bmo spaces associated with semigroups of operators we study bmo spaces associated with semigroup of operators on noncommutative function spaces i e von neumann algebras and apply the results to boundedness of fourier multipliers on non abelian discrete groups we prove an interpolation theorem for bmo spaces and prove the boundedness of a class of fourier multipliers on noncommutative lp spaces for all 1 p with optimal constants in p

duality of hardy and bmo spaces associated with operators with heat kernel bounds the introduction and development of hardy and bmo spaces on euclidean spaces r in the 1960s and 1970s played an important role in modern harmonic analysis and applications in partial differential equations these spaces were studied extensively in 32 22 18 19 31 and many others an l function f on r is in the hardy spaceh r if the area integral function of the poisson integral e t f satisfies

global regularity and formation of singularities of solutions to first order quasilinear hyperbolic systems for the cauchy problem for strictly hyperbolic systems with general eigenvalues we obtain existence of global smooth solutions under certain conditions on the composition of the eigenvalues and the initial data on the other hand we give a sufficient condition which guarantees that singularities of the solution must occur in a finite time and describe certain applications the present paper includes the corresponding results in earlier papers by several authors as special cases

global existence of classical solutions to the mixed initial boundary value problem for quasilinear hyperbolic systems of diagonal form with large bv data abstract in this paper we investigate the mixed initial boundary value problem with large bv data for linearly degenerate quasilinear hyperbolic systems of diagonal form with general nonlinear boundary conditions in the half space t x t 0 x 0 as the result in a bressan contractive metrics for nonlinear hyperbolic systems indiana univ math j 37 1988 409 421 suggests that one may achieve global smoothness even if the c 1 norm of the initial data is large we prove that if the c 1 norm and the bv norm of the initial and boundary data are bounded but possibly large then the solution remains c 1 globally in time and possesses uniformly bounded total variation in x for all t 0 as an application we apply the result to the system describing the motion of relativistic closed strings in the minkowski space r 1 n

tent spaces associated with semigroups of operators abstract we study tent spaces on general measure spaces we assume that there exists a semigroup of positive operators on l p satisfying a monotone property but do not assume any geometric metric structure on the semigroup plays the same role as integrals on cones and cubes in euclidean spaces we then study bmo spaces on general measure spaces and get an analogue of fefferman s h 1 bmo duality theory we also get a h 1 bmo duality inequality without assuming the monotone property all the results are proved in a more general setting namely for noncommutative l p spaces

diffusion in a granular fluid i theory many important properties of granular fluids can be represented by a system of hard spheres with inelastic collisions traditional methods of nonequilibrium statistical mechanics are effective for analysis and description of the inelastic case as well this is illustrated here for diffusion of an impurity particle in a fluid undergoing homogeneous cooling an appropriate scaling of the liouville equation is described such that the homogeneous cooling ensemble and associated time correlation functions map to those of a stationary state in this form the familiar methods of linear response can be applied leading to green kubo and einstein representations of diffusion in terms of the velocity and mean square displacement correlation functions these correlation functions are evaluated approximately using a cumulant expansion and from kinetic theory providing the diffusion coefficient as a function of the density and the restitution coefficients comparisons with results from molecular dynamics simulation are given in the following companion paper

factorization and dilation problems for completely positive maps on von neumann algebras we study factorization and dilation properties of markov maps between von neumann algebras equipped with normal faithful states i e completely positive unital maps which preserve the given states and also intertwine their automorphism groups the starting point for our investigation has been the question of existence of non factorizable markov maps as formulated by c anantharaman delaroche we provide simple examples of non factorizable markov maps on m n mathbb c for all n 3 as well as an example of a one parameter semigroup t t t 0 of markov maps on m 4 mathbb c such that t t fails to be factorizable for all small values of t 0 as applications we solve in the negative an open problem in quantum information theory concerning an asymptotic version of the quantum birkhoff conjecture as well as we sharpen the existing lower bound estimate for the best constant in the noncommutative little grothendieck inequality

noncommutative maximal ergodic theorems this paper is devoted to the study of various maximal ergodic theorems in noncommutative spaces in particular we prove the noncommutative analogue of the classical dunford schwartz maximal ergodic inequality for positive contractions on and the analogue of stein s maximal inequality for symmetric positive contractions we also obtain the corresponding individual ergodic theorems we apply these results to a family of natural examples which frequently appear in von neumann algebra theory and in quantum probability

