Showing papers in "American Journal of Mathematics in 1985"

Multipliers of the Fourier Algebras of Some Simple Lie Groups and Their Discrete Subgroups

Abstract: For any amenable locally compact group G, the space of multipliers MA(G) of the Fourier algebra A(G) coincides with the space B(G) of functions on G that are linear combinations of continuous positive definite functions. We prove that MA(G)\B(G) * 0 for many non-amenable connected groups. More specifically we prove that MOA(G)\B(G) * 0 for the classical complex Lie groups, and the general Lorentz groups SOO(n, 1), n > 2. MOA(G) is a certain subspace of MA(G), which we call the space of completely bounded multipliers of A(G). Unlike MA(G), the space MOA(G) has nice stability properties with respect to direct products of groups. It is known that the Fourier algebra of the free group on N generators (N 2 2) admits an unbounded approximate unit ((Pn), which is bounded in the multiplier norm. We extend this result to any closed subgroup of the general Lorentz group SOO(n, 1). Moreover we show that for these groups ((Pn) can be chosen to be bounded with respect to the MOA(G)-norm. By a duality argument we obtain that the reduced C*-algebra of every discrete subgroup of SOO(n, 1) has "the completely bounded approximation property. " In particular this property holds for C* (F2), the reduced C*-algebra of the free group on two generators. We also prove

Forward and reverse carleson inequalities for functions in bergman spaces and their derivatives

Abstract: is satisfied. In the process, a simple proof of certain inequalities on derivatives of functions in HP is obtained (Section 3) as well as some information on interpolation sequences for LP (Section 5). Inequalities like (1.1) have already been used to provide information on Bergman-Toeplitz operators [L1] and are useful for obtaining representations of functions in Lp (Section 4, Theorem 4.6). The inequality which reverses the roles of A and m in (1.1) is relatively easy-even if an appropriate weighting factor is included with m (Theorem A below). It was obtained by Oleinik and Pavlov [OP] and independently (when p = q > 1) by Stegenga [S] and (with c = 0) Hastings [Ha]. The results of Oleinik and Pavlov (which are more general than Theorem A) were extended somewhat by Oleinik in [0]. Cima and Wogen [CW], using Stegenga's methods, proved the analogue of Theorem A in several variables. The methods of [L3] more or less supercede all these results.

Weighted poincare and sobolev inequalities and estimates for weighted peano maximal functions

Abstract: 1. Introduction. In this paper, we derive local Poincare and Sobolev inequalities involving two weight functions. The estimates we obtain include those proved by Fabes, Kenig and Serapioni in [8] for the one weight function case. We also consider the notion of weighted Peano derivatives and show that the Peano derivative formed with a given weight function can be estimated in terms of a maximal function involving a second weight function. In the special case of ordinary Lebesgue measure, results of this kind lead to the theorems of A. P. Calderon and A. Zygmund about local differentiability of functions in Sobolev spaces and of functions which are of bounded variation in the sense of Tonelli (see [3] and [4]). All our results are based on variants of the basic estimate in [5], but we do not presuppose any facts from [5]. Let v(x) be a weight function on R"; i.e., let v(x) be nonnegative and locally integrable with respect to Lebesgue measure. We shall use the notation v(E) = SE v(x) dx; ordinary Lebesgue measure will be denoted IE l. If Bh(x) is the open ball with center x and radius h, we say v satisfies the doubling condition (with respect to Lebesgue measure) if

Smooth linearization near a fixed point

Abstract: In this paper we extend a theorem of Sternberg and Bi- leckii. We study a vector field, or a diffeomorphism, in the vicinity of a hyperbolic fixed point. We assume that the eigenvalues of the linear part A (at the fixed point) satisfy Qth order algebraic inequalities, where Q 2 2, then there is CK-linearization in the vicinity of the fixed point, where K is the Q-smoothness of A. We give an explicit and simple algorithm for com- puting K. An illustrative example from celestial mechanics is included which shows that our main theorem is an improvement over previously known theories. We also show that if the vector field or diffeomorphism depends smoothly on a parameter, then the linearizing conjugation de- pends smoothly on the parameter.

