subelliptic p harmonic maps into spheres and the ghost of hardy spaces the story begins with the paper of muller 59 who for the sake of an application to nonlinear elasticity proved that if the jacobian determinant ju of a sobolev map u w 1 n loc rn rn is nonnegative then it belongs locally to l log l the result is quite intriguing since a priori holder inequality implies only that ju l1 and one does not suspect any higher integrability if one does not assume that ju is nonnegative then as coifman lions meyer and semmes 14 have proved ju belongs to the local hardy space h 1 loc since a nonnegative function belongs to the local hardy space if and only if it belongs locally to l log l the result of coifman lions meyer and semmes generalizes that of muller in fact coifman lions meyer and semmes proved more namely for 1 p the scalar product of a divergence free vector field e lp rn rn and a curl free vector field b lp p 1 rn rn belongs to the hardy space h 1 a priori holder inequality implies integrability of this expression only note that the jacobian is of this form the discovery that many algebraic expressions involving partial derivatives belong to the hardy space turned out to be important and widely applicable in nonlinear partial differential equations this was first shown by helein 40 41 42 let us describe briefly his result consider maps u bn s m from the n dimensional ball to the mdimensional sphere such that the p energy of u given by the functional

estimationslpdes operateurs de schrodinger sur les groupes nilpotents resume dans cet article on se propose d etudier la continuite dans les espaces l p des groupes nilpotents des operateurs 2 w 1 w 1 2 nabla w 1 et w 1 2 ou est un sous laplacien et w un potentiel qui verifie d assez bonnes conditions

frechet spaces with nuclear kothe quotients each separable frechet non banach space x with a continuous norm is shown to have a quotient y with a continuous norm and a basis if in addition y can be chosen to be nuclear we say that x has a nuclear kothe quotient we obtain a slightly technical characterization of those separable frechet spaces with nuclear kothe quotients in particular separable reflexive frh secondary 46a12 46a14 46a20 46a25 46a35 research partially supported by the national science foundation 01982 american mathematical society 0002 9947 82 0000 0090 05 00

small time heat kernel asymptotics at the sub riemannian cut locus for a sub riemannian manifold provided with a smooth volume we relate the small time asymptotics of the heat kernel at a point y of the cut locus from x with roughly how much y is conjugate to x this is done under the hypothesis that all minimizers connecting x to y are strongly normal i e all pieces of the trajectory are not abnormal our result is a renement of the one of leandre 4t logpt x y d 2 x y for t 0 in which only the leading exponential term is detected our results are obtained by extending an idea of molchanov from the riemannian to the sub riemannian case and some details we get appear to be new even in the riemannian context these results permit us to obtain properties of the sub riemannian distance starting from those of the heat kernel and vice versa for the grushin plane endowed with the euclidean volume we get the expansion pt x y t 5 4 exp d 2 x y 4t where y is reached from a riemannian point x by a minimizing geodesic which is conjugate at y

metric structures and probabilistic computation continuous first order logic is used to apply model theoretic analysis to analytic structures e g hilbert spaces banach spaces probability spaces etc classical computable model theory is used to examine the algorithmic structure of mathematical objects that can be described in classical first order logic the present paper shows that probabilistic computation sometimes called randomized computation and continuous logic stand in a similar close relationship the main result of this paper is an effective completeness theorem showing that every decidable continuous first order theory has a probabilistically decidable model we also show that probabilistically computable structures give rise to a model of aca 0in a natural way and describe a connection with complexity theory

the aronsson euler equation for absolutely minimizing lipschitz extensions with respect to carnot caratheodory metrics we derive the euler lagrange equation also known in this setting as the aronsson euler equation for absolute minimizers of the l variational problem formula math where c g is an open subset of a carnot group 0 u denotes the horizontal gradient of u r and the lipschitz class is defined in relation to the carnot caratheodory metric in particular we show that absolute minimizers are infinite harmonic in the viscosity sense as a corollary we obtain the uniqueness of absolute minimizers in a large class of groups this result extends previous work of jensen and of crandall evans and gariepy we also derive the aronsson euler equation for more regular absolutely minimizing lipschitz extensions corresponding to those carnot caratheodory metrics which are associated to free systems of vector fields

