bounds for partial derivatives necessity of umd and sharp constants we prove the necessity of the umd condition with a quantitative estimate of the umd constant for any inequality in a family of l p bounds between different partial derivatives partial beta u of u in c infty c mathbb r n x in particular we show that the estimate vert u xy vert p le k vert u xx vert p vert u yy vert p characterizes the umd property and the best constant k is equal to one half of the umd constant this precise value of k seems to be new even for scalar valued functions

an operator valued t1 theory for symmetric czos abstract we provide a natural bmo criterion for the l 2 boundedness of calderon zygmund operators with operator valued kernels satisfying a symmetric property our arguments involve both classical and quantum probability theory in the appendix we give a proof of the l 2 boundedness of the commutators r j b whenever b belongs to the bourgain s vector valued bmo space where r j is the j th riesz transform a common ingredient is the operator valued haar multiplier studied by blasco and pott

the navier stokes limit of the boltzmann equation for bounded collision kernels the present work establishes a navier stokes limit for the boltzmann equation considered over the infinite spatial domain r3 appropriately scaled families of diperna lions renormalized solutions are shown to have fluctuations whose limit points in the w l1 topology are governed by leray solutions of the limiting navier stokes equations this completes the arguments in bardos golse levermore commun pure appl math 46 5 667 753 1993 for the steady case and in lions masmoudi arch ration mech anal 158 3 173 193 2001 for the time dependent case

some good dilutions of fuzzy compactness abstract the obvious generalisation of compactness to fuzzy topology is known not to be a good extension in the sense of lowen following lowen s approach to defining a good fuzzy compactness definitions are suggested for fuzzy almost nearly and weak compactness which are proved to satisfy goodness fuzzy s closedness and rs compactness are treated to a similar approach

rate of convergence to equilibrium for the spatially homogeneous boltzmann equation with hard potentials for the spatially homogeneous boltzmann equation with hard potentials and grad s cutoff e g hard spheres we give quantitative estimates of exponential convergence to equilibrium and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation on which we provide a lower bound our approach is based on establishing spectral gap like estimates valid near the equilibrium and then connecting the latter to the quantitative nonlinear theory this leads us to an explicit study of the linearized boltzmann collision operator in functional spaces larger than the usual linearization setting

carleson measures and vector valued bmo martingales we study the relationship between vector valued bmo martingales and carleson measures let omega mathcal f p be a probability space and 2 q let x be a banach space given a stopping time let widehat tau denote the tent over widehat tau w k in omega times mathbb n tau w leq k tau w infty we prove that there exists a positive constant c such that sup tau frac 1 p tau infty int limits widehat tau df k qdp otimes dm leq c q f bmo x q for any finite martingale with values in x iff x admits an equivalent norm which is q uniformly convex the validity of the converse inequality is equivalent to the existence of an equivalent p uniformly smooth norm and then we also give a characterization of umd banach lattices

γ radonifying operators and umd valued littlewood paley stein functions in the hermite setting on bmo and hardy spaces abstract in this paper we study littlewood paley stein functions associated with the poisson semigroup for the hermite operator on functions with values in a umd banach space b if we denote by h the hilbert space l 2 0 d t t h b represents the space of radonifying operators from h into b we prove that the hermite square function defines bounded operators from bmo l r n b respectively h l 1 r n b into bmo l r n h b respectively h l 1 r n h b where bmo l and h l 1 denote bmo and hardy spaces in the hermite setting also we obtain equivalent norms in bmo l r n b and h l 1 r n b by using littlewood paley stein functions as a consequence of our results we establish new characterizations of the umd banach spaces

a geometrical characterization of banach spaces in which martingale difference sequences are unconditional we study banach space valued martingale transforms and in particular characterize those banach spaces for which the classical theorems of the real valued case carry over for example if b is a banach space and 1 0 and zeta x y leq x y text if x leq 1 leq y

pointwise convergence for semigroups in vector valued l p spaces suppose that tt t 0 is a symmetric diffusion semigroup on l2 x and denote by widetilde t t t geq0 its tensor product extension to the bochner space l p x mathcal b where mathcal b belongs to a certain broad class of umd spaces we prove a vector valued version of the hopf dunford schwartz ergodic theorem and show that this extends to a maximal theorem for analytic continuations of widetilde t t t geq 0 on l p x mathcal b as an application we show that such continuations exhibit pointwise convergence

fractional square functions and potential spaces abstract carlos segovia and richard wheeden defined fractional square functions involving fractional derivatives they obtained characterizations of potential spaces via square functions our aim in this paper is to reconsider the ideas of segovia and wheeden under the light of the semigroups of operators we develop a quite general theory of fractional square functions associated to certain classes of operators we present some examples of differential operators where our theory applies we recover in a more compact way the results of segovia and wheeden and we obtain new characterizations of the potential spaces associated to the harmonic oscillator and ornstein uhlenbeck operators

bmo spaces related to schrodinger operators with potentials satisfying a reverse holder inequality we identify the dual space of the hardy type space related to the time independent schrodinger operator v with v a potential satisfying a reverse holder inequality as a bmo type space we prove the boundedness in this space of the versions of some classical operators associated to hardy littlewood semigroup and poisson maximal functions square function fractional integral operator we also get a characterization of in terms of carlesson measures