admissible decomposition for spectral multipliers on gaussian l p l p this paper concerns harmonic analysis of the ornstein uhlenbeck operator l on the euclidean space we examine the method of decomposing a spectral multiplier phi l into three parts according to the notion of admissibility which quantifies the doubling behaviour of the underlying gaussian measure gamma we prove that the above mentioned admissible decomposition is bounded in l p gamma for 1 p le 2 in a certain sense involving the gaussian conical square function the proof relates admissibility with e nelson s hypercontractivity theorem in a novel way

h calculus for semigroup generators on bmo abstract we prove that the negative generator l of a semigroup of positive contractions on l has bounded h s calculus on the associated poisson semigroup bmo space for any angle 2 provided l satisfies bakry emery s 2 0 criterion our arguments only rely on the properties of the underlying semigroup and work well in the noncommutative setting a key ingredient of our argument is a type of quasi monotone properties for the subordinated semigroup t t e t l 0 1 that is proved in the first part of this article

modified sonine approximation for the navier stokes transport coefficients of a granular gas abstract motivated by the disagreement found at high dissipation between simulation data for the heat flux transport coefficients and the expressions derived from the boltzmann equation by the standard first sonine approximation j j brey m j ruiz montero phys rev e 70 2004 051301 j j brey m j ruiz montero p maynar m i garcia de soria j phys condens matter 17 2005 s2489 we implement in this paper a modified version of the first sonine approximation in which the maxwell boltzmann weight function is replaced by the homogeneous cooling state hcs distribution the structure of the transport coefficients is common in both approximations the distinction appearing in the coefficient of the fourth cumulant a 2 comparison with computer simulations shows that the modified approximation significantly improves the estimates for the heat flux transport coefficients at strong dissipation in addition the slight discrepancies between simulation and the standard first sonine estimates for the shear viscosity and the self diffusion coefficient are also partially corrected by the modified approximation finally the extension of the modified first sonine approximation to the transport coefficients of the enskog kinetic theory is presented

cauchy problem for weakly linearly degenerate hyperbolic systems in diagonal form abstract we study the cauchy problem for quasi linear strictly hyperbolic systems of diagonal form with weakly linearly degenerate characteristics the global existence and uniqueness of c1 solution with initial data in c b 1 r w 1 1 r are proved when the l 1 r norm of or the c 0 r norm of is small we also give some examples to show that if the initial data are not in w 1 1 r or not small then the solution to the cauchy problem may blow up in a finite time

vector valued littlewood paley stein theory for semigroups abstract we develop a generalized littlewood paley theory for semigroups acting on l p spaces of functions with values in uniformly convex or smooth banach spaces we characterize in the vector valued setting the validity of the one sided inequalities concerning the generalized littlewood paley stein g function associated with a subordinated poisson symmetric diffusion semigroup by the martingale cotype and type properties of the underlying banach space we show that in the case of the usual poisson semigroup and the poisson semigroup subordinated to the ornstein uhlenbeck semigroup on r n this general theory becomes more satisfactory and easier to be handled in virtue of the theory of vector valued calderon zygmund singular integral operators

harmonic analysis on quantum tori this paper is devoted to the study of harmonic analysis on quantum tori we consider several summation methods on these tori including the square fej r means square and circular poisson means and bochner riesz means we first establish the maximal inequalities for these means then obtain the corresponding pointwise convergence theorems in particular we prove the noncommutative analogue of the classical stein theorem on bochner riesz means the second part of the paper deals with fourier multipliers on quantum tori we prove that the completely bounded l p fourier multipliers on a quantum torus are exactly those on the classical torus of the same dimension finally we present the littlewood paley theory associated with the circular poisson semigroup on quantum tori we show that the hardy spaces in this setting possess the usual properties of hardy spaces as one can expect these include the quantum torus analogue of fefferman s h 1 bmo duality theorem and interpolation theorems our analysis is based on the recent developments of noncommutative martingale ergodic inequalities and littlewood paley stein theory

steady state representation of the homogeneous cooling state of a granular gas the properties of a dilute granular gas in the homogeneous cooling state are mapped to those of a stationary state by means of a change in the time scale that does not involve any internal property of the system the new representation is closely related with a general property of the granular temperature in the long time limit the physical and practical implications of the mapping are discussed in particular simulation results obtained by the direct simulation monte carlo method applied to the scaled dynamics are reported this includes ensemble averages and also the velocity autocorrelation function as well as the self diffusion coefficient obtained from the latter by means of the green kubo representation in all cases the obtained results are compared with theoretical predictions

