metriques de carnot cartheodory et quasiisometries des espaces symetriques de rang un we exhibit a rigidity property of the simple groups sp n 1 and f7 20 which implies mostow rigidity this property does not extend to o n 1 and u n 1 the proof relies on quasiconformal theory applied in the cr setting extensions are given to a class of solvable lie groups as a byproduct a result on quasiisometries of infinite nilpotent groups is obtained dans cet article on etablit une propriete de rigidite des groupes simples de rang un sp n 1 n 2 2 et fqui entraine la rigidite de mostow 1 theoreme toute quasiisornwtrie de 1 espace hyperbolique quaternionien hhn n 2 resp du plan hyperbolique de cayley cah2 est d distance bornee d une isometrie i e differe d une isomnetrie par une application qui diplace les points d une distance bornee une application f entre espaces metriques est une quasiisometrie s il existe des constantes l et c telles que l image de f soit c dense et que pour tous x y c d x y d fx fy ld x y c l une quasiisometrie entre des g et g est une sorte d isomorphisme virtuel dans la categorie topologique en effet cela correspond a une action de g sur un fibre principal c de groupe g sur une base compacte cf ra un isomorphisme entre sous groupes cocompacts covolume fini suffit pour les groupes simples de rang un de groupes de lie s etend en une quasiisometrie des espaces symetriques ou des groupes de lie si celle ci est proche d une isometrie des espaces symetriques resp un isomorphisme des groupes de lie les sous groupes sont conjugues c est la rigidite de mostow m2 la propriete ci dessus ne s etend pas aux groupes o n 1 et u n 1 en effet paragraphe 11 7 ceux ci ont beaucoup de quasiisometries au moins this content downloaded from 207 46 13 92 on sun 20 nov 2016 04 26 46 utc all use subject to http about jstor org terms

some new characterizations of sobolev spaces in this paper we present some new characterizations of sobolev spaces here is a typical result let g l p r n 1 p we prove that g w 1 p r n if and only if

on the convergence rate of some nonlocal energies abstract we study the rate of convergence of some nonlocal functionals recently considered by bourgain brezis and mironescu and after a suitable rescaling we establish the convergence of the corresponding rate functionals to a limit functional of second order

jump type hunt processes generated by lower bounded semi dirichlet forms let e be a locally compact separable metric space and m be a positive radon measure on it given a nonnegative function k defined on e e off the diagonal whose anti symmetric part is assumed to be less singular than the symmetric part we construct an associated regular lower bounded semi dirichlet form on l2 e m producing a hunt process x0 on e whose jump behaviours are governed by k for an arbitrary open subset d e we also construct a hunt process xd 0 on d in an analogous manner when d is relatively compact we show that xd 0 is censored in the sense that it admits no killing inside d and killed only when the path approaches to the boundary when e is a d dimensional euclidean space and m is the lebesgue measure a typical example of x0 is the stable like process that will be also identified with the solution of a martingale problem up to an polar set of starting points approachability to the boundary d in finite time of its censored process xd 0 on a bounded open subset d will be examined in terms of the polarity of d for the symmetric stable processes with indices that bound the variable exponent x

characterization of sobolev and bv spaces abstract the main results of this paper are new characterizations of w 1 p 1 p and bv for r n an arbitrary open set using these results we answer some open questions of brezis 2002 10 and ponce 2004 25

some properties of fuzzy groups on definit des sous groupes flous normaux des sous groupes flous normaux de niveau et leurs homomorphismes et on donne certaines de leurs proprietes

metric differentiation monotonicity and maps to l1 this is one of a series of papers on lipschitz maps from metric spaces to l1 here we present the details of results which were announced in cheeger and kleiner ann math 2006 to appear arxiv math mg 0611954 sect 1 8 a new approach to the infinitesimal structure of lipschitz maps into l1 and as a first application an alternative proof of the main theorem of cheeger and kleiner ann math 2006 to appear arxiv math mg 0611954 that the heisenberg group does not admit a bi lipschitz embedding in l1 the proof uses the metric differentiation theorem of pauls commun anal geom 9 5 951 982 2001 and the cut metric description in cheeger and kleiner ann math 2006 to appear arxiv math mg 0611954 to reduce the nonembedding argument to a classification of monotone subsets of the heisenberg group a quantitative version of this classification argument is used in our forthcoming joint paper with assaf naor cheeger et al in arxiv 0910 2026 2009