isoperimetric inequalities and the homology of groups an isoperimetric function for a finitely presented group g bounds the number of defining relations needed to prove that a word w g 1 in terms of the length of w suppose g f r is a finitely presented group where f is a finitely generated free group and r is the normal closure of a finite set of relators for each w r we define the area w to be the least number of uses of the given relators needed to prove that w g 1 the isoperimetric function g n is defined to be the maximum of the areas of all the words in r of length n one compares the size of such functions using a suitable order 1 on functions different presentations give rise to equivalent isoperimetric functions where is the equivalence relation determined by 1 interest in such functions was stimulated by gromov 13 who characterized word hyperbolic groups as those with linear isoperimetric function that is g n n further information was obtained by cannon et al 8 who showed that automatic groups satisfy g 1 n and that asynchronously automatic groups satisfy g 1 2 they also showed the free nilpotent group of class 2 and rank 2 has g n n gersten 10 used geometric arguments to obtain interesting lower bounds of exponential type for certain one relator groups he exploited the fact that for a torsion free one relator groupg the usual 2 dimensional complex having fundamental group isomorphic to g is aspherical the general technique for giving upper bounds for g n is to describe a solution to the word problem for g and analyze its use of defining relations it seems quite difficult to establish lower bounds for g n in this manner in this paper we begin a more algebraic study of isoperimetric inequalities and introduce new methods for establishing lower bounds for g instead of trying to analyze r directly we consider some of its quotient groups for which additional machinery is available for instance we consider for each w r its image w r f in r r f and define a suitable centralized isoperimetric function g n such that g n 1 g n using the hopf formula it can be shown that r r f is a direct sum of the second homology group h2 g z with a free abelian

the intrinsic hypoelliptic laplacian and its heat kernel on unimodular lie groups we present an invariant definition of the hypoelliptic laplacian on sub riemannian structures with constant growth vector using the popp s volume form introduced by montgomery this definition generalizes the one of the laplace beltrami operator in riemannian geometry in the case of left invariant problems on unimodular lie groups we prove that it coincides with the usual sum of squares we then extend a method first used by hulanicki on the heisenberg group to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular lie group of type i the main tool is the noncommutative fourier transform we then study some relevant cases su 2 so 3 sl 2 with the metrics inherited by the killing form and the group se 2 of rototranslations of the plane our study is motivated by some recent results about the cut and conjugate loci on these sub riemannian manifolds the perspective is to understand how singularities of the sub riemannian distance reflect on the kernel of the corresponding hypoelliptic heat equation

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

convergence of arithmetic means of operators in frechet spaces abstract every kothe echelon frechet space x that is montel and not isomorphic to a countable product of copies of the scalar field admits a power bounded continuous linear operator t such that i t does not have closed range but the sequence of arithmetic means of the iterates of t converges to 0 uniformly on the bounded sets in x on the other hand if x is a frechet space which does not have a quotient isomorphic to a nuclear kothe echelon space with a continuous norm then the sequence of arithmetic means of the iterates of any continuous linear operator t for which 1 n t n converges to 0 on the bounded sets converges uniformly on the bounded subsets of x i e t is uniformly mean ergodic if and only if the range of i t is closed this result extends a theorem due to lin for such operators on banach spaces the connection of browder s equality for power bounded operators on frechet spaces to their uniform mean ergodicity is exposed an analysis of the mean ergodic properties of the classical cesaro operator on banach sequence spaces is also given

regular submanifolds graphs and area formula in heisenberg groups we describe intrinsically regular submanifolds in heisenberg groups h low dimensional and low codimensional submanifolds turn out to be of a very different nature the first ones are legendrian surfaces while low codimensional ones are more general objects possibly non euclidean rectifiable nevertheless we prove that they are graphs in a natural group way as well as that an area formula holds for the intrinsic hausdorff measure finally they can be seen as federer fleming currents given a natural complex of differential forms on h

