estimates for riesz kernels of eigenfunction expansions of elliptic differential operators on compact manifolds let 2 and be the eigenvalues and an orthonormal system of eigenvectors of a second order elliptic differential operator on a compact n dimensional manifold m the riesz means of order of an integrable function on m are defined by r x m 1 2 2 x y y d y in this paper we study the kernels and the operator norms of he operators r on lp m 1 p 2 n 1 n 3 we also prove that if 1 p 2 n 1 n 3 and np n 1 2 then the operators r are of weak type p p uniformly with respect to

on almost everywhere divergence of bochner riesz means on compact lie groups let g be a connected simply connected compact semisimple lie group of dimension n it has been shown by clerc ann inst fourier grenoble 24 1 149 172 1974 that for any f in l 1 g the bochner riesz mean s r delta f converges almost everywhere to f provided delta n 1 2 in this paper we show that at the critical index delta n 1 2 there exists an f in l 1 g such that begin aligned limsup r rightarrow infty big s r n 1 2 f x big infty quad a e x in g end aligned this is an analogue of a well known result of kolmogoroff fund math 4 1 324 328 1923 for fourier series on the circle and a result of stein ann math 2 74 140 170 1961 for bochner riesz means on the tori mathbb t n n ge 2 we also study localization properties of the bochner riesz mean s r n 1 2 f for f in l 1 g

on the asymptotics of the bochner riesz kernel on compact lie groups suppose g is an n dimensional compact connected lie group and for 0 let dr r 0 be the bochner riesz kernel on g we prove that the lebesgue constants dr 1 are of the same order as r n 1 2 as r tends to infinity 0 n 1 2 while dr 1 log r if n 1 2 as a convergence we find best conditions for the norm convergence of the bochner riesz means on the spaces l1 g and c g

norms of harmonic projection operators on compact lie groups in order to simplify the notation we will assume throughout that g is connected simply connected and semisimple sharp estimates for vp z 0 when g su 2 have been obtained by sogge 6 who proved that vp zt d tl v where y t is the function which is affine on 1 2 3 4 and on 3 4 1 and is such that 7 1 2 0 7 3 4 1 4 7 1 1 two results in the literature give crucial estimates from below for vp n in the general case the first estimate concernes the lp norm of the character x if is the highest weight of n and 0 is half the sum of the positive roots then ii x llp 011 dimg p 1 2

hardy space estimates for the wave equation on compact lie groups abstract let l be the laplacian in this paper we prove that on a compact lie group g of dimension n the multiplier operator e i s l 1 l 2 s 0 1 extends to a bounded operator on the hardy space h p g 0 p if and only if 1 p 1 2 n 1 the result is an analogue of a well known theorem in euclidean space

central oscillating multipliers on compact lie groups fefferman and stein and wainger proved optimal l p boundedness for certain oscillating multipliers on mathbb r d in this article we prove analogues of their results for a compact lie group

sharp estimates for lebesgue constants on compact lie groups suppose g is a n dimensional compact connected semisimple lie group and dr is the spherical dirichlet kernel on g we prove the existence of a positive constant k such that dr 1 kr n 1 2 this complements the known result dr 1 hr n 1 2 we also prove that for a polyhedral dirichlet kernel dn the above inequalities hold with np in place of r n 1 2 p is the number of positive roots of g

on the convergence of riesz means on compact manifolds the purpose of this paper is to study the lp mapping properties of the riesz means of eigenfunction expansions associated to elliptic differential operators on compact manifolds in order to be more specific let us first introduce some notation let m be a compact connected c manifold of dimension n 2 and p an elliptic differential operator on m with c00 coefficients we shall assume that p is self adjoint with respect to some positive c00 density dx kept fixed throughout that is pu v u pv where as usual

effective symbolic dynamics random points statistical behavior complexity and entropy we consider the dynamical behavior of martin lof random points in dynamical systems over metric spaces with a computable dynamics and a computable invariant measure we use computable partitions to define a sort of effective symbolic model for the dynamics through this construction we prove that such points have typical statistical behavior the behavior which is typical in the birkhoff ergodic theorem and are recurrent we introduce and compare some notions of complexity for orbits in dynamical systems and prove i that the complexity of the orbits of random points equals the kolmogorov sinai entropy of the system ii that the supremum of the complexity of orbits equals the topological entropy