optimal mass transportation in the heisenberg group in this paper we consider the problem of optimal transportation of absolutely continuous masses in the heisenberg group hn in the case when the cost function is either the square of the carnot caratheodory distance or the square of the koranyi norm in both cases we show existence and uniqueness of an optimal transport map in the former case the proof requires a delicate analysis of minimizing geodesics of the group and of the differentiability properties of the squared distance function in the latter case the proof requires some fine properties of bv functions in the heisenberg group

a class of high order nonlocal operators we study a class of nonlocal operators that may be seen as high order generalizations of the well known nonlocal diffusion operators we present properties of the associated nonlocal functionals and nonlocal function spaces including nonlocal versions of sobolev inequalities such as the nonlocal poincare and nonlocal gagliardo nirenberg inequalities nonlocal characterizations of high order sobolev spaces in the spirit of bourgain brezis mironescu are provided applications of nonlocal calculus of variations to the well posedness of linear nonlocal models of elastic beams and plates are also considered

the unique games conjecture integrality gap for cut problems and embeddability of negative type metrics into l 1 in this article we disprove a conjecture of goemans and linial namely that every negative type metric embeds into e1 with constant distortion we show that for an arbitrarily small constant 0 for all large enough n there is an n point negative type metric which requires distortion at least log log n 1 6 to embed into e1 surprisingly our construction is inspired by the unique games conjecture ugc establishing a previously unsuspected connection between probabilistically checkable proof systems pcps and the theory of metric embeddings we first prove that the ugc implies a super constant hardness result for the nonuniform s parsest c ut problem though this hardness result relies on the ugc we demonstrate nevertheless that the corresponding pcp reduction can be used to construct an integrality gap instance for s parsest c ut towards this we first construct an integrality gap instance for a natural sdp relaxation of u nique g ames then we simulate the pcp reduction and translate the integrality gap instance of u nique g ames to an integrality gap instance of s parsest c ut this enables us to prove a log log n 1 6 integrality gap for s parsest c ut which is known to be equivalent to the metric embedding lower bound

weak continuity and lower semicontinuity results for determinants weak continuity properties of minors and lower semicontinuity properties of functionals with polyconvex integrands are addressed in this paper in particular it is shown that if un is bounded in open image in new window and if u bv n are such that un u in l1 n and

the gauss green theorem in stratified groups abstract we lay the foundations for a theory of divergence measure fields in noncommutative stratified nilpotent lie groups such vector fields form a new family of function spaces which generalize in a sense the bv fields they provide the most general setting to establish gauss green formulas for vector fields of low regularity on sets of finite perimeter we show several properties of divergence measure fields in stratified groups ultimately achieving the related gauss green theorem

action minimization and sharp interface limits for the stochastic allen cahn equation we study the action minimization problem which is formally associated to phase transformation in the stochastically perturbed allen cahn equation the sharp interface limit is related to but di erent from the sharp interface limits of the related energy functional and deterministic gradient flows in the sharp interface limit of the action minimization problem we find distinct most likely switching pathways depending on the relative costs of nucleation and propagation of interfaces this competition is captured by the limit of the action functional which we derive formally and propose as the natural candidate for the limit guided by the reduced functional we prove upper and lower bounds for the minimal action which agree on the level of scaling this is a preprint of an article accepted for publication in comm pure app math october 2005