sub riemannian geometry of the coefficients of univalent functions abstract we consider coefficient bodies m n for univalent functions based on the lowner kufarev parametric representation we get a partially integrable hamiltonian system in which the first integrals are kirillov s operators for a representation of the virasoro algebra then m n are defined as sub riemannian manifolds given a lie poisson bracket they form a grading of subspaces with the first subspace as a bracket generating distribution of complex dimension two with this sub riemannian structure we construct a new hamiltonian system to calculate regular geodesics which turn to be horizontal lagrangian formulation is also given in the particular case m 3

a strong geometric hyperbolicity property for directed graphs and monoids abstract we introduce and study a strong thin triangle condition for directed graphs which generalises the usual notion of hyperbolicity for a metric space we prove that finitely generated left cancellative monoids whose right cayley graphs satisfy this condition must be finitely presented with polynomial dehn functions and hence word problems in np under the additional assumption of right cancellativity or in some cases the weaker condition of bounded indegree they also admit algorithms for more fundamentally semigroup theoretic decision problems such as green s relations l r j d and the corresponding pre orders in contrast we exhibit a right cancellative but not left cancellative finitely generated monoid in fact an infinite class of them whose cayley graph is essentially a tree hence hyperbolic in our sense and probably any reasonable sense but which is not even recursively presentable this seems to be strong evidence that no geometric notion of hyperbolicity will be strong enough to yield much information about finitely generated monoids in absolute generality

on weighted inductive limits of spaces of continuous functions the general problem of a projective description of weighted inductive limits of spaces of continuous functions was raised by bierstedt eise and summers in 3 similar questions were studied in 4 in the setting of sequence spaces on a general index set i the study was recently continued by bierstedt and meise in 5 the aim of the present article is to treat some questions proposed to the author by k d bierstedt on this subject our notation for locally convex spaces 1 c s is standard and we refer the reader to 13 and 14 if e is a 1 c s cs e denotes the family of all the continuous seminorms on e following floret 10 a 1 c s is said to satisfy the countable neighbourhood property c n p if given p ces e there are a 0 and p cs e such that p x 0 such that u 0 there is v v such that avi 0 i e if given any x x there is v v with v x 0 then v will be refered to as a nachbin family on x in what follows if e is a 1 c s c x e indicates the collection of all continuous e valued functions on x and cc x e the subspace of c x e of all the functions with compact support corresponding to a nachbin family v and to a 1 c s e we associate the following nachbin spaces

sub riemannian interpolation inequalities we prove that ideal sub riemannian manifolds i e admitting no non trivial abnormal minimizers support interpolation inequalities for optimal transport a key role is played by sub riemannian jacobi fields and distortion coefficients whose properties are remarkably different with respect to the riemannian case as a byproduct we characterize the cut locus as the set of points where the squared sub riemannian distance fails to be semiconvex answering to a question raised by figalli and rifford geom funct anal 20 1 124 159 2010 as an application we deduce sharp and intrinsic borell brascamp lieb and geodesic brunn minkowski inequalities in the aforementioned setting for the case of the heisenberg group we recover in an intrinsic way the results recently obtained by balogh et al calc var part differ equ 57 2 61 2018 and we extend them to the class of generalized h type carnot groups our results do not require the distribution to have constant rank yielding for the particular case of the grushin plane a sharp measure contraction property and a sharp brunn minkowski inequality

hausdorff volume in non equiregular sub riemannian manifolds in this paper we study the hausdorff volume in a non equiregular sub riemannian manifold and we compare it with a smooth volume we first give the lebesgue decomposition of the hausdorff volume then we study the regular part show that it is not commensurable with the smooth volume and give conditions under which it is a radon measure we finally give a complete characterization of the singular part we illustrate our results and techniques on numerous examples and cases e g to generic sub riemannian structures