Heights and arakelov's intersection theory

Abstract: Introduction. This paper relates the Neron-Tate height on Jacobian varieties to an intersection theory on arithmetic surfaces constructed by S. Arakelov [A1 ]. Here I extend and prove a statement of Arakelov to the effect that his intersection pairing is a Neron pairing [A2I. I use this to prove an analogue of the Hodge Index theorem for arithmetic surfaces, a result due independently to G. Faltings. The results of this paper have been extracted from my thesis, written under the direction of Michael Artin. It differs from the thesis in three ways: Firstly it employs the language of local height functions as defined by A. Neron and J. Tate and expounded by S. Lang [LI ]. In the thesis I used Neron's language of local intersection pairings [N1 ]. The second major difference is in the proof of Theorem 1.3. In my thesis I proved this result by using work of Raynaud and Picard Functors [R 1 ], this method was suggested by M. Artin. In this paper I provide a simpler proof observed by S. Lang (and independently) by B. Gross. Finally the development of Arakalov's theory in the context of Neron Functions in section 2 is due to S. Lang, who has kindly given me leave to publish it. This exposition of my results is due in considerable part to S. Lang; it is essentially an edited transcript of a series of conversations between us, which he provided. It is my great pleasure to thank both him and Michael Artin for their help and encouragement. 1. The Neron pairing. Neron [Ne] has given a pairing between divisors and points on abelian varieties, and also on arbitrary varieties, under suitable conditions. We shall give a complement to the Neron theory on curves. Let C be a curve, by which we mean a complete regular curve, geometrically irreducible over a field K with an absolute value v. Let

Solving homogeneous linear differential equations in terms of second order linear differential equations

Abstract: Soit F un corps differentiel de caracteristique zero et soit L(y)=0 une equation differentielle lineaire homogene d'ordre n a coefficients dans F. On developpe des conditions necessaires et suffisantes pour que L(y)=0 soit resoluble en termes d'equations differentielles lineaires d'ordre 2

Stably Free Modules

Abstract: In [Su. Prob. 3], Suslin had asked the following question: Let A be any affine algebra of dimension n over an algebraically closed field. What is the smallest integer m such that all stably free projective modules of rank bigger than m are free? All the examples in the literature of stably free non-free modules have rank less than or equal to (n 1)/2. The aim of this note is to construct examples of such modules of large rank. We construct rank p stably free non-free modules over (p + 2)-dimensional affine algebras over algebraically closed fields, wherep is any prime. These varieties are actually smooth and rational. Over C, these are trivial as holomorphic vector bundles. [Forp > 2, this is classical. Forp = 2, see [MS]]. So these are strictly algebraic examples. I had described this construction in [MK 1] and proved the result for p = 2. We will reproduce the construction with necessary modifications in this note. Let p be any prime number and k any field. Letf (x) be any polynomial of degree p over k. Letf (0) = a E k* and Fi (xo, xl) = F(xo,x1) = xP *f (x0 /x1 ). Also let

A Symbol Class for Some Schrodinger Equations on R n

Abstract: On considere l'equation de Schrodinger dans R n : idψ t /dt=Hψ t , et on interprete certains resultats de Zelditch d'une facon plus geometrique On suit les singularites de Ψ t a travers l'espace (x, ξ) en introduisant une notion d'ensemble front d'onde qui est metaplectiquement covariant

Submanifolds and homology of nonsingular real algebraic varieties

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.

The inverse problem for ordinary differential operators on the line

Abstract: On etudie le probleme inverse sur la ligne pour l'operateur P=D n +q n−2 D n−2 +...+q 0 , D=d/idx, pour tout ordre n≥2 et sans hypothese de propriete d'auto-adjoint

Automorphisms of Tube Domains and Spherical Hypersurfaces

Abstract: 0. Introduction. Let Q C R' be an open convex set containing no lines and let D = Q X iR' = {x + iy E Cn, y E R' } be the tube domain over U. The domain D is biholomorphically equivalent to a bounded pseudoconvex domain and Aut(D) = {f: D -DI f a biholomorphic map } is a Lie group. When Q is a bounded smooth strictly convex domain, the group Aut(D) was shown ([5]) to consist of real affine linear transformations. In that same paper the Cauchy-Riemann structure of the boundary AD = aQ + iR' was studied in terms of the affine structure of an.

An inverse problem for the wave equation with first order perturbation

Abstract: Pour l'equation d'onde sur R n avec perturbation du potentiel on montre que l'on peut retrouver le potentiel a partir du comportement asymptotique de l'amplitude de diffusion. On montre qu'un resultat semblable est vrai quand on ajoute des termes d'ordre 1 a l'equation