semi dirichlet forms feynman kac functionals and the cauchy problem for semilinear parabolic equations abstract in the first part of the paper we prove various results on regularity of feynman kac functionals of hunt processes associated with time dependent semi dirichlet forms in the second part we study the cauchy problem for semilinear parabolic equations with measure data involving operators associated with time dependent forms model examples are non symmetric divergence form operators and fractional laplacians with possibly variable exponents we first introduce a definition of a solution resembling stampacchia s definition in the sense of duality and then using the results of the first part we prove the existence uniqueness and regularity of solutions of the problem under mild assumptions on the data

fine properties of functions with bounded variation in carnot caratheodory spaces abstract we study properties of functions with bounded variation in carnot caratheodory spaces we prove their almost everywhere approximate differentiability and we examine their approximate discontinuity set and the decomposition of their distributional derivatives under an additional assumption on the space called property r we show that almost all approximate discontinuities are of jump type and we study a representation formula for the jump part of the derivative

the dirichlet problem for nonlocal operators in this note we set up the elliptic and the parabolic dirichlet problem for linear nonlocal operators as opposed to the classical case of second order differential operators here the boundary data are prescribed on the complement of a given bounded set we formulate the problem in the classical framework of hilbert spaces and prove unique solvability using standard techniques like the fredholm alternative

mathematical models for local nontexture inpaintings dedicated to stanley osher on the occasion of his 60th birthday abstract inspired by the recent work of bertalmio et al on digital inpaintings siggraph 2000 we develop general mathematical models for local inpaintings of nontexture images on smooth regions inpaintings are connected to the harmonic and biharmonic extensions and inpainting orders are analyzed for inpaintings involving the recovery of edges we study a variational model that is closely connected to the classical total variation tv denoising model of rudin osher and fatemi phys d 60 1992 pp 259 268 other models are also discussed based on the mumford shah regularity comm pure appl math xlii 1989 pp 577 685 and curvature driven diffusions cdd of chan and shen j visual comm image rep 12 2001 the broad applications of the inpainting models are demonstrated through restoring scratched old photos disocclusion in vision analysis text removal digital zooming and edge based image coding

non local non convex functionals converging to sobolev norms abstract we study the pointwise convergence and the convergence of a family of non local non convex functionals in l p for p 1 we show that the limits are multiples of u p this is a continuation of our previous work where the case p 1 was considered

isoperimetric and sobolev inequalities for carnot caratheodory spaces and the existence of minimal surfaces after hormander s fundamental paper on hypoellipticity 54 the study of partial differential equations arising from families of noncommuting vector fields has developed significantly in this paper we study some basic functional and geometric properties of general families of vector fields that include the hormander type as a special case similar to their classical counterparts such properties play an important role in the analysis of the relevant differential operators both linear and nonlinear to motivate our results we recall some classical inequalities let e c r be a caccioppoli set a measurable set having a locally finite perimeter then one has the isoperimetric inequality

lipschitz continuity aleksandrov theorem and characterizations for h convex functions in the geometries of stratified groups we show that h convex functions locally bounded from above are locally lipschitz continuous and that the class of v convex functions exactly corresponds to the class of upper semicontinuous h convex functions as a consequence v convex functions are locally lipschitz continuous in every stratified group in the class of step 2 groups we characterize locally lipschitz h convex functions as measures whose distributional horizontal hessian is positive semidefinite in euclidean space the same results were obtained by dudley and reshetnyak we prove that a continuous h convex function is a e twice differentiable whenever its second order horizontal derivatives are radon measures

isomorphisms between spaces of lipschitz functions abstract we develop tools for proving isomorphisms of normed spaces of lipschitz functions over various doubling metric spaces and banach spaces in particular we show that lip 0 z d lip 0 r d for all d n more generally we e g show that lip 0 lip 0 g where is from a large class of finitely generated nilpotent groups and g is its mal cev closure or that lip 0 l p lip 0 l p for all 1 p we leave a large area for further possible research

