su n tensor product multiplicities and virtual berenstein zelevinsky triangles information on su n tensor product multiplicities is neatly encoded in berenstein zelevinsky triangles here we study a generalization of these triangles by allowing negative as well as non negative integer entries for a fixed triple product of weights these generalized berenstein zelevinsky triangles span a lattice in which one may move by adding integer linear combinations of so called virtual triangles inequalities satisfied by the coefficients of the virtual triangles describe a polytope the tensor product multiplicities may be computed as the number of integer points in this convex polytope as our main result we present an explicit formula for this discretised volume as a multiple sum as an application we also address the problem of determining when a tensor product multiplicity is non vanishing the solution is represented by a set of inequalities in the dynkin labels we also allude to the question of when a tensor product multiplicity is greater than a given non negative integer

gradient estimate for solutions to poisson equations in metric measure spaces abstract let x d be a complete pathwise connected metric measure space with a locally ahlfors q regular measure where q 1 suppose that x d supports a local 1 2 poincare inequality and a suitable curvature lower bound for the poisson equation u f on x d moser trudinger and sobolev inequalities are established for the gradient of u the local holder continuity with optimal exponent of solutions is obtained

the geometry of the universal teichmuller space and the euler weil petersson equation on the identity component of the universal teichmuller space endowed with the takhtajan teo topology the geodesics of the weil petersson metric are shown to exist for all time this component is naturally a subgroup of the quasisymmetric homeomorphisms of the circle viewed this way the regularity of its elements is shown to be h3 2 epsilon for all epsilon 0 the evolutionary pde associated to the spatial representation of the geodesics of the weil petersson metric is derived using multiplication and composition below the critical sobolev index 3 2 geodesic completeness is used to introduce special classes of solutions of this pde analogous to peakons our setting is used to prove that there exists a unique geodesic between each two shapes in the plane in the context of the application of the weil petersson metric in imaging c 2015 elsevier inc all rights reserved

local and global well posedness of the fractional order epdiff equation on r d abstract of concern is the study of fractional order sobolev type metrics on the group of h diffeomorphism of r d and on its sobolev completions d q r d it is shown that the h s sobolev metric induces a strong and smooth riemannian metric on the banach manifolds d s r d for s 1 d 2 as a consequence a global well posedness result of the corresponding geodesic equations both on the banach manifold d s r d and on the smooth regular frechet lie group of all h diffeomorphisms is obtained in addition a local existence result for the geodesic equation for metrics of order 1 2 s 1 d 2 is derived

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

abstract hardy sobolev spaces and interpolation the purpose of this work is to describe an abstract theory of hardy sobolev spaces on doubling riemannian manifolds via an atomic decomposition we study the real interpolation of these spaces with sobolev spaces and finally give applications to riesz inequalities

hajlasz gradients are upper gradients abstract let x d be a metric measure space with a borel regular measure in this article we prove that if u l loc 1 x and g is a hajlasz gradient of u then there exists u such that u u almost everywhere and 4g is a p weak upper gradient of u this result avoids a priori assumption on the quasi continuity of u used in shanmugalingam 2000 19 we also introduce the notion of local hajlasz gradients and investigate the relations between the local hajlasz gradient and the upper gradient

the camassa holm equation as a geodesic flow on the diffeomorphism group misiolek j geom phys 24 203 208 1998 has shown that the camassa holm equation is a geodesic flow on the bott virasoro group in this paper it is shown that the camassa holm equation for the case 0 is the geodesic spray of the weak riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating the h1 inner product over the entire group this paper uses the right trivialization technique to rigorously verify that the euler poincare theory for lie groups can be applied to diffeomorphism groups the observation made in this paper has led to physically meaningful generalizations of the ch equation to higher dimensional manifolds