fractional isoperimetric inequalities and subgroup distortion isoperimetric inequalities measure the complexity of the word problem in finitely presented groups by giving a bound on the number of relators that one must apply in order to show that a word w in the given generators represents the identity such bounds are given in terms of the length of w and the function describing the optimal bound is known as the dehn function of the group modulo a standard equivalence relation the dehn function is an invariant of the group not just the given finite presentation about seven years ago as the result of efforts by a number of authors 9 11 14 it was established that for every positive integer d one can construct finitely presented groups whose dehn function is polynomial of degree d the question of whether or not there exist groups whose dehn functions are of fractional degree has attracted a good deal of interest e g 18 25 according to a theorem of m gromov one does not get fractional exponents less than 2 because if a group satisfies a sub quadratic isoperimetric inequality then it actually satisfies a linear isoperimetric inequality see 13 10 19 22 the main purpose of this article is to prove the following

on some integral inequalities similar to hardy s inequality abstract the aim of the present paper is to establish some new integral inequalities which claim their origin in the well known hardy s integral inequality the method by which we obtain our results is quite elementary and based on the idea used by levinson to obtain the generalizations of hardy s integral inequality

some hardy type inequalities in this paper some new hardy type inequalities involving many functions are obtained these on the one hand generalize and on the other hand improve some existing results by isumi and isumi levinson and pachpatte on this famous type of inequalities

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

the campbell baker hausdorff dynkin formula and solutions of differential equations the campbell baker hausdorff dynkin formula is a special case of a simpler and more general formula for the solution of nonautonomous systems of first order ordinary differential equations in terms of autonomous systems specifically suppose u t takes values in a c manifold and satisfies the initial value problem u t a t u t u 0 a where a t is a vector field on the manifold depending continuously on t then u t exp z t a here exp z a means the solution at s 1 to v s z v s v 0 a for z t r 1 pr e r2r 1e tr t x a s 1 a s 2 a s r ds as t 0 where tr t s rr 0 s1 s2 sr t pr is the set of permutations on 1 r e is the number of errors in ordering consecutive terms in 1 r and is the usual commutator of vector fields under appropriate analyticity assumptions the series for z t is convergent for small t this formula gives an explicit formulation of results of k t chen published in 1957 applications are given to problems in sub riemannian geometry and to improving convergence estimates for the campbell baker hausdorff dynkin formula in the context of banach algebras our formula can be thought of as a noncommutative generalization of the familiar formula u t a exp 0t a s ds in the scalar linear case in the same way that the campbell baker hausdorff dynkin formula is a noncommutative generalization of the familiar formula exey ex y

filling invariants of stratified nilpotent lie groups filling invariants are measurements of a metric space describing the behaviour of isoperimetric inequalities in this article we examine filling functions and higher divergence functions we prove for a class of stratified nilpotent lie groups that in the low dimensions the filling functions grow as fast as the ones of the euclidean space and in the high dimensions slower than the filling functions of the euclidean space we do this by developing a purely algebraic condition on the lie algebra of a stratified nilpotent lie group further we find a sufficient criterion for such groups to have a filling function in a special dimension with faster growth as the appropriate filling function of the euclidean space further we bound the higher divergence functions of stratified nilpotent lie groups

examples of non archimedean frechet spaces without nuclear kothe quotients let k be a spherically complete non archimedean valued field we prove that the dual space l of the banach space c0 has a total strongly non norming subspace m using this subspace m we construct a non normable frechet space f of countable type with a continuous norm such that its strong dual fb is a strict lb space next we show that f has no nuclear kothe quotient

sub riemannian vs euclidean dimension comparison and fractal geometry on carnot groups we solve gromov s dimension comparison problem for hausdorff and box counting dimension on carnot groups equipped with a carnot caratheodory metric and an adapted euclidean metric the proofs use sharp covering theorems relating optimal mutual coverings of euclidean and carnot caratheodory balls and elements of sub riemannian fractal geometry associated to horizontal self similar iterated function systems on carnot groups inspired by falconer s work on almost sure dimensions of euclidean self affine fractals we show that carnot caratheodory self similar fractals are almost surely horizontal as a consequence we obtain explicit dimension formulae for invariant sets of euclidean iterated function systems of polynomial type jet space carnot groups provide a rich source of examples