the gibbs thompson relation within the gradient theory of phase transitions this paper discusses the asymptotic behavior as 0 of the chemical potentials associated with solutions of variational problems within the van der waals cahn hilliard theory of phase transitions in a fluid with free energy per unit volume given by 2 2 w where is the density the main result is that is asymptotically equal to e d o with e the interfacial energy per unit surface area of the interface between phases the constant sum of principal curvatures of the interface and d the density jump across the interface this result is in agreement with a formula conjectured by m gurtin and corresponds to the gibbs thompson relation for surface tension proved by g caginalp within the context of the phase field model of free boundaries arising from phase transitions

weak differentiability of bv functions on stratified groups we prove the almost everywhere approximate differentiability of functions with bounded variation in stratified groups and we study their approximate discontinuity set we introduce functions of bounded higher order variation and obtain a weak version of alexandroff differentiability theorem in this context we present a nontrivial class of functions with second order bounded variation arising from inf convolution formula of a suitable cost function

two subtle convex nonlocal approximations of the bv norm abstract inspired by the bbm formula and by work of g leoni and d spector we analyze the asymptotic behavior of two sequences of convex nonlocal functionals n u and n u which converge formally to the bv norm of u we show that pointwise convergence when u is not smooth can be delicate by contrast convergence to the bv norm is a robust and very useful mode of convergence

gamma convergence sobolev norms and bv functions we prove that the family of functionals i defined by i g rn rn g x g y x y n p dx dy g l r for p 1 and 0 converges in l r as goes to 0 when p 1 hereafter denotes the euclidean norm of r we also introduce a characterization for bv functions which has some advantages in comparison with the classic one based on the notion of essential variation on a e line

compression bounds for lipschitz maps from the heisenberg group to l 1 we prove a quantitative bi lipschitz non embedding theorem for the heisenberg group with its carnot caratheodory metric and apply it to give a lower bound on the integrality gap of the goemans linial semidefinite relaxation of the sparsest cut problem

some rules on categories of l fuzzy subgroups abstract the aim of this note is to study the l fuzzy subgroups through categorial methods for this we construct the category g l whose objects are l fuzzy subgroups and whose morphisms are l admissible homomorphisms by applying certain conditions we obtain the sub categories g 1 l g 2 l g n l and g com l of the category g l we observe that these categories are complete

notes on limits of sobolev spaces and the continuity of interpolation scales we extend lemmas by bourgain brezis mironescu 2001 and maz ya shaposhnikova 2002 on limits of sobolev spaces to the setting of interpolation scales this is achieved by means of establishing the continuity of real and complex interpolation scales at the end points a connection to extrapolation theory is developed and a new application to limits of sobolev scales is obtained we also give a new approach to the problem of how to recognize constant functions via sobolev conditions

sequences of non local functionals which approximate free discontinuity problems we prove by means of a generalized version of the well known fundamental estimate that the limit in the sense of convergence of a wide class of families of non local integral functionals is a functional related to a free discontinuity problem

a new characterization of sobolev spaces on mathbb r n in this paper we present a new characterization of sobolev spaces on mathbb r n our characterizing condition is obtained via a quadratic multiscale expression which exploits the particular symmetry properties of euclidean space an interesting feature of our condition is that depends only on the metric of mathbb r n and the lebesgue measure so that one can define sobolev spaces of any order of smoothness on any metric measure space

new sobolev spaces via generalized poincare inequalities on metric measure spaces in this paper we introduce some new function spaces of sobolev type on metric measure spaces these new function spaces are defined by variants of poincare inequalities associated with generalized approximations of the identity and they generalize the classical sobolev spaces on euclidean spaces we then obtain two characterizations of these new sobolev spaces including the characterization in terms of a variant of local sharp maximal functions associated with generalized approximations of the identity for the well known hajlasz sobolev spaces on metric measure spaces we also establish some new characterizations related to generalized approximations of the identity finally we clarify the relations between the sobolev type spaces introduced in this paper and the hajlasz sobolev spaces on metric measure spaces