gradient estimates for heat kernels and harmonic functions abstract let x d be a doubling metric measure space endowed with a dirichlet form e deriving from a carre du champ assume that x d e supports a scale invariant l 2 poincare inequality in this article we study the following properties of harmonic functions heat kernels and riesz transforms for p 2 i g p l p estimate for the gradient of the associated heat semigroup ii r h p l p reverse holder inequality for the gradients of harmonic functions iii r p l p boundedness of the riesz transform p iv g b e a generalised bakry emery condition we show that for p 2 i ii iii are equivalent while for p i ii iv are equivalent moreover some of these equivalences still hold under weaker conditions than the l 2 poincare inequality our result gives a characterisation of li yau s gradient estimate of heat kernels for p while for p 2 it is a substantial improvement as well as a generalisation of earlier results by auscher coulhon duong hofmann 7 and auscher coulhon 6 applications to isoperimetric inequalities and sobolev inequalities are given our results apply to riemannian and sub riemannian manifolds as well as to non smooth spaces and to degenerate elliptic parabolic equations in these settings

a t 1 theorem for non integral operators let x be a space of homogeneous type and let l be a sectorial operator with bounded holomorphic functional calculus on l 2 x we assume that the semigroup e tl t 0 satisfies davies gaffney estimates associated with l are certain approximations of the identity we call an operator t a non integral operator if compositions involving t and these approximations satisfy certain weighted norm estimates the davies gaffney and the weighted norm estimates are together a substitute for the usual kernel estimates on t in calderon zygmund theory in this paper we show under the additional assumption that a vertical littlewood paley stein square function associated with l is bounded on l 2 x that a non integral operator t is bounded on l 2 x if and only if t 1 in bmo l x and t 1 in bmo l x here bmo l x and bmo l x denote the recently defined bmo x spaces associated with l that generalize the space bmo x of john and nirenberg generalizing a recent result due to f bernicot we show a second version of a t 1 theorem under weaker off diagonal estimates which gives a positive answer to a question raised by him as an application we prove l 2 x boundedness of a paraproduct operator associated with l we moreover study criterions for a t b theorem to be valid

bounded compositions on scaling invariant besov spaces abstract for 0 s 1 q we characterize the homeomorphisms r n r n for which the composition operator f f is bounded on the homogeneous scaling invariant besov space b n s q s r n where the emphasis is on the case q n s left open in the previous literature we also establish an analogous result for besov type function spaces on a wide class of metric measure spaces as well and make some new remarks considering the scaling invariant triebel lizorkin spaces f n s q s r n with 0 s 1 and 0 q

a characterization of hajlasz sobolev and triebel lizorkin spaces via grand littlewood paley functions abstract in this paper we establish the equivalence between the hajlasz sobolev spaces or classical triebel lizorkin spaces and a class of grand triebel lizorkin spaces on euclidean spaces and also on metric spaces that are both doubling and reverse doubling in particular when p n n 1 we give a new characterization of the hajlasz sobolev spaces m 1 p r n via a grand littlewood paley function

sharp estimates for pseudo differential operators with symbols of limited smoothness and commutators we consider here pseudo differential operators whose symbol x is not infinitely smooth with respect to x decomposing such symbols into four sometimes five components and using tools of paradifferential calculus we derive sharp estimates on the action of such pseudo differential operators on sobolev spaces and give explicit expressions for their operator norm in terms of the symbol x we also study commutator estimates involving such operators and generalize or improve the so called kato ponce and calderon coifman meyer estimates in various ways

pointwise characterizations of besov and triebel lizorkin spaces and quasiconformal mappings abstract in this paper the authors characterize in terms of pointwise inequalities the classical besov spaces b p q s and triebel lizorkin spaces f p q s for all s 0 1 and p q n n s both in r n and in the metric measure spaces enjoying the doubling and reverse doubling properties applying this characterization the authors prove that quasiconformal mappings preserve f n s q s on r n for all s 0 1 and q n n s a metric measure space version of the above morphism property is also established

integral energy characterization of hajlasz sobolev spaces abstract we provide an integral energy characterization of hajlasz sobolev spaces m 1 p x e with p 1 where x is a metric measure space that satisfies weak poincare inequality and e is a banach space with the radon nikodym property this result extends previously known results which were originally established on r n

