violation of the fluctuation dissipation theorem in glassy systems basic notions and the numerical evidence this review reports on the research done during past years on violations of the fluctuation dissipation theorem fdt in glassy systems it is focused on the existence of a quasi fluctuation dissipation theorem qfdt in glassy systems and the current supporting knowledge gained from numerical simulation studies it covers a broad range of non stationary aging and stationary driven systems such as structural glasses spin glasses coarsening systems ferromagnetic models at criticality trap models models with entropy barriers kinetically constrained models sheared systems and granular media the review is divided into four main parts 1 an introductory section explaining basic notions related to the existence of the fdt in equilibrium and its possible extension to the glassy regime qfdt 2 a description of the basic analytical tools and results derived in the framework of some exactly solvable models 3 a detailed report of the current evidence in favour of the qfdt and 4 a brief digression on the experimental evidence in its favour this review is intended for inexpert readers who want to learn about the basic notions and concepts related to the existence of the qfdt as well as for the more expert readers who may be interested in more specific results

a complete classification of bifurcation diagrams of a dirichlet problem with concave convex nonlinearities abstract we study the bifurcation diagrams of positive solutions of the two point boundary value problem u x f u x 0 1 u 1 u 1 0 where f u g u h u g h c 0 c2 0 and 0 is a bifurcation parameter we assume that functions g and h satisfy hypotheses h1 h3 under hypotheses h1 h3 we give a complete classification of bifurcation diagrams and we prove that on the u plane each bifurcation diagram consists of exactly one curve which is either a monotone curve or has exactly one turning point where the curve turns to the left hence the problem has at most two positive solutions for each 0 more precisely we prove the exact multiplicity of positive solutions

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

uniqueness results for the one dimensional p laplacian this paper is concerned with positive solutions of the boundary value problem y p 2y f y 0 y b 0 y b where p 1 b is a positive parameter assume that f is continuous on 0 changes sign from nonpositive to positive and f y yp 1 is nondecreasing in the interval of f 0 the uniqueness results are proved using a time mapping analysis

exact number of solutions of stationary reaction diffusion equations we study both existence and the exact number of positive solutions of the problem p l l u p 2u f u 0in 0 1 u 0 u 1 0 where l is a positive parameter p 1 the nonlinearity f is positive in 0 1 and f 0 f 1 0 assuming that f satisfies the condition lim s 1 f s 1 s email protected 0 where email protected 0 p 1 we study its behavior near zero and we obtain existence and exactness results for positive solutions we prove the results using the shooting method we show that there always exist solutions with a flat core for l sufficiently small as an application we prove the existence of a non negative solution for a class of singular quasilinear elliptic problems in a bounded domain in r n having a flat core in a ball

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

mapping properties of operator valued pseudo differential operators abstract in this paper we investigate the mapping properties of pseudo differential operators with operator valued symbols we prove the boundedness of regular symbols on sobolev spaces h 2 r d l 2 m and besov spaces b p q r d l p m for r and 1 p q as well as the boundedness of forbidden symbols on h 2 r d l 2 m and b p q r d l p m for 0 and 1 p q thanks to the smooth atomic decomposition of the operator valued triebel lizorkin spaces f 1 c r d m obtained in our previous paper we establish the f 1 c regularity of regular symbols for every r and the f 1 c regularity of forbidden symbols for 0 as applications we obtain the same results on the usual and quantum tori

characterizations of operator valued hardy spaces and applications to harmonic analysis on quantum tori this paper studies the operator valued hardy spaces introduced and studied by tao mei our principal result shows that the poisson kernel in mei s definition of these spaces can be replaced by any reasonable test function as an application we get a general characterization of hardy spaces on quantum tori the latter characterization plays a key role in our recent study of triebel lizorkin spaces on quantum tori

stationary profiles of degenerate problems with inhomogeneous saturation values abstract this work is concerned with the profiles of solutions for stationary problems between the degenerate diffusion of the p laplacian and a reaction with an inhomogeneous saturation value it is shown that if a parameter for the diffusion is sufficiently small then the solution attains the saturation value of the reaction in each region where the saturation value is constant the associated evolutional problem is also mentioned

interpolation between non commutative bmo and non commutative lp spaces we prove that for 1 p q the analogue of the classical result bmo lp pq lq holds in the setting of a finite von neumann algebra m equipped with an increasing filtration mn n 1 of von neumann subalgebras we also obtain the corresponding results for the real method of interpolation we discuss the appropriate operator space matrix norms and show that these interpolation results hold in the category of operator spaces