ergodic theorems for individual random sequences in the framework of the kolmogorov s approach to the substantiation of the probability theory and information theory on the base of the theory of algorithms we try to formulate probabilistic laws i e statements of the form p a 1 where a is some formula in pointwise form if is random then a holds nevertheless not all proofs of such laws can be directly translated into the algorithmic form in 11 two examples have been distinguished birkhoff s 2 ergodic theorem and shannon mcmillan breiman theorem of information theory 1 in this paper an analysis of algorithmic effectiveness of these theorems is given we prove that birkhoff s ergodic theorem is indeed in some strong sense nonconstructive at the same time the claim to formulate probabilistic laws for algorithmically random sequences is not so restrictive we present the versions of these laws for individual random sequences

concerning the lp norm of spectral clusters for second order elliptic operators on compact manifolds in this paper we study the lp l2 mapping properties of a spectral projection operator for riemannian manifolds this operator is a generalization of the harmonic projection operator for spherical harmonics on s see c d sogge duke math j 53 1986 43 65 among other things we generalize the l2 restriction theorems of c fefferman e m stein and p tomas bull amer math sot 81 1975 477478 for the fourier transform in r to the setting of riemannian manifolds we obtain these results for the spectral projection operator as a corollary of a certain sobolev inequality involving d r2 for large r this sobolev inequality generalizes certain results for r of c kenig a ruiz and the author duke math j 55 1987 329 347 the main tools in the proof of the sobolev inequalities for riemannian manifolds are the hadamard parametrix cf l hbrmander acta mad 88 1968 341 370 and the analysis of linear partial differential equations vol iii springer verlag new york 1985 and oscillatory integral theorems of l carleson and p sjolin studia math 44 19x 287 299 and stein arm math stud 112 1986 307 357 0 1988 academic pnss iw

norms of characters and fourier series on compact lie groups abstract we prove new estimates for the p norms of irreducible characters of compact lie groups these estimates are applied to give negative results on p mean convergence of fourier series on compact lie groups

from exact sciences to life phenomena following schrodinger and turing on programs life and causality this text presents a survey and a conceptual analysis of a path which goes from programming to physics and biology schrodinger s early reflections on coding and the genome will be a starting point by his and turing s remarks a link is explicitly made between the notion of program and the analysis of causality and determination in physics in particular turing s work in computing and in morphogenesis his 1952 paper on continuous dynamics will be seen as part of a scientific path which goes from laplace s understanding of deterministic predictability to the developments of poincare s analysis of unpredictability in non linear systems at the core of turing s 1952 work the relevance of planetary resonance in poincare s three body theorem and its analogies and differences with logical circularities will then be discussed on these grounds some recent technical results will be mentioned relating algorithmic randomness a strong form of logical undecidability and physical deterministic unpredictability this will be a way to approach the issue of resonances and circularities in system biology where these notions have a deeply different nature in spite of some confusion which is often made finally three aspects of the author s and his collaborators recent work in system biology will be surveyed they concern an approach to biological structural stability as extended criticality the structure of time and of biological rhythms and the role of a proper biological observable organization this is described in terms of anti entropy a new notion inspired by a remark by schrodinger

positive solutions of elliptic equations with nonlinear boundary conditions abstract this paper deals with the nonlinear elliptic equation u u f x u in a bounded smooth domain r n with a nonlinear boundary value condition the existence results are obtained by the sub supersolution method and the mountain pass lemma and nonexistence is also considered

l2 singular dichotomy for orbital measures of classical simple lie algebras we prove that the g invariant orbital measures supported on adjoint orbits in the lie algebra of a classical compact connected simple lie group satisfy a smoothness dichotomy either k is singular to lebesgue measure or k l 2 the minimum k for which k l 2 is specified and is also the minimum k such that the k fold sum of the orbit has positive measure