regularly open sets in fuzzy topological spaces abstract this paper is devoted to the study of the role of fuzzy regularly open sets we prove some properties of fuzzy almost continuous mappings and define fuzzy almost open mappings we prove that under a fuzzy almost continuous and fuzzy almost open map the inverse image of a fuzzy regularly open set is fuzzy regularly open further we define a new type of fuzzy separation axioms fuzzy almost separation axioms it is interesting that there are some deviations in the behaviour of these axioms as compared to those in general topology for example in a fuzzy almost t1 space not every fuzzy singleton is closed also a fuzzy space which is fuzzy almost t 2 1 2 as well as fuzzy almost t0 is fuzzy almost regular while in general topology we have to take an almost t2 space in place of almost t0 space

uniformly convex and smooth banach spaces and lp boundedness properties of littlewood paley and area functions associated with semigroups abstract in this paper we obtain new characterizations of the uniformly convex and smooth banach spaces these characterizations are established by using l p boundedness properties of littlewood paley functions and area integrals associated with heat semigroups and involving fractional derivatives our results apply for instance by considering the heat semigroups defined by hermite and laguerre operators that do not satisfy the markovian property

conical square functions associated with bessel laguerre and schrodinger operators in umd banach spaces abstract in this paper we consider conical square functions in the bessel laguerre and schrodinger settings where the functions take values in umd banach spaces following a recent paper of hytonen van neerven and portal 36 in order to define our conical square functions we use radonifying operators we obtain new equivalent norms in the lebesgue bochner spaces l p 0 b and l p r n b 1 p in terms of our square functions provided that b is a umd banach space our results can be seen as banach valued versions of known scalar results for square functions

some remarks on lo rs fuzzy compactness abstract in this paper we define lo rs fuzzy compact fuzzy sets we study some properties of lo rs fuzzy compact spaces and lo rs fuzzy compact fuzzy sets we give some characterizations of these notions by fuzzy regularly open sets and fuzzy regularly closed sets we prove that lo rs fuzzy compactness is preserved under a fuzzy almost continuous and fuzzy almost open mapping

diffusive behavior from a quantum master equation we study a general class of translation invariant quantum markov evolutions for a particle on zd the evolution consists of free flow interrupted by scattering events we assume spatial locality of the scattering events and exponentially fast relaxation of the momentum distribution it is shown that the particle position diffuses in the long time limit this generalizes standard results about central limit theorems for classical non quantum markov processes

operator valued fourier multiplier theorems on lp x and geometry of banach spaces abstract this paper gives the optimal order l of smoothness in the mihlin and hormander conditions for operator valued fourier multiplier theorems this optimal order l is determined by the geometry of the underlying banach spaces e g fourier type this requires a new approach to such multiplier theorems which in turn leads to rather weak assumptions formulated in terms of besov norms

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

the phonon boltzmann equation properties and link to weakly anharmonic lattice dynamics for low density gases the validity of the boltzmann transport equation is well established the central object is the one particle distribution function f which in the boltzmann grad limit satisfies the boltzmann equation grad and much refined cercignani argue for the existence of this limit on the basis of the bbgky hierarchy for hard spheres at least for a short kinetic time span the argument can be made mathematically precise following the seminal work of lanford in this article a corresponding program is undertaken for weakly nonlinear both discrete and continuum wave equations our working example is the harmonic lattice with a weakly nonquadratic on site potential we argue that the role of the boltzmann f function is taken over by the wigner function which is a very convenient device to filter the slow degrees of freedom the wigner function so to speak labels locally the covariances of dynamically almost stationary measures one route to the phonon boltzmann equation is a gaussian decoupling which is based on the fact that the purely harmonic dynamics has very good mixing properties as a further approach the expansion in terms of feynman diagrams is outlined both methods are extended to the quantized version of the weakly nonlinear wave equation the resulting phonon boltzmann equation has been hardly studied on a rigorous level as one novel contribution we establish that the spatially homogeneous stationary solutions are precisely the thermal wigner functions for three phonon processes such a result requires extra conditions on the dispersion law we also outline the reasoning leading to fourier s law for heat conduction

transference between laguerre and hermite settings in this paper we present a new method in order to transfer boundedness results for operators associated with hermite functions to boundedness results for operators associated with laguerre functions the technique relies on an exact point wise identity relating the heat kernels of both systems the method that we present here has the novelty that can be used backwards that is boundedness results for laguerre systems can be also transfered to boundedness results for hermite systems we apply our method in order to get new properties of some operators in the laguerre setting among others we mention the description of riesz transforms as principal value operators as an application of the reversibility of the method we characterize the class of banach spaces b for which the riesz transforms in the laguerre setting are bounded from lbp into itself it is shown that this class coincides with the umd class

approach to equilibrium for the phonon boltzmann equation we study the asymptotics of solutions of the boltzmann equation describing the kinetic limit of a lattice of classical interacting anharmonic oscillators we prove that if the initial condition is a small perturbation of an equilibrium state and vanishes at infinity the dynamics tends diffusively to equilibrium the solution is the sum of a local equilibrium state associated to conserved quantities that diffuse to zero and fast variables that are slaved to the slow ones this slaving implies the fourier law which relates the induced currents to the gradients of the conserved quantities

towards a derivation of fourier s law for coupled anharmonic oscillators we consider a hamiltonian system made of weakly coupled anharmonic oscillators arranged on a three dimensional lattice mathbb z 2n times mathbb z 2 and subjected to stochastic forcing mimicking heat baths of temperatures t1 and t2 on the hyperplanes at 0 and n we introduce a truncation of the hopf equations describing the stationary state of the system which leads to a nonlinear equation for the two point stationary correlation functions we prove that these equations have a unique solution which for n large is approximately a local equilibrium state satisfying fourier law that relates the heat current to a local temperature gradient the temperature exhibits a nonlinear profile