characterization of polynomials and higher order sobolev spaces in terms of functionals involving difference quotients abstract the aim of this paper which deals with a class of singular functionals involving difference quotients is twofold deriving suitable integral conditions under which a measurable function is polynomial and stating necessary and sufficient criteria for an integrable function to belong to a k th order sobolev space one of the main theorems is a new characterization of w k p k n and p 1 for arbitrary open sets r n in particular we provide natural generalizations of the results regarding sobolev spaces summarized in brezis overview article brezis 2002 to the higher order case and extend the work borghol 2007 to a more general setting

fractional sobolev norms and bv functions on manifolds abstract the bounded variation seminorm and the sobolev seminorm on compact manifolds are represented as a limit of fractional sobolev seminorms this establishes a characterization of functions of bounded variation and of sobolev functions on compact manifolds as an application the special case of sets of finite perimeter is considered

quasiconformal mappings on the heisenberg group on etudie les applications conformes et quasi conformes sur le groupe de heisenberg la metrique invariante a gauche naturelle sur cet espace est non isotrope sous des dilatations et ne peut se decrire par une metrique de riemann ceci engendre de nouveaux phenomenes

can the nonlocal characterization of sobolev spaces by bourgain et al be useful for solving variational problems we question whether the recent characterization of sobolev spaces by bourgain brezis and mironescu 2001 could be useful to solve variational problems on w 1 p omega to answer this we introduce a sequence of functionals so that the seminorm is approximated by an integral operator involving a differential quotient and a radial mollifier then for the approximated formulation we prove existence uniqueness and convergence of the solution to the unique solution of the initial formulation we show that these results can also be extended in the bv case interestingly this approximation leads to a unified implementation for sobolev spaces including with high p values and for the bv space finally we show how this theoretical study can indeed lead to a numerically tractable implementation and we give some image diffusion results as an illustration

rectifiability and perimeter in the heisenberg group in this paper we fully extend to the heisenberg group endowed with its intrinsic carnot caratheodory metric and perimeter the classical de giorgi s rectifiability divergence theorems

on formulae decoupling the total variation of bv functions abstract in this paper we prove several formulae that enable one to capture the singular portion of the measure derivative of a function of bounded variation as a limit of non local functionals one special case shows that rescalings of the fractional laplacian of a function u s b v converge strictly to the singular portion of d u

equivalent definitions of bv space and of total variation on metric measure spaces abstract in this paper we introduce a new definition of bv based on measure upper gradients and prove the equivalence of this definition and the coincidence of the corresponding notions of total variation with the definitions based on relaxation of l 1 norm of the slope of lipschitz functions or upper gradients as in the previous work by the first author with gigli and savare in the sobolev case the proof requires neither local compactness nor doubling and poincare

limits of higher order besov spaces and sharp reiteration theorems abstract we compute the limits of higher order besov norms and derive the sharp constants for certain forms of the sobolev embedding theorem our results extend to higher order spaces the recent work by brezis bourgain mironescu and maz ya shaposhnikova the interpolation methods we develop are of interest on their own and could have applications to related inequalities

trace theorems for vector fields in the setting of carnot caratheodory spaces we prove some trace theorems for sobolev functions we consider the trace on a non characteristic surface for hormander vector fields of step r ge 1 and the trace on the boundary of a class of domains in the grushin plane

a characterization of fuzzy subgroups abstract an ordinary subgroup of a group g is 1 a subset of g 2 closed under the group operation in a fuzzy subgroup it is precisely these two notions that lose their deterministic character a fuzzy subgroup of a group g associates with each group element a number the larger the number the more certainly that element belongs to the fuzzy subgroup the closure property is captured by the inequality x y t x y in a rosenfeld s original definition t was the function minimum however any t norm t provides a meaningful generalization of the closure property two classes of fuzzy subgroups are investigated the fuzzy subgroups in one class are subgroup generated those in the other are function generated each fuzzy subgroup in these classes satisfies the above inequality with t given by t a b max a b 1 0 while the two classes look different each fuzzy subgroup in either is isomorphic to one in the other it is shown that a fuzzy subgroup satisfies the above inequality with t minimum if and only if it is subgroup generated of a very special type finally these notions are applied to some abstract pattern recognition problems

the wulff theorem revisited the parametrized indicator measures and the brunn minkowski inequality are used to prove that the wulff set w is a minimizer for the surface energy where the density is the support function of w the support of the indicator measures associated to minimizing sequences is characterized it is shown that if w is polyhedral then minimizing sequences cannot oscillate