a constructive borel cantelli lemma constructing orbits with required statistical properties in the general context of computable metric spaces and computable measures we prove a kind of constructive borel cantelli lemma given a sequence constructive in some way of sets a i with effectively summable measures there are computable points which are not contained in infinitely many a i as a consequence of this we obtain the existence of computable points which follow the typical statistical behavior of a dynamical system they satisfy the birkhoff theorem for a large class of systems having computable invariant measure and a certain logarithmic speed of convergence of birkhoff averages over lipschitz observables this is applied to uniformly hyperbolic systems piecewise expanding maps systems on the interval with an indifferent fixed point and it directly implies the existence of computable numbers which are normal with respect to any base

the weak type l1 convergence of eigenfunction expansions for pseudodifferential operators soit m une variete compacte connexe lisse sans bord de dimension n 2 soit p d un operateur pseudodifferentiel elliptique d ordre 1 sur m qui est auto adjoint par rapport a une densite positive lisse dx on demontre la convergence faible de type l 1 pour les developpements en fonctions propres

norms of zonal spherical functions and fourier series on compact symmetric spaces abstract new estimates are given for the p norms of zonal spherical functions on compact symmetric spaces these estimates are applied to give a cohen type inequality for convolutor norms which leads to negative results on p mean convergence of fourier series on compact symmetric spaces of arbitrary rank

weak type estimates for bochner riesz spherical summation multipliers we consider the bochner riesz multiplier tofw 1 1 12 tjw 8 0 where denotes the fourier transform it is shown that the multiplier operator to is weak type po po acting on lpo rn radial functions where po is the critical value 2n n 1 28

necessary and sufficient conditions of existence for a system involving the p laplacian 1 p n in this work we study the existence of positive solutions for the system p pu f x u v inrn pv g x u v inrn lim x u x 0 lim x v x where pu div u p 2 u is the p laplacian 1 in some case we can have because these parameters depend on p n and the growth of f and g the existence result is related to a sub super solution methods and the nonexistence result is a consequence of picone s identity

l p norms of certain kernels on the n dimensional torus in this paper we study a class of kernels fr which generalize the bochner riesz kernels on the n dimensional torus our main result consists in upper estimates for the lp norms of fr as r tends to infinity as a consequence we prove a convergence theorem for means of functions belonging to suitable besov spaces 1 throughout this paper we identify the n dimensional torus tn n 2 with the n dimensional cube qn x e r n 1 2 0 we form the means fr g t f r m g m exp 2 7imt 1 m where t e tn m ranges on the integer lattice zn and denotes the convolution in tn in several cases one is able to estimate the l 1 tn norm of the kernel fr t e f r 1m exp 2 srimt 2 m and if 0 e s and f o 1 deduce convergence results for the means 1 as r x oc when g belongs to certain classes of periodic functions see e g 1 3 7 8 12 for instance if f x 1 i x12 8 when lxl 1 then fr k a is the familiar bochner riesz kernel in this case k i babenko cf 1 has shown that iikriil is of the same order as r n 1 2 as r tends to infinity n 2 0 8 n 1 2 while stein proved the estimate iikr il log r if 8 n 1 2 9 recently yudin 12 obtained the estimate lf o r n1 2 if f is the characteristic function of a closed balanced set c whose boundary has finite upper minkowski measure the method of yudin which also yields estimates for the lp tn norms of fr uses jackson approximation theorem which in tum involves estimates for the l2 r n modulus of continuity of f such a method can be adapted to a more general situation as we show in this paper in 2 4 we consider kernels of the type 2 associated with functionsf whose derivatives of certain orders are controlled by not necessarily positive received by the editors may 13 1980 1980 mathematics subject classification primary 42b99 secondary 46e35

a multiplicity result for generalized laplacian systems with multiparameters abstract in this paper we consider the existence nonexistence and multiplicity of a positive solution for a gelfand type generalized laplacian system with a singular indefinite weight and a vector parameter by using the upper and lower solution method and fixed point index theory we obtain a global multiplicity result with respect to the parameter