limit sets of discrete groups of isometries of exotic hyperbolic spaces let be a geometrically finite discrete group of isometries of hyperbolic space hf where f r c h or o in which case n 2 we prove that the critical exponent of equals the hausdorff dimension of the limit sets and that the smallest eigenvalue of the laplacian acting on square integrable functions is a quadratic function of either of them when they are sufficiently large a generalization of hopf ergodicity theorem for the geodesic flow with respect to the bowen margulis measure is also proven

the isoperimetric problem on surfaces of revolution of decreasing gauss curvature we prove that the least perimeter way to enclose prescribed area in the plane with smooth rotationally symmetric complete metric of nonincreasing gauss curvature consists of one or two circles bounding a disc the complement of a disc or an annulus we also provide a new isoperimetric inequality in general surfaces with boundary

computable algebra and group embeddings this work grew out of our investigations of the infinitely generated subgroups of finitely presented groups a useful technique for embedding countable groups with sufficiently nice local properties into finitely presented groups was introduced in l this technique was then applied in 1 21 to prove that every countable locally polycyclic group and every countable metabelian group can be embedded in a finitely presented group it became apparent from 2 that effective methods in commutative algebra could be extended to group rings of polycyclic groups and then used to embed certain groups in finitely presented groups in this paper these extensions are carried out with the consequent application that every countable locally abelian by nilpotent group can be embedded in a finitely presented group much of the effective commutative algebra we require seems to be known see for instance 6 9 14 171 unfortunately a systematic account is not available in particular we need a proof of the hilbert basis theorem which

the dehn function of richard thompson s group f is quadratic we prove that the dehn function that is the smallest isoperimetric function of the r thompson s group f is quadratic

gromov hyperbolic spaces and the sharp isoperimetric constant in this article we exhibit the largest constant in a quadratic isoperimetric inequality which ensures that a geodesic metric space is gromov hyperbolic as a particular consequence we obtain that euclidean space is a borderline case for gromov hyperbolicity in terms of the isoperimetric function we prove similar results for the linear filling radius inequality our results strengthen and generalize theorems of gromov papasoglu and others

the conjugacy and isomorphism problems for combable groups there exist combable groups in which the conjugacy problem is unsolvable the isomorphism problem is unsolvable for certain recursive sequences of finite presentations of combable groups

generalizations of hardy s integral inequality extensions of the integral version of hardy s inequality were given by kadlec and kufner 1967 and by copson 1976 this paper provides several levels of further generalization of their results obtained mostly by specializing four main inequalities most of the inequalities have the form k f c f where k is an integral transform and is a generalized l p norm some have the inequality sign reversed best possible constants c are obtained in several cases under mild extra hypotheses

large deviation on a covering graph with group of polynomial growth we discuss a large deviation principle of a periodic random walk on a covering graph with its transformation group of polynomial volume growth in view of geometry as we shall observe the behavior of a random walk at infinity is closely related to the gromov hausdorff limit of an infinite graph and in the case where the graph admits an action of a group of polynomial volume growth the carnot caratheodory metric shows up in its limit space

solvable groups with polynomial dehn functions given a finitely presented group h finitely generated subgroup b of h and a monomorphism b h we obtain an upper bound of the dehn function of the corresponding hnn extension g hh t j t i1 bt b i in terms of the dehn function of h and the distortion of b in g using such a bound we construct first examples of non polycyclic solvable groups with polynomial dehn functions the constructed groups are metabelian and contain the solvable baumslag solitar groups in particular this answers a question posed by birget ol shanskii rips and sapir