the poincare inequality is an open ended condition let p 1 and let x d be a complete metric measure space with borel and doubling that admits a 1 p poincare inequality then there exists e 0 such that x d admits a 1 q poincare inequality for every q p e quantitatively

diffusion processes and heat kernels on metric spaces processes x p on any given locally compact metric space x d tx equipped with a radon measure m these processes are associated with local regular dirichlet forms which are obtained as limits of approxi mating nonlocal dirichlet forms this general method works without any restrictions on x d m and yields processes which are well defined for quasi every starting point the second main topic of this paper is to formulate and exploit the so called measure contraction property this is a condition on the original data x d m which can be regarded as a generalization of curvature bounds on the metric space x d it is a bound for distortions of the measure m under contractions of the state space x along suitable geodesics or quasi geodesics w r t the metric d in the case of rieman nian manifolds this condition is always satisfied several other examples will be discussed including uniformly elliptic operators operators with weights certain subelliptic operators manifolds with boundaries or cor ners and glueing together of manifolds the measure contraction property implies upper and lower gaussian estimates for the heat kernel and a harnack inequality for the associated harmonic functions therefore the above mentioned diffusion processes are strong feller processes and are well defined for every starting point

riesz transforms through reverse holder and poincare inequalities we study the boundedness of riesz transforms in l p for p 2 on a doubling metric measure space endowed with a gradient operator and an injective omega accretive operator l satisfying davies gaffney estimates if l is non negative self adjoint we show that under a reverse holder inequality the riesz transform is always bounded on l p for p in some interval 2 2 epsilon and that l p gradient estimates for the semigroup imply boundedness of the riesz transform in l q for q in 2 p this improves results of auscher et al ann sci ecole norm sup 37 4 911 957 2004 and auscher and coulhon ann scuola norm sup pisa 4 531 555 2005 where the stronger assumption of a poincare inequality and the assumption e tl 1 1 were made the poincare inequality assumption is also weakened in the setting of a sectorial operator l in the last section we study elliptic perturbations of riesz transforms

p poincare inequality versus poincare inequality some counterexamples we point out some of the differences between the consequences of p poincare inequality and that of poincare inequality in the setting of doubling metric measure spaces based on the geometric characterization of poincare inequality given in durand cartagena et al mich math j 60 2011 we obtain a geometric property implied by the support of a p poincare inequality and demonstrate by examples that an analogous geometric characterization for finite p is not possible the examples we give are metric measure spaces which are doubling and support an poincare inequality but support no finite p poincare inequality in particular these examples show that one cannot expect a self improving property for poincare inequality in the spirit of keith zhong ann math 167 2 575 599 2008 we also show that the persistence of poincare inequality under measured gromov hausdorff limits fails for poincare inequality

existence of parabolic minimizers on metric measure spaces abstract the objects of our studies are vector valued parabolic minimizers u associated to a convex caratheodory integrand f obeying a p growth assumption from below and a certain monotonicity condition in the gradient variable here the functions being considered are defined on a metric measure space x d for such parabolic minimizers that coincide with a time independent cauchy dirichlet datum u 0 on the parabolic boundary of a space time cylinder 0 t with an open subset x and t 0 we prove existence in the parabolic newtonian space l p 0 t n 1 p r n in this paper we generalize results from bogelein et al 2014 2015 to the metric setting and argue completely on a variational level

geometry and curvature of diffeomorphism groups withh1metric and mean hydrodynamics holm marsden and ratiu adv in math 137 1998 1 81 derived a new model for the mean motion of an ideal fluid in euclidean space given by the equationv t u t v t 2 u t t u t grad p t where div u 0 andv 1 2 u in this model the momentumvis transported by the velocityu with the effect that nonlinear interaction between modes corresponding to length scales smaller than is negligible we generalize this equation to the setting of ann dimensional compact riemannian manifold the resulting equation is the euler poincare equation associated with the geodesic flow of theh1right invariant metric on ds the group of volume preserving hilbert diffeomorphisms of classhs we prove that the geodesic spray is continuously differentiable fromtds m intottds m so that a standard picard iteration argument proves existence and uniqueness on a finite time interval our goal in this paper is to establish the foundations for lagrangian stability analysis following arnold ann inst grenoble16 1966 319 361 to do so we use submanifold geometry and prove that the weak curvature tensor of the right invarianth1metric on ds is a bounded trilinear map in thehstopology from which it follows that solutions to jacobi s equation exist using such solutions we are able to study the infinitesimal stability behavior of geodesics