exact transformation of a langevin equation to a fluctuating response equation we demonstrate that a langevin equation that describes the motion of a brownian particle under non equilibrium conditions can be exactly transformed to a special equation that explicitly exhibits the response of the velocity to a time dependent perturbation this transformation is constructed on the basis of an operator formulation originally used in nonlinear perturbation theory for differential equations by extending it to stochastic analysis we find that the expression obtained is useful for the calculation of fundamental quantities of the system and that it provides a physical basis for the decomposition of the forces in the langevin description into effective driving dissipative and random forces in a large scale description

asymptotic behaviour of non linear parabolic equations with monotone principal part abstract we consider non linear parabolic equations with subdifferential principal part and give conditions under which they posses global attractors in spite of considering non lipschitz perturbations the case of globally lipschitz perturbations of a maximal monotone operator has been addressed in boll un mat ital b 8 2 2000 693 706 in the case of perturbations which are not globally lipschitz the main difficulty is the lack of uniqueness of solutions which at first does not even allow us to define attractors we overcome this difficulty for problems enjoying certain regularity and absorption properties that allow uniqueness of solutions after some time has been elapsed the results developed here are applied to the case when the subdifferential operator is the p laplacian to obtain existence of attractors and the existence of periodic solutions

uniqueness theorems for positive radial solutions of quasilinear elliptic equations in a ball we establish a new pohozaev type identity and use it to prove a theorem on the uniqueness of positive radial solutions to the quasilinear elliptic problem div u m 2emsp14 u f u 0 inb andu 0 on b wherebis a finite ball in rn n 3 and 1 0 and 1 p q n 2 n 2 has a unique positive solution whenn 6 this gives a complete answer to an open problem raised by brezis and nirenberg in 1983 in the casen 6 we shall also derive some partial results to the open problem in the casesn 3 4 5

derivation of stokes law from kirkwood s formula and the green kubo formula via large deviation theory we study the friction coefficient of a macroscopic sphere in a viscous fluid at low reynolds number first kirkwood s formula for the friction coefficient is reviewed on the basis of the hamiltonian description of particle systems according to this formula the friction coefficient is expressed in terms of the stress correlation on the surface of the macroscopic sphere then with the aid of large deviation theory we relate the surface stress correlation to the stress correlation in the bulk of the fluid where the latter is characterized by the viscosity in the green kubo formula by combining kirkwood s formula and the green kubo formula in large deviation theory we derive stokes law without explicitly employing the hydrodynamic equations

hardy littlewood theory for semigroups soit t t t 0 un semi groupe symetrique sousmarkovien et soit n 2 alors on a les proprietes equivalentes a dim t t n b il existe c 2 0 s t t t f c 2 t n 2 f 1 f l 1 x t 0 en fait a b meme dans le cas n 2

noncommutative martingale deviation and poincare type inequalities with applications we prove a deviation inequality for noncommutative martingales by extending oliveira s argument for random matrices by integration we obtain a burkholder type inequality with satisfactory constant using continuous time we establish noncommutative poincare type inequalities for nice semigroups with a positive curvature condition these results allow us to prove a general deviation inequality and a noncommutative transportation inequality due to bobkov and gotze in the commutative case to demonstrate our setting is general enough we give various examples including certain group von neumann algebras random matrices and classical diffusion processes among others

exact scaling functions for one dimensional stationary kpz growth we determine the stationary two point correlation function of the one dimensional kpz equation through the scaling limit of a solvable microscopic model the polynuclear growth model the equivalence to a directed polymer problem with specific boundary conditions allows one to express the corresponding scaling function in terms of the solution to a riemann hilbert problem related to the painleve ii equation we solve these equations numerically with very high precision and compare our up to numerical rounding exact result with the prediction of colaiori and moore 1 obtained from the mode coupling approximation

harmonic analysis approach to gromov hausdorff convergence for noncommutative tori we show that the rotation algebras are limits of matrix algebras in a very strong sense of convergence for algebras with additional lipschitz structure our results generalize to higher dimensional noncommutative tori and operator valued coefficients in contrast to previous results by rieffel li kerr and latremoliere we use lipschitz norms induced by the carre du champ of certain natural dynamical systems including the heat semigroup