isoperimetric functions of groups and computational complexity of the word problem we prove that the word problem of a finitely generated group g is in np solvable in polynomial time by a nondeterministic turing machine if and only if this group is a subgroup of a finitely presented group h with polynomial isoperimetric function the embedding can be chosen in such a way that g has bounded distortion in h this completes the work started in 6 and 25

on a new class of hardy type inequalities in this paper we establish a new class of integral inequalities which originate from the well known hardy s inequality the analysis used in the proofs is quite elementary and is based on the idea used by levinson to obtain generalisations of hardy s inequality

smoothness of subriemannian isometries we show that the group of isometries i e distance preserving homeomorphisms of an equiregular subriemannian manifold is a finite dimensional lie group of smooth transformations the proof is based on a new pde argument in the spirit of harmonic coordinates establishing that in an arbitrary subriemannian manifold there exists an open dense subset where all isometries are smooth

definitions of quasiconformality summary we establish that the infinitesimal h definition for quasiconforreal mappings on carnot groups implies global quasisymmetry and hence the absolute continuity on almost all lines our method is new even in r where we obtain that the limsup condition in the h definition can be replaced by a liminf condition this leads to a new removability result for quasi conformal mappings in euclidean spaces an application to parametrizations of chord arc surfaces is also given

collapsing of riemannian manifolds and eigenvalues of laplace operator on etudie la perturbation des valeurs propres de l operateur de laplace sous des deformations des varietes qui ne conservent pas les dimensions

pauls rectifiable and purely pauls unrectifiable smooth hypersurfaces abstract this paper is related to the problem of finding a good notion of rectifiability in sub riemannian geometry in particular we study which kind of results can be expected for smooth hypersurfaces in carnot groups our main contribution will be a consequence of the following result there exists a c hypersurface s without characteristic points that has uncountably many pairwise non isomorphic tangent groups on every positive measure subset the example is found in a carnot group of topological dimension 8 it has hausdorff dimension 12 and so we use on it the hausdorff measure h 12 as a consequence we show that any lipschitz map defined on a subset of a carnot group of hausdorff dimension 12 with values in s has negligible image with respect to the hausdorff measure h 12 in particular we deduce that s cannot be lipschitz parametrizable by countably many maps each defined on some subset of some carnot group of hausdorff dimension 12 as main consequence we have that a notion of rectifiability proposed by s pauls is not equivalent to one proposed by b franchi r serapioni and f serra cassano at least for arbitrary carnot groups in addition we show that given a subset u of a homogeneous subgroup of hausdorff dimension 12 of a carnot group every bi lipschitz map f u s satisfies h 12 f u 0 finally we prove that such an example does not exist in heisenberg groups we prove that all c hypersurfaces in h n with n 2 are countably h n 1 r rectifiable according to pauls definition even with bi lipschitz maps

the derivational complexity of string rewriting systems we study the derivational complexities of string rewriting systems we discuss the following fundamental question which functions can be derivational complexities of terminating finite string rewriting systems they are recursive but for any recursive function there is a derivational complexity larger than it we relate them to the time functions of turing machines in particular we show that the functions n a a 2 and a n a 1 for a real number a are equivalent to the derivational complexity of some finite string rewriting systems if the computational complexity of a is relatively low for example a is rational algebraic p or e on the other hand they cannot be equivalent to any derivational complexities if the complexity of a is high

c0 semigroups and mean ergodic operators in a class of frechet spaces abstract it is shown that the generator of every exponentially equicontinuous uniformly continuous c 0 semigroup of operators in the class of quojection frechet spaces x which includes properly all countable products of banach spaces is necessarily everywhere defined and continuous if in addition x is a grothendieck space with the dunford pettis property then uniform continuity can be relaxed to strong continuity two results one of m lin and one of h p lotz both concerned with uniformly mean ergodic operators in banach spaces are also extended to the class of frechet spaces mentioned above they fail to hold for arbitrary frechet spaces