a compactness result for fano manifolds and kahler ricci flows we obtain a compactness result for fano manifolds and kahler ricci flows comparing to the more general riemannian versions in anderson invent math 102 2 429 445 1990 and hamilton am j math 117 545 572 1995 in this fano case the curvature assumption is much weaker and is preserved by the kahler ricci flows one assumption is the c 1 boundedness of the ricci potential and the other is the smallness of perelman s entropy as one application we obtain a new local regularity criteria and structure result for kahler ricci flows the proof is based on a holder estimate for the gradient of harmonic functions and mixed derivative of green s function

higher genus su n fusion multiplicities as polytope volumes we show how higher genus su n fusion multiplicities may be computed as the discretized volumes of certain polytopes the method is illustrated by explicit analyses of some su 3 and su 4 fusions but applies to all higher point and higher genus su n fusions it is based on an extension of the realm of berenstein zelevinsky triangles by including so called gluing and loop gluing diagrams the identification of the loop gluing diagrams is our main new result since they enable us to characterize higher genus fusions in terms of polytopes also the genus 2 0 point su 3 fusion multiplicity is found to be a simple binomial coefficient in the affine level

existence and almost uniqueness for p harmonic green functions on bounded domains in metric spaces abstract we study p harmonic singular functions defined by means of upper gradients in bounded domains in metric measure spaces it is shown that singular functions exist if and only if the complement of the domain has positive capacity and that they satisfy very precise capacitary identities for superlevel sets suitably normalized singular functions are called green functions uniqueness of green functions is largely an open problem beyond unweighted r n but we show that all green functions in a given domain and with the same singularity are comparable as a consequence for p harmonic functions with a given pole we obtain a similar comparison result near the pole various characterizations of singular functions are also given our results hold in complete metric spaces with a doubling measure supporting a p poincare inequality or under similar local assumptions

density of lipschitz mappings in the class of sobolev mappings between metric spaces we prove that lipschitz mappings are dense in the newtonian sobolev classes n1 p x y of mappings from spaces x supporting p poincare inequalities into a finite lipschitz polyhedron y if and only if y is p connected 1 y 2 y p y 0 where p is the largest integer less than or equal to p

gagliardo nirenberg inequalities on manifolds abstract we prove gagliardo nirenberg inequalities on some classes of manifolds lie groups and graphs

affine su 3 and su 4 fusion multiplicities as polytope volumes affine su 3 and su 4 fusion multiplicities are characterized as discretized volumes of certain convex polytopes the volumes are measured explicitly resulting in multiple sum formulae these are the first polytope volume formulae for higher rank fusion multiplicities the associated threshold levels are also discussed for any simple lie algebra we derive an upper bound on the threshold levels using a refined version of the gepner witten depth rule

polynomial control on stability inversion and powers of matrices on simple graphs abstract spatially distributed networks of large size arise in a variety of science and engineering problems such as wireless sensor networks and smart power grids most of their features can be described by properties of their state space matrices whose entries have indices in the vertex set of a graph in this paper we introduce novel algebras of beurling type that contain matrices on a connected simple graph having polynomial off diagonal decay and we show that they are banach subalgebras of b l p 1 p the space of all bounded operators on the space l p of all p summable sequences the l p stability of state space matrices is an essential hypothesis for the robustness of spatially distributed networks in this paper we establish the equivalence among l p stabilities of matrices in beurling algebras for different exponents 1 p with quantitative analysis for the lower stability bounds admission of norm control inversion plays a crucial role in some engineering practice in this paper we prove that matrices in beurling subalgebras of b l 2 have norm controlled inversion and we find a norm controlled polynomial with close to optimal degree polynomial estimate to powers of matrices is important for numerical implementation of spatially distributed networks in this paper we apply our results on norm controlled inversion to obtain a polynomial estimate to powers of matrices in beurling algebras the polynomial estimate is a noncommutative extension about convolution powers of a complex function and is applicable to estimate the probability of hopping from one agent to another agent in a stationary markov chain on a spatially distributed network