global existence of classical solutions to the cauchy problem on a semi bounded initial axis for a nonhomogeneous quasilinear hyperbolic system abstract it is proven that if the leftmost eigenvalue is weakly linearly degenerate then the cauchy problem for a class of nonhomogeneous quasilinear hyperbolic systems with small and decaying initial data given on a semi bounded axis admits a unique global c 1 solution on the domain t x t 0 x x n t where x x n t is the fastest forward characteristic emanating from the origin as an application of our result we prove the existence of global classical c 1 solutions of the flow equations of a model class of fluids with viscosity induced by fading memory with small smooth initial data given on a semi bounded axis

shock reflection for general quasilinear hyperbolic systems of conservation laws abstract this paper concerns the reflection of shock waves for general quasilinear hyperbolic systems of conservation laws in one space dimension it is shown that the mixed initial boundary value problem for general quasilinear hyperbolic systems of conservation laws with nonlinear boundary conditions on the quarter plane t x t 0 x 0 admits a unique global piecewise c 1 solution u u t x containing only shock waves with small amplitude and this solution possesses a global structure similar to that of the riemann solution u u x t of the corresponding riemann problem if the positive eigenvalues are genuinely nonlinear and the riemann solution has only shock waves and no rarefaction waves and contact discontinuities our result indicates that the riemann solution u u x t consisting of only shock waves possesses a semi global nonlinear structure stability

some new function spaces and their applications to harmonic analysis abstract in this paper a family of spaces is introduced which seems well adapted for the study of a variety of questions related to harmonic analysis and its applications these spaces are the tent spaces they provide the natural setting for the study of such things as maximal functions the relevant space here is t p and also square functions where the space t2p is relevant as such these spaces lead to unifications and simplifications of some basic techniques in harmonic analysis thus they are closely related to lp and hardy spaces important parts of whose theory become corollaries of the description of tent spaces also as proc conf harmonic analysis cortona lect notes in math vol 992 springer verlag berlin new york 1983 already indicated where these spaces first appeared explicitly the tent spaces can be used to simplify some of the results related to the cauchy integral on lipschitz curves and multilinear analysis in retrospect one can recognize that various ideas important for tent spaces had been used if only implicitly for quite some time here one should mention carleson s inequality its simplifications and extensions the theory of hardy spaces and atomic decompositions

non commutative martingale inequalities we prove the analogue of the classical burkholder gundy inequalites for non commutative martingales as applications we give a characterization for an ito clifford integral to be an lp martingale via its integrand and then extend the ito clifford integral theory in l2 developed by barnett streater and wilde to lp for all 1 p we include an appendix on the non commutative analogue of the classical fefferman duality between h1 and bmo

noncommutative riesz transforms a probabilistic approach for 2 p 0 in the classical setting this is certainly satisfied for the maps tt f f 1 t where t is a semigroup of measure preserving maps on a probability space in that case n l then the maps tt act on all noncommutative lp spaces lp n and in particular on the hilbert space l2 l2 n let a be the negative generator i e tt e ta for technical reasons we have to assume that there is a weakly dense algebra a n which lies in the domain of the unbounded operator a i e a a a the algebra a plays the role of smooth functions in the classical setting this assumption might be too restrictive for some classical applications but it is easily verified in the new noncommutative examples for elements x y a we may define the gradient form 2 x y a x y x a y a x y the correct definition of is the one given by sauvageot as a bilinear form with values in l1 again our arguments require stronger assumptions in fact in all our examples we will assume that the tt s are selfadjoint i e ttxy xtty and unital i e tt 1 1 then a is indeed a positive unbounded operator under these circumstances we can formulate p a meyer s problem is it true that x x 2 p c p a 1 2 x p 0 1 holds for all selfadjoint elements x a let us illustrate this question by considering the laplace operator a f f where i 2 x2 i then it is easily verified that

green kubo expressions for a granular gas the transport coefficients for a gas of smooth inelastic hard spheres are obtained from the boltzmann equation in the form of green kubo relations the associated time correlation functions are not simply those constructed from the fluxes of conserved densities instead fluxes constructed from the reference local homogeneous distribution occur as well the analysis exposes some complexities to be expected in the application of linear response methods to granular systems

a note on the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems abstract this paper is concerned with the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems with linearly degenerate characteristic fields on the basis of the existence result for the global classical solution we prove that when t tends to the infinity the solution approaches a combination of c 1 traveling wave solutions provided that the c 1 norm and the bv norm of the initial data are bounded but possibly large in contrast to former results obtained by liu and zhou j liu y zhou asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems math methods appl sci 30 2007 479 500 ours do not require their assumption that the system is rich in the sense of serre applications include that to the one dimensional born infeld system arising in string theory and high energy physics