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

some remarks on rearrangement for nonlocal functionals we prove that a nonlocal functional approximating the standard dirichlet p norm fails to decrease under two point rearrangement furthermore we get other properties related to this functional such as decay and compactness

differentiating maps into l1 and the geometry of bv functions this is one of a series of papers examining the interplay between differentiation theory for lipschitz maps x v and bi lipschitz nonembeddability where x is a metric measure space and v is a banach space here we consider the case v l 1 where differentiability fails we establish another kind of differentiability for certain x including n and h the heisenberg group with its carnot caratheodory metric it follows that does not bi lipschitz embed into l 1 as conjectured by j lee and a naor when combined with their work this provides a natural counterexample to the goemans linial conjecture in theoretical computer science the first such counterexample was found by khot vishnoi kv05 a key ingredient in the proof of our main theorem is a new connection between lipschitz maps to l 1 and functions of bounded variation which permits us to exploit results on the structure of bv functions on the heisenberg group fssc01

a uniqueness proof for the wulff theorem the wulff problem is a generalisation of the isoperimetric problem and is relevant for the equilibrium of small elastic crystals it consists in minimising the generally anisotropic surface energy among sets of given volume a solution of this problem is given by a geometric construction due to wulff in the class of sets of finite perimeter this was first shown by j e taylor who using methods of geometric measure theory also proved uniqueness here a more analytic uniqueness proof is presented the main ingredient is a sharpened version of the brunn minkowski inequality

characterizations of sobolev spaces via averages on balls abstract in this paper the authors prove several equivalent characterizations of sobolev spaces of even integer orders on r n using the average b t f x 1 b x t b x t f y d y of a function f over the ball b x t y r n y x t with x r n and t 0 these characterizations rely only on the metric and the lebesgue measure on r n and are simpler than those obtained recently by alabern et al 2012 moreover these results may shed new light on the theory of high order sobolev spaces on spaces of homogeneous type

bmo type seminorms from escher type tessellations abstract the paper is about a representation formula introduced by fusco moscariello and sbordone in 14 the formula permits to characterize the gradient norm of a sobolev function defined on the whole space as the limit of non local energies bmo type seminorms defined on tessellations of generated by cubic cells with arbitrary orientation we improve the main result in 14 in three different regards we give a new concise proof of the representation formula we analyze the case of a generic open subset and consider general tessellations of by means of cells more general than cubes again arbitrarily oriented inspired by the creative mind of the graphic artist m c escher

characterizations of second order sobolev spaces abstract second order sobolev spaces are important in applications to partial differential equations and geometric analysis in particular to equations such as the bi laplacian the main purpose of this paper is to establish some new characterizations of the second order sobolev spaces w 2 p r n in euclidean spaces we will present here several types of characterizations by second order differences by the taylor remainder of first order and by the differences of the first order gradient such characterizations are inspired by the works of bourgain et al 2001 and nguyen 2006 2008 on characterizations of first order sobolev spaces in the euclidean space

on the relaxation in bv ω rm of quasi convex integrals abstract given a quasi convex function f with linear growth we find the integral representation in bv r m of the functional f arising from the relaxation of f u f u dx u c 1 r m in the l loc 1 r m topology

uniqueness in law for pure jump markov processes let a be the operator defined on c2 functions by af left x right smallint left f left x h right f left x right f left x right h 1 1 1 left h right right v left x dh right sufficient conditions are given for existence and uniqueness for the martingale problem associated with a in the case of stable like processes where v x dh is equal to the levy measure for the stable symmetric process of index x for each x the conditions reduce to x continuous for existence and x dini continuous for uniqueness