algorithms and geometry for graph products of groups recent work of gromov epstein cannon thurston and many others has generated strong interest in the geometric and algorithmic structure of finitely generated infinite groups see 16 17 and 14 many of these structures are preserved by taking graph products the graph product of groups not to be confused with the fundamental group of a graph of groups is a product mixing direct and free products whether the product between two groups in the graph product is free or direct is determined by a simplicial graph given a simplicial graph we say that two vertices are adjacent if they are joined by a single edge

isoperimetric and isodiametric functions of groups this is the first of two papers devoted to connections between asymptotic functions of groups and computational complexity in particular we show how to construct a finitely presented group with np complete word problem one of the main results of this paper states that if a real number cx 4 is computable in time 0 then na is equivalent big o to the dehn function of a finitely presented group the smallest isodiametric function of this group is n3 8 4 on the other hand if na is equivalent to the dehn function of a finitely presented group then oe is computable in time 4 are equivalent to dehn functions of some finitely presented groups and that n1r and na for all rational numbers oe 3 are equivalent to the smallest isodiametric functions of finitely presented groups moreover we describe all dehn functions of finitely presented groups f n4 as time functions of turing machines modulo two conjectures

commutators of hardy operators in this paper we establish hardy s integral inequalities for commutators generating hardy operators with one sided dyadic cmo functions

local reflexivity of normed spaces operators and frechet spaces a new and stronger proof of the principle of local reflexivity is given applications to normed spaces local reflexivity of operators and frechet spaces are given a partial answer to a question of pelczynski is obtained the proof is based on an unpublished proof of r c james

on 2 step corank 2 nilpotent sub riemannian metrics in this paper we study the nilpotent 2 step corank 2 sub riemannian metrics that are nilpotent approximations of general sub riemannian metrics we exhibit optimal syntheses for these problems it turns out that in general the cut time is not equal to the first conjugate time but has a simple explicit expression as a by product of this study we get some smoothness properties of the spherical hausdorff measure in the case of a generic 6 dimensional 2 step corank 2 sub riemannian metric

on some generalizations of hardy s integral inequality in this paper we establish some new generalizations of the hardy s inequality by using a fairly elementary analysis the inequalities given here contain in the special cases some of the recent generalizations of hardy s integral inequality appearing in the literature

intrinsic random walks and sub laplacians in sub riemannian geometry on a sub riemannian manifold we define two type of laplacians the macroscopic laplacian as the divergence of the horizontal gradient once a volume is fixed and the microscopic laplacian as the operator associated with a geodesic random walk we consider a general class of random walks where all sub riemannian geodesics are taken in account this operator depends only on the choice of a complement c to the sub riemannian distribution and is denoted l c we address the problem of equivalence of the two operators this problem is interesting since on equiregular sub riemannian manifolds there is always an intrinsic volume e g popp s one p but not a canonical choice of complement the result depends heavily on the type of structure under investigation on contact structures for every volume there exists a unique complement c such that l c on carnot groups if h is the haar volume then there always exists a complement c such that h l c however this complement is not unique in general for quasi contact structures in general p l c for any choice of c in particular l c is not symmetric w r t popp s measure this is surprising especially in dimension 4 where in a suitable sense p is the unique intrinsic macroscopic laplacian a crucial notion that we introduce here is the n intrinsic volume i e a volume that depends only on the set of parameters of the nilpotent approximation when the nilpotent approximation does not depend on the point a n intrinsic volume is unique up to a scaling by a constant and the corresponding n intrinsic sub laplacian is unique this is what happens for dimension smaller or equal than 4 and in particular in the 4 dimensional quasi contact structure mentioned above

symmetric space bounded computation abstract a symmetric turing machine is one such that the yields relation between configurations is symmetric the space complexity classes for such machines are found to be intermediate between the corresponding deterministic and nondeterministic space complexity classes certain natural problems are shown to be complete for symmetric space complexity classes and the relationship of symmetry to determinism and nondeterminism is investigated

the word problem of the brin thompson group is conp complete abstract we prove that the word problem of the brin thompson group nv over a finite generating set is conp complete for every n 2 it is known that n v n 1 is an infinite family of infinite finitely presented simple groups we also prove that the word problem of the thompson group v over a certain infinite set of generators related to boolean circuits is conp complete