isoperimetric inequality from the poisson equation via curvature in this paper we establish an isoperimetric inequality in a metric measure space via the poisson equation let x d be a complete pathwise connected metric space with locally ahlfors q regular measure where q 1 that supports a local l2 poincare inequality we show that for the poisson equation u g if the local l norm of the gradient du can be bounded by the lorentz norm lq 1 of g then we obtain an isoperimetric inequality and a sobolev inequality in x d with optimal exponents by assuming a suitable curvature lower bound we establish such optimal bounds on 2011 wiley periodicals inc

local holder continuity for some doubly nonlinear parabolic equations in measure spaces abstract we establish the local holder continuity for the nonnegative weak solutions of certain doubly nonlinear parabolic equations possessing a singularity in the time derivative part and a degeneracy in the principal part the proof involves the method of intrinsic scaling and the setting is a measure space equipped with a doubling non trivial borel measure supporting a poincare inequality

sobolev spaces laplacian and heat kernel on alexandrov spaces we prove the compactness of the imbedding of the sobolev space w 1 2 0 omega into l 2 omega for any relatively compact open subset omega of an alexandrov space as a corollary the generator induced from the dirichlet energy form has discrete spectrum the generator can be approximated by the laplacian induced from the dc structure on the alexandrov space we also prove the existence of the locally holder continuous heat kernel

weighted sobolev spaces on metric measure spaces abstract we investigate weighted sobolev spaces on metric measure spaces x d m denoting by the weight function we compare the space w x d m which always concides with the closure h x d m of lipschitz functions with the weighted sobolev spaces w 1 p x d m and h 1 p x d m defined as in the euclidean theory of weighted sobolev spaces under mild assumptions on the metric measure structure and on the weight we show that w x d m h x d m we also adapt the results proved by muckenhoupt and the ones proved by zhikov to the metric measure setting considering appropriate conditions on that ensure the equality w 1 p x d m h x d m

a geometric description of tensor product decompositions in su 3 the direct sum decomposition of tensor products for su 3 has many applications in physics and the problem has been studied extensively this has resulted in many decomposition methods each with its advantages and disadvantages the description given here is geometric in nature and it describes both the constituents of the direct sum and their multiplicities in addition to providing decompositions of specific tensor products this approach is very well suited to studying tensor products as the parameters vary and helping draw general conclusions after a description and proof of the method several consequences are discussed and proved in particular questions regarding multiplicities are considered

harnack s inequality for a nonlinear eigenvalue problem on metric spaces abstract we prove harnack s inequality for first eigenfunctions of the p laplacian in metric measure spaces the proof is based on the famous moser iteration method which has the advantage that it only requires a weak 1 p poincare inequality as a by product we obtain the continuity and the fact that first eigenfunctions do not change signs in bounded domains

poincare inequalities and newtonian sobolev functions on noncomplete metric spaces abstract let x be a noncomplete metric measure space satisfying the usual local assumptions of a doubling property and a poincare inequality we study extensions of newtonian sobolev functions to the completion x of x and use them to obtain several results on x itself in particular concerning minimal weak upper gradients lebesgue points quasicontinuity regularity properties of the capacity and better poincare inequalities we also provide a discussion about possible applications of the completions and extension results to p harmonic functions on noncomplete spaces and show by examples that this is a rather delicate issue opening for various interpretations and new investigations

generating function method for fusion rules this is the second of two articles devoted to an exposition of the generating function method for computing fusion rules in affine lie algebras the present paper focuses on fusion rules using the machinery developed for tensor products in the companion article although the kac walton algorithm provides a method for constructing a fusion generating function from the corresponding tensor product generating function we describe a more powerful approach which starts by first defining the set of fusion elementary couplings from a natural extension of the set of tensor product elementary couplings a set of inequalities involving the level are derived from this set using farkas lemma these inequalities taken in conjunction with the inequalities defining the tensor products define what we call the fusion basis given this basis the machinery of our previous paper may be applied to construct the fusion generating function new generating functions for sp 4 and su 4 together with a closed form expres