from local to global in quasiconformal structures abstract we exhibit a large class of metric spaces whose infinitesimal quasiconformal structure is strong enough to capture the global quasiconformal structure a sufficient condition for this to happen is described in terms of a poincare type inequality

on the bourgain brezis and mironescu theorem concerning limiting embeddings of fractional sobolev spaces the article is concerned with the bourgain brezis and mironescu theorem on the asymptotic behaviour of the norm of the sobolev type embedding operator ws p lpn n sp as s 1 and s n p thei

a geometrical approach to monotone functions in r n this paper is concerned with the fine properties of monotone func tions on r n we study the continuity and differentiability properties of these functions the approximability properties the structure of the distributional derivatives and of the weak jacobians moreover we exhibit an example of a monotone function u which is the gradient of a c 1 convex function and

characterization of generalized gradient young measures generated by sequences in w1 1 and bv generalized young measures as introduced by diperna and majda commun math phys 108 667 689 1987 provide a quantitative tool for studying the one point statistics of oscillation and concentration in sequences of functions in this work after developing a functional analytic framework for such measures including a compactness theorem and results on the generation of such young measures by l1 bounded sequences or even by sequences of bounded radon measures we turn to investigation of those young measures that are generated by bounded sequences of w1 1 gradients or bv derivatives we provide several techniques to manipulate such measures including shifting averaging and approximation by piecewise homogeneous young measures and then establish the main new result of this work the duality characterization of the set of bv or w1 1 gradient young measures in terms of jensen type inequalities for quasiconvex functions with linear growth at infinity this result is the natural generalization of the kinderlehrer pedregal theorem arch ration mech anal 115 329 365 1991 j geom anal 4 59 90 1994 for classical young measures to the w1 1 and bv case and contains its version for weakly converging sequences in w1 1 as a special case finally we give an application to a new lower semicontinuity theorem in bv

lower semicontinuity of surface energies using the theory of indicator measures a lower semicontinuity result for quasiconvex functions in w 1 1 and assuming only l 1 convergence is obtained

coarse differentiation and quantitative nonembeddability for carnot groups abstract we give lower bound estimates for the macroscopic scale of coarse differentiability of lipschitz maps from a carnot group with the carnot caratheodory metric g d c c to a few different classes of metric spaces using this result we derive lower bound estimates for quantitative nonembeddability of lipschitz embeddings of g into a metric space x d x if x is either an alexandrov space with nonpositive or nonnegative curvature a superreflexive banach space or another carnot group that does not admit a bilipschitz homomorphic embedding of g for the same targets we can further give lower bound estimates for the bilipschitz distortion of every embedding f b n x where b n is the ball of radius n of a finitely generated nonabelian torsion free nilpotent group g we also prove an analogue of bourgain s discretization theorem for carnot groups and show that carnot groups have nontrivial markov convexity these give the first examples of metric spaces that have nontrivial markov convexity but cannot bilipschitzly embed into banach spaces of nontrivial markov convexity

sub elliptic global high order poincare inequalities in stratified lie groups and applications sharp poincare inequalities on balls or chain type bounded domains have been extensively studied both in classical euclidean space and carnot caratheodory spaces associated with sub elliptic vector fields e g vector fields satisfying hormander s condition in this paper we investigate the validity of sharp global poincare inequalities of both first order and higher order on the entire nilpotent stratified lie groups or on unbounded extension domains in such groups we will show that simultaneous sharp global poincare inequalities also hold and weighted versions of such results remain to be true more precisely let g be a nilpotent stratified lie group and f be in the localized non isotropic sobolev space w m p loc g w here 1 p q mand q is the homogeneous dimension of the lie group g suppose that the mth sub elliptic derivatives of f is globally l p integrable i e g x m f x p dx is finite but assume that lower order sub elliptic derivatives are only locally l p integrable we denote the space of such functions as b m p g we prove a high order poincare inequality for f minus a polynomial of order m 1 over the entire space g or unbounded extension domains as applications we will prove a density theorem stating that smooth functions with compact support are dense in b m p g modulus a finite dimensional subspace