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

classification of bifurcation diagrams of a p laplacian dirichlet problem with examples abstract we study bifurcation diagrams of positive solutions of the p laplacian dirichlet problem p u x f u x 0 1 x 1 u 1 u 1 0 where p y y p 2 y p u is the one dimensional p laplacian and p 1 and 0 are two bifurcation parameters assume that f u g u h u where g h c 0 c 2 0 satisfy hypotheses h1 h5 presented herein for different values p with 1 p 2 and with p 2 we give a classification of totally six different bifurcation diagrams we prove that on the u plane each possible bifurcation diagram consists of exactly one curve with exactly one turning point where the curve turns to the right hence we are able to determine the exact multiplicity of positive solutions in addition for 1 p 2 and for p 2 we give interesting examples f u k u p 1 u q u r satisfying r q p 1 and k 0 and show complete evolution of bifurcation diagrams as evolution parameter k varies from 0 to

on the shape of the nonnegative solutions to a singularly perturbed quasilinear dirichlet problem abstract we study the shape of the nontrivial nonnegative solutions for a singularly perturbed quasilinear dirichlet problem with non lipschitz nonlinearity we find that there exists at least one nontrivial nonnegative solution of the problem which consists of some bubbles the profile of each bubble is discussed

the set of positive solutions of semilinear equations in large balls the exact number of positive solutions of u f u 0 on finite balls in n is determined the assumptions about f u are similar to those imposed by serrin and the second author in a previous study of uniqueness of the positive solution when the spatial domain is all of n see 7 8 for finite balls of sufficiently large radius it is shown here that there are exactly two positive and hence radial solutions to this end we first prove the linear nondegeneracy of the positive solution of n this is obtained by applying the technique of monotone separation of graphs 7 to the linearised equations somewhat sharper estimates are required here see part i section 2

exact multiplicity for some quasilinear elliptic dirichlet problems where the non linearity changes sign abstract the exact multiplicity of positive solutions is obtained for a class of quasilinear elliptic dirichlet problems where the non linearity changes sign meanwhile the uniqueness of positive ground state on r n is also obtained

continuity properties on p for p laplacian parabolic problems abstract in this work we obtain some continuity properties on the parameter p at p 2 for the takeuchi yamada problem which is a degenerate p laplacian version of the chafee infante problem we prove the continuity of the flows and the equilibrium sets and the upper semicontinuity of the global attractors

coincidence sets in quasilinear elliptic problems of monostable type abstract this paper concerns the formation of a coincidence set for the positive solution of the boundary value problem e p u u q 1 f a x u in with u 0 on where e is a positive parameter p u div u p 2 u 1 q p f s s 1 s s 0 for some 0 and a x is a positive smooth function satisfying p a 0 in with inf a 0 it is proved in this paper that if 0 1 the coincidence set o e x u e x a x has a positive measure for small e and converges to with order o e 1 p as e 0 moreover it is also shown that if 1 then o e is empty for any e 0 the proofs rely on comparison theorems and the energy method for obtaining local comparison functions

complete boundedness of heat semigroups on the von neumann algebra of hyperbolic groups we prove that lambda g mapsto e t g r lambda g t 0 defines a completely bounded semigroup of multipliers on the von neuman algebra of hyperbolic groups for all real number r one ingredient in the proof is the observation that a construction of ozawa allows to characterize the radial multipliers that are bounded on every hyperbolic graph partially generalizing results of haagerup steenstrup szwarc and wysocza nski another ingredient is an upper estimate of trace class norms for hankel matrices which is based on peller s characterization of such norms

on ergodic theorems for free group actions on noncommutative spaces we extend in a noncommutative setting the individual ergodic theorem of nevo and stein concerning measure preserving actions of free groups and averages on spheres s 2 n of even radius here we study state preserving actions of free groups on a von neumann algebra a and the behaviour of s 2 n x for x in noncommutative spaces l p a for the cesaro means this problem was solved by walker our approach is based on ideas of bufetov we prove a noncommutative version of rota alternierende verfahren theorem to this end we introduce specific dilations of the powers of some noncommutative markov operators

asymmetric doob inequalities in continuous time abstract the present paper is devoted to the second part of our project on asymmetric maximal inequalities where we consider martingales in continuous time let m be a noncommutative probability space equipped with a continuous filtration of von neumann subalgebras m t 0 t 1 whose union is weak dense in m let e t denote the corresponding family of conditional expectations as for discrete filtrations we shall prove that for 1 p 2 and x l p m one can find a b l p m and contractions u t v t m such that e t x a u t v t b and max a p b p c p x p moreover a u t and v t b converge in the row column hardy spaces h p r m and h p c m respectively we also confirm in the continuous setting the validity of related asymmetric maximal inequalities which we recently found for discrete filtrations including p 1 as for other results in noncommutative martingale theory the passage from discrete to continuous index is quite technical and requires genuinely new methods our approach towards asymmetric maximal inequalities is based on certain construction of conditional expectations for a sequence of projective systems of l p modules the convergence in h p r m and h p c m also imposes new algebraic atomic decompositions