holder estimates of p harmonic extension operators it is now a well known fact that for 1 p the p harmonic functions on domains in metric measure spaces equipped with a doubling measure supporting a 1 p poincare inequality are locally holder continuous in this note we provide a characterization of domains in such metric spaces for which p harmonic extensions of holder continuous boundary data are globally holder continuous we also provide a link between this regularity property of the domain and the uniform p fatness of the complement of the domain

on equivalent conditions for the validity of poincare inequality on weighted sobolev space with applications to the solvability of degenerated pdes involving p laplacian abstract let r n be a given subset p 1 and w 1 p f l loc 1 f f x 1 f x n l p be the weighted sobolev space subordinated to the weight function we derive a chain of equivalent conditions for the validity of poincare inequality p f x p x d x c f x p x d x the conditions include isoperimetric inequalities representation of functionals as well as solvability of degenerated elliptic problem s div x u p 2 u x where x belongs to the dual space to the completion of c 0 in w 1 p furthermore we adopt recent technique due to skrzypczak to construct one parameter families of weights such that the problem s is solvable with

self improvement of weighted pointwise inequalities on open sets abstract we prove a general self improvement property for a family of weighted pointwise inequalities on open sets including pointwise hardy inequalities with distance weights for this purpose we introduce and study the classes of p poincare and p hardy weights for an open set x where x is a metric measure space we also apply the self improvement of weighted pointwise hardy inequalities in connection with usual integral versions of hardy inequalities

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

sphericalization and p harmonic functions on unbounded domains in ahlfors regular spaces abstract we use sphericalization to study the dirichlet problem perron solutions and boundary regularity for p harmonic functions on unbounded sets in ahlfors regular metric spaces boundary regularity for the point at infinity is given special attention in particular we allow for several approach directions towards infinity and take into account the massiveness of their complements in 2005 llorente manfredi wu showed that the p harmonic measure on the upper half space r n n 2 is not subadditive on null sets when p 2 using their result and spherical inversion we create similar bounded examples in the unit ball b r n showing that the n harmonic measure is not subadditive on null sets when n 3 and neither are the p harmonic measures in b generated by certain weights depending on p 2 and n 2

space filling with metric measure spaces we show a sharp relationship between the existence of space filling mappings with an upper gradient in a lorentz space and the poincare inequality in a general metric setting as key examples we consider these phenomena in cantor diamond spaces and the heisenberg groups

equivalence and self improvement of p fatness and hardy s inequality and association with uniform perfectness we present an easy proof that p hardy s inequality implies uniform p fatness of the boundary when p n the proof works also in metric space setting and demonstrates the self improving phenomenon of the p fatness we also explore the relationship between p fatness p hardy inequality and the uniform perfectness for all p 1 and demonstrate that in the ahlfors q regular metric measure space setting with p q these three properties are equivalent when p 2 our results are new evenin the euclidean setting

analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid we establish the existence of three new subgroups of the group of volumepreserving diffeomorphisms of a compact n dimensional n 2 riemannian manifold which are associated with the dirichlet neumann and mixed type boundary conditions that arise in second order elliptic pdes we prove that when endowed with the hs hilbert class topologies for s n 2 1 these subgroups are c differential manifolds we consider these new diffeomorphism groups with an h1 equivalent right invariant metric and prove the existence of unique smooth geodesics t of this metric as well as existence and uniqueness of the jacobi equations associated to this metric geodesics on these subgroups are in fact the flows of a timedependent velocity vector field u t x so that t t u t t with 0 x x and remarkably the vector field u t x solves the so called lagrangian averaged euler lae equations on m these equations and their viscous counterparts the lagrangian averaged navier stokes lans equations model the motion of a fluid at scales larger than an a priori fixed parameter 0 while averaging or filtering out the small scale motion and this is achieved without the use of artificial viscosity we prove that for divergence free initial data satisfying u 0 on m the lae equations are well posed globally when n 2 we also find the boundary conditions that make the lans equations well posed globally when n 3 and prove that solutions of the lans equations converge when n 2 3 for almost all t in some fixed time interval 0 t in hs s n 2 1 3 to solutions of the lae equations thus confirming the scaling arguments of barenblatt chorin received april 26 2000 1991 mathematics subject classification 35q35 35q53 58b20 58d05

