is a bounded operator on L2(R ~) and 2 ER ~, let e~(A) denote the conjugate of A with the function e *~'x acting as a multiplier denoted e~. We first s tudy the C*-algebra of those A for which the function 2~+e~(A) has a strongly continuous extension to the radial compactification of R ~. The restriction of this function to the complement of R ~ then gives the usual (principal) symbol a(A) when A is a pseudo-differential operator of order zero (of a suitable type). We characterize the Fourier multipliers in this algebra and the image of the symbol map. We give sufficient conditions for the usual construction of a pseudo-differential operator as well as one of Friedrichs' constructions to give an element of this algebra. In particular, the lat ter gives a positive linear right inverse for the symbol m a p a t least when the symbol is sufficiently smooth. I n fact, we show in w 3 tha t the Friedrichs map is a right inverse to the symbol map in the almost periodic case. We expect this to be true in the general case also.

The structure of a von Neumann algebra with a homogeneous periodic state: Dedicated to Professor M. Fukamiya on his 60th birthday

Abstract: Table of

Operators of principal type with interior boundary conditions

Abstract: In this paper we shall prove results, extending slightly those announced in [16]. The background is some work of HSrmander [9] and Egorov and Kondratev [5], which we shall first describe briefly. We shall always use the same notations for function spaces as HSrmander [7]. Let ~ be a paracompact C ~ manifold without boundary, T*(~) the cotangent space, T*(~)~,0 the space of non zero cotangent vecors and Lm(~) the space of pseudodifferential operators of type 1,0, introduced by HSrmander [8, 10]. In [9] HSrmander studied a pseudodifferential operator PELm(C2) with a principal symbol p E C ~ ( T * ( ~ ) ~ O ) , positively homogeneous of degree m, such that C~ # 0 everywhere on the set of zeros of p. Here C~,EC~(T * ( ~ ) ~ 0 ) is defined by

The fourier transform on semisimple lie groups of real rank one

Abstract: SO~E RESULTS OF H~RIsH-CItA~DRA . . . . . . . . . . . . . . . . . . . . . . 3 The structure of (~ and G . . . . . . . . . . . . . . . . . . . . . . . . 3 The invariant integral on G . . . . . . . . . . . . . . . . . . . . . . . . 4 The characters of the discrete series . . . . . . . . . . . . . . . . . . . . 6 The characters of the principal series . . . . . . . . . . . . . . . . . . . 7 FOURIER T R A N S I ~ O R ~ OF A R E G U L A R O R B I T . . . . . . . . . . . . . . . . . 8

How a minimal surface leaves an obstacle

Abstract: We announce that the function of least area among all functions defined in a convex domain, vanishing on its boundary, and constrained to lie above a concave analytic obstacle leaves the obstacle along an analytic curve. We announce a result about the curve of separation determined by the solution to a variational inequality. A strictly convex domain Q with smooth boundary <3Q is given in the z = xx + ix2 plane together with a smooth function \\j/(z) which assumes a positive maximum in Q and is negative on 3Q. Let K denote the closed convex set of Lipschitz functions v satisfying v ^ i// in Q and v = 0 on <9Q. Let us denote by u the function of K which minimizes area among all functions of K\\ that is (1) ueK: (\\ _L \\u | 2 \\ l /2 n (I + \\ux\\ ) (v — u)x dx ^ 0, v e K. The existence of such w, actually satisfying ueH(Q) n C(Q), 1 ^ q < oo, 0 < A < 1, was shown in the work of H. Lewy and G. Stampacchia [7] and also in M. Giaquinta and L. Pepe [1]. For u there is a set of coincidence / consisting of the points zeQ where u(z) = il/(z). Let us call (2) r(u) = r = {(xi,X2,x3):x3 = u(z) = \\\\j{z\\zedl} the \"curve\" of separation. Up to this time it has only been known that when i// is smooth and strictly concave, T is a Jordan curve [2], On the other hand, the corresponding problem for the ueK minimizing the Dirichlet integral has been thoroughly studied by H. Lewy and G. Stampacchia [6]. We wish to announce here the THEOREM. Let ij/ be analytic and strictly concave. Let u be the solution of(\\). Then F(u) is an analytic Jordan curve (as a function of its arc length parameter). The demonstration relies on the resolution of a system of differential equations and the utilization of the system to extend analytically a conAMS 1970 subject classifications. Primary 35J20; Secondary 53A10.