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

generalized jacobian elliptic functions and their application to bifurcation problems associated with p laplacian abstract the jacobian elliptic functions are generalized and applied to bifurcation problems associated with p laplacian the values of bifurcation parameter and the corresponding solutions are represented in terms of common parameters and a complete description of the bifurcation diagram and a closed form representation of the corresponding solutions are obtained as a by product of the representation it turns out that a kind of solution is also a solution of an eigenvalue problem of p 2 laplacian

branches of positive and free boundary solutions for some singular quasilinear elliptic problems we study the existence and multiplicity of solutions strictly positive or nonnegative having a free boundary the boundary of the set where the solution vanishes of some one dimensional quasilinear problems of eigenvalue type with possibly singular nonlinear terms

uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large existence and uniqueness results are proved for positive solutions of a class of quasilinear elliptic equations in a domain n via a generalisation of serrin s sweeping principle in the case when is an annulus it is shown that the solution is radially symmetric

multiplicity result for a degenerate elliptic equation with logistic reaction abstract a degenerate elliptic equation pu uq 1 1 ur 0 with zero dirichlet boundary condition where is a positive parameter 2 0 is studied it has been known that there exists a positive number such that if then the problem has no positive solution if then there exists a maximal solution and for sufficiently small

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

infinitely many nodal solutions for generalized logistic equations without odd symmetry on reaction abstract we consider nonlinear parametric logistic type equations driven by a nonlinear nonhomogeneous differential operator the reaction term is a caratheodory function which is not assumed to be odd the subdiffusive equidiffusive and superdiffusive cases are all treated we prove the existence of infinitely many nodal solutions we also prove a bifurcation type result describing the dependence of the set of constant sign solutions on the parameter in the superdiffusive case our approach uses variational methods together with upper lower solutions and truncation techniques and flow invariance arguments

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

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

bifurcation phenomena associated to the p laplace operator on determine la structure de l ensemble des solutions u de u x p 2 u x x f u u p 2 u sur 0 1 telles que u 0 u 1 0 ou p 1 et r on demontre que les solutions avec k zeros sont uniques quand 1 2

twisted hilbert transforms vs kakeya sets of directions given a discrete group g and an orthogonal action g o n we study lp convergence of fourier integrals which are frequency supported on the semidirect product ro g given a unit u rn and 1 p 6 2 our main result shows that the twisted directional hilbert transform hu o idg is lp bounded iff the orbit o u is finite this is in sharp contrast with twisted riesz transforms ru o idg which are always bounded our result characterizes fourier summability in lp for this class of groups we also extend de leeuw s compactification theorem to this setting and obtain stronger estimates for functions with lacunary frequency support

harmonic analysis on semigroups wherever t 0 and 1 p so e m stein st 3 developed a littlewoodpaley theory for such semigroups with some additional hypotheses r r coifman r rochberg and g weiss crw pointed out that some of stein s estimates could be obtained more simply by the use of transference techniques which were developed by coifman and weiss cw our object is to push these transference methods to present an alternative and simpler approach to some of the principal results of stein s book we shall also use these results to prove estimates for the wave equation

ground states and free boundary value problems for the n laplacian in n dimensional space using a new gradient estimate we prove several theorems on the existence of radial ground states for the n laplace equation div u n 2 u f u 0 in rn n 1 and the existence of positive radial solutions for the associated dirichlet neumann free boundary value problem in a ball we treat exponentially subcritical critical and supercritical nonlinearities f u which also are allowed to have singularities at zero moreover we show that the local behavior of f at zero affects the existence in a crucial way this allows us to prove the existence of ground states for a large class of functions f u without imposing any restriction on their growth for large u

linear response theory in stochastic many body systems revisited abstract the green kubo relation the einstein relation and the fluctuation response relation are representative universal relations among measurable quantities that are valid in the linear response regime we provide pedagogical proofs of these universal relations for stochastic many body systems through these simple proofs we characterize the three relations as follows the green kubo relation is a direct result of the local detailed balance condition the fluctuation response relation represents the dynamic extension of both the green kubo relation and the fluctuation relation in equilibrium statistical mechanics and the einstein relation can be understood by considering thermodynamics we also clarify the interrelationships among the universal relations

on p laplacian differential inclusions global existence compactness properties and asymptotic behavior abstract in this work we consider coupled systems of p laplacian differential inclusions we obtain the global existence of solutions and good compactness properties in such way that the set of all possible solutions compose a generalized semiflow which has a global attractor for each pair of positive diffusion coefficients we also prove that the attractors are upper semicontinuous on positive finite diffusion parameters