on quadratic dehn functions we confirm with new examples that solvable groups of high rank are expected to satisfy a polynomial isoperimetric inequality gro93 5a9 to that end we study invariant quasi geodesic foliations in simply connected solvable lie groups endowed with left invariant riemannian metrics whose leaves are isometric to closed subgroups we establish a decomposition theorem which implies upper bounds on the dehn or filling function of loops by disks of the solvable group in terms of the dehn functions of the leaves we obtain examples of metabelian polycyclic groups with exponential growth and quadratic dehn functions we also deduce that the horospheres in sl 4 so 4 which bound an invariant core for sl 4 and that the horospheres which bound an invariant core for hilbert modular groups in open image in new window have quadratic filling functions the main theorem also applies to some solvable lie groups which are not quasi isometric to horospheres in symmetric spaces

the weighted isoperimetric type and sobolev type inequalities for hypersurfaces in carnot groups abstract this paper mainly studies hypersurfaces immersed in a sub riemannian step k carnot group g with a density e f where f is a smooth function on g we give the first variation formula about the weighted horizontal area of hypersurfaces furthermore on the base of some work we derive the weighted isoperimetric type and sobolev type inequalities for hypersurfaces

on the geometry of normal forms in discrete groups we study normal forms in finitely generated groups from the geometric viewpoint of combings we introduce notions of combability considerably weaker than those commonly in use we prove that groups which satisfy these conditions are finitely presented and satisfy isoperimetric and isodiametric inequalities of a controlled nature

sharp a priori estimates for div curl systems in heisenberg groups abstract in this paper we prove a family of inequalities for differential forms in heisenberg groups h 1 and h 2 that are the natural counterpart of a class of div curl inequalities in de rham s complex proved by lanzani stein and bourgain brezis

the subelliptic heat kernel on su 2 representations asymptotics and gradient bounds the lie group su 2 endowed with its canonical subriemannian structure appears as a three dimensional model of a positively curved subelliptic space the goal of this work is to study the subelliptic heat kernel on it and some related functional inequalities

cheeger constants of surfaces and isoperimetric inequalities we show that if the isoperimetric profile of a bounded genus non compact surface grows faster than vt then it grows at least as fast as a linear function this generalizes a result of gromov for simply connected surfaces we study the isoperimetric problem in dimension 3 we show that if the filling volume function in dimension 2 is euclidean while in dimension 3 it is sub euclidean and there is a g such that minimizers in dimension 3 have genus at most g then the filling function in dimension 3 is almost linear

asymptotic cyclic expansion and bridge groups of formal proofs abstract formal proofs even simple ones may hide an unexpected intricate combinatorics we define a new combinatorial invariant the bridge group of a proof which encodes the cyclic structure of proofs in the sequent calculus we compute the bridge groups of two infinite families of proofs and identify them with the baumslag solitar and gersten groups we observe that the distortion of cyclic subgroups in these groups equals the asymptotic growth of the procedure of elimination of lemmas from the proofs

groups finitely presented in burnside varieties abstract for all sufficiently large odd integers n the following version of higman s embedding theorem is proved in the variety b n of all groups satisfying the identity x n 1 a finitely generated group g from b n has a presentation g a r with a finite set of generators a and a recursively enumerable set r of defining relations if and only if it is a subgroup of a group h finitely presented in the variety b n it follows that there is a universal 2 generated finitely presented in b n group containing isomorphic copies of all finitely presented in b n groups as subgroups

on some new inequalities related to hardy s integral inequality some hardy type inequalities related to those derived by pachpatte and love are given mathematics subject classification 2000 26d15 26d20

on some variants of hardy s inequality abstract the present paper deals with some new generalizations and extensions of a certain variant of hardy s inequality given by izumi and izumi the method employed in our analysis is quite elementary and the results established in this paper provide new estimates on this type of integral inequalities