new characterizations of sobolev spaces on the heisenberg group abstract the main aim of this paper is to present some new characterizations of sobolev spaces on the heisenberg group h first among several results theorem 1 1 theorem 1 2 we prove that if f l p h p 1 then f w 1 p h if and only if sup 0 1 f u f v p u 1 v q p d u d v this characterizations is in the spirit of the work by bourgain brezis and mironescu 5 in particular the work by hoai minh nguyen 29 in the euclidean spaces our work extends that of nguyen to sobolev spaces w 1 p h for p 1 in the setting of heisenberg group second corresponding to the case p 1 we give a characterizations of bv functions on the heisenberg group theorem 4 1 theorem 4 2 third we give some more generalized characterizations of sobolev spaces on the heisenberg groups theorem 5 1 theorem 5 2 it is worth to note that the underlying geometry of the euclidean spaces such as that any two points in r n can be connected by a line segment plays an important role in the proof of the main theorems in 29 thus one of the main techniques in 29 is to use the uniformity in every directions of the unit sphere in the euclidean spaces more precisely to deal with the general case s n 1 it is often assumed that e n 0 0 1 and hence one just needs to work on the one dimensional case this can be done by using the rotation in the euclidean spaces due to the non commutative nature of the heisenberg group the absence of this uniformity on the heisenberg group creates extra difficulties for us to handle hence we need to find a different approach to establish this characterization

some equivalent definitions of high order sobolev spaces on stratified groups and generalizations to metric spaces recently in the article lw the authors use the notion of polynomials in metric spaces mathcal s rho mu of homogeneous type in the sense of coifman weiss to prove a relationship between high order poincare inequalities and representation formulas involving fractional integrals of high order assuming only that mu is a doubling measure and that geodesics exist motivated by this and by recent work in h fhk ks and flw about first order sobolev spaces in metric spaces we define sobolev spaces of high order in such metric spaces mathcal s rho mu we prove that several definitions are equivalent if functions of polynomial type exist in the case of stratified groups where polynomials do exist we show that our spaces are equivalent to the sobolev spaces defined by folland and stein in fs our results also give some alternate definitions of sobolev spaces in the classical euclidean case

high order total variation based image restoration the total variation tv denoising method is a pde based technique that preserves edges well but has the sometimes undesirable staircase effect namely the transformation of smooth regions ramps into piecewise constant regions stairs in this paper we present an improved model constructed by adding a nonlinear fourth order diffusive term to the euler lagrange equations of the variational tv model our technique substantially reduces the staircase effect while preserving sharp jump discontinuities edges we show numerical evidence of the power of resolution of this novel model with respect to the tv model in some 1d and 2d numerical examples

censored stable processes we present several constructions of a censored stable process in an open set d rn i e a symmetric stable process which is not allowed to jump outside d we address the question of whether the process will approach the boundary of d in a finite time we give sharp conditions for such approach in terms of the stability index and the thickness of the boundary as a corollary new results are obtained concerning besov spaces on non smooth domains including the critical exponent case we also study the decay rate of the corresponding harmonic functions which vanish on a part of the boundary we derive a boundary harnack principle in c1 1 open sets

lp compression traveling salesmen and stable walks we show that if h is a group of polynomial growth whose growth rate is at least quadratic then the lp compression of the wreath product z oh equals max 1 p 1 2 we also show that the lp compression of z oz equals max p 2p 1 2 3 and the lp compression of z oz 0 the zero section of z oz equipped with the metric induced from z o z equals max p 1 2p 3 4 the fact that the hilbert compression exponent of z o z equals 3 while the hilbert compression exponent of z o z 0 equals 3 4 is used to show that there exists a lipschitz function f z o z 0 l2 which cannot be extended to a lipschitz function defined on all of z o z

new characterizations of sobolev metric spaces abstract we provide new characterizations of sobolev ad bv spaces in doubling and poincare metric spaces in the spirit of the bourgain brezis mironescu and nguyen limit formulas holding in domains of r n

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159