noncommutative burkholder rosenthal inequalities we investigate martingale inequalities in noncommutative lp spaces associated with a von neumann algebra equipped with a faithful normal state we prove the noncommutative analogue of the classical burkholder inequality on the conditioned or little square function and extend the noncommutative burkholder gundy inequalities from comm math phys 189 1997 667 698 to this nontracial setting we include several related results

bmo estimates for non commutative vector valued lipschitz functions abstract we construct markov semi groups t and associated bmo spaces on a finite von neumann algebra m and obtain results for perturbations of commutators and non commutative lipschitz estimates in particular we prove that for any a m self adjoint and f r r lipschitz there is a markov semi group t such that for x m f a x bmo m t c a b s f a x we obtain an analogue of this result for more general von neumann valued functions f r n n by imposing hormander mikhlin type assumptions on f in establishing these result we show that markov dilations of markov semi groups have certain automatic continuity properties we also show that markov semi groups of double operator integrals admit standard and reversed markov dilations

exact multiplicity results for boundary value problems with nonlinearities generalizing cubic using techniques of bifurcation theory we present two exact multiplicity results for boundary value problems of the type the first result concerns the case when the nonlinearity is independent of x and behaves like a cubic in u the second one deals with a class of nonlinearities with explicit x dependence

a noncommutative version of the john nirenberg theorem we prove a noncommutative version of the john nirenberg theorem for nontracial filtrations of von neumann algebras as an application we obtain an analogue of the classical large deviation inequality for elements of the associated bmo space

on certain second order ordinary differential equations associated with sobolev poincare type inequalities on considere l equation e i 1 n u x i p 2 u x i x i u q 2 u 0 dans u x 0 sur on donne les proprietes des solutions de e pour le cas n 1 et a b mais pour p q 1 avec p q regularite existence et unicite de solutions positives

thermodynamic formula for the cumulant generating function of time averaged current the cumulant generating function of time averaged current is studied from an operational viewpoint specifically for interacting brownian particles under nonequilibrium conditions we show that the first derivative of the cumulant generating function is equal to the expectation value of the current in a modified system with an extra force added where the modified system is characterized by a variational principle the formula reminds us of einstein s fluctuation theory in equilibrium statistical mechanics furthermore since the formula leads to the fluctuation dissipation relation when the linear response regime is focused on it is regarded as an extension of the linear response theory to that valid beyond the linear response regime the formula is also related to previously known theories such as the donsker varadhan theory the additivity principle and the least dissipation principle but it is not derived from them examples of its application are presented for a driven brownian particle on a ring subject to a periodic potential

structures and evolution of bifurcation diagrams for a one dimensional diffusive generalized logistic problem with constant yield harvesting abstract we study the one dimensional diffusive generalized logistic problem with constant yield harvesting u x g u 0 1 x 1 u 1 u 1 0 where 0 we assume that nonlinearity g satisfies g 0 g 1 0 g u 0 on 0 1 and g either is concave on 0 1 or is concave convex on 0 1 and satisfies a certain condition we prove that for any fixed 0 on the u plane the bifurcation diagram consists of a shaped curve and then we study the structures and evolution of bifurcation diagrams for varying 0 we also prove that for any fixed 2 4 g 0 on the u plane the bifurcation diagram consists of a reversed shaped curve and then we study the structures and evolution of bifurcation diagrams for varying 2 4 g 0

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

new function spaces of bmo type the john nirenberg inequality interpolation and applications in this paper we introduce and develop some new function spaces of bmo bounded mean oscillation type on spaces of homogeneous type or measurable subsets of spaces of homogeneous type the new function spaces are defined by variants of maximal functions associated with generalized approximations to the identity and they generalize the classical bmo space we show that the john nirenberg inequality holds on these spaces and they interpolate with lp spaces by the complex interpolation method we then give applications on lp boundedness of singular integrals whose kernels do not satisfy the hormander condition 2005 wiley periodicals inc

multiple positive solutions of the one dimensional p laplacian in this work we investigate the existence of positive solutions of the p laplacian using the quadrature method we prove the existence of multiple solutions of the one dimensional p laplacian for 0 and determine their exact number for 0 2005 elsevier inc all rights reserved

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

lap number properties for p laplacian problems investigated by lyapunov methods abstract in this work we consider a one dimensional quasilinear parabolic equation and we prove that the lap number of any solution cannot increase through orbits as the time passes if the initial data is a continuous function we deal with the lap number functional as a lyapunov function and apply lap number properties to reach an understanding on the asymptotic behavior of a particular problem