nonlinear balayage on metric spaces we develop a theory of balayage on complete doubling metric measure spaces supporting a poincare inequality in particular we are interested in continuity and p harmonicity of the balayage we also study connections to the obstacle problem as applications we characterize regular boundary points and polar sets in terms of balayage

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

fredholm properties of riemannian exponential maps on diffeomorphism groups we prove that exponential maps of right invariant sobolev h r metrics on a variety of diffeomorphism groups of compact manifolds are nonlinear fredholm maps of index zero as long as r is sufficiently large this generalizes the result of ebin et al geom funct anal 16 2006 for the l 2 metric on the group of volume preserving diffeomorphisms important in hydrodynamics in particular our results apply to many other equations of interest in mathematical physics we also prove an infinite dimensional morse index theorem settling a question raised by arnold and khesin topological methods in hydrodynamics springer new york 1998 on stable perturbations of flows in hydrodynamics finally we include some applications to the global geometry of diffeomorphism groups

an axiomatic approach to gradients with applications to dirichlet and obstacle problems beyond function spaces abstract we develop a framework for studying variational problems in banach spaces with respect to gradient relations which encompasses many of the notions of generalized gradients that appear in the literature we stress the fact that our approach is not dependent on function spaces and therefore applies equally well to functions on metric spaces as to operator algebras in particular we consider analogues of dirichlet and obstacle problems as well as first eigenvalue problems and formulate conditions for the existence of solutions and their uniqueness moreover we investigate to what extent a lattice structure may be introduced on ordered banach spaces via a norm minimizing variational problem a multitude of examples is provided to illustrate the versatility of our approach

sharp capacity estimates for annuli in weighted mathbf r n and in metric spaces we obtain estimates for the nonlinear variational capacity of annuli in weighted mathbf r n and in metric spaces we introduce four different pointwise exponent sets show that they all play fundamental roles for capacity estimates and also demonstrate that whether an end point of an exponent set is attained or not is important as a consequence of our estimates we obtain for instance criteria for points to have zero resp positive capacity our discussion holds in rather general metric spaces including carnot groups and many manifolds but it is just as relevant on weighted mathbf r n indeed to illustrate the sharpness of our estimates we give several examples of radially weighted mathbf r n which are based on quasiconformality of radial stretchings in mathbf r n

self improvement of uniform fatness revisited we give a new proof for the self improvement of uniform p fatness in the setting of general metric spaces our proof is based on rather standard methods of geometric analysis and in particular the proof avoids the use of deep results from potential theory and analysis on metric spaces that have been indispensable in the previous proofs of the self improvement a key ingredient in the proof is a self improvement property for local hardy inequalities

spectral convergence under bounded ricci curvature abstract for a noncollapsed gromov hausdorff convergent sequence of riemannian manifolds with a uniform bound of ricci curvature we establish two spectral convergence one of them is on the hodge laplacian acting on differential one forms the other is on the connection laplacian acting on tensor fields of every type which include all differential forms these are sharp generalizations of cheeger colding s spectral convergence of the laplacian acting on functions to the cases of tensor fields and differential forms these spectral convergence have two direct corollaries one of them is to give new bounds on such eigenvalues in terms of bounds on volume diameter and the ricci curvature the other is that we show the upper semicontinuity of the first betti numbers with respect to the gromov hausdorff topology and give the equivalence between the continuity of them and the existence of a uniform spectral gap on the other hand we also define measurable curvature tensors of the noncollapsed gromov hausdorff limit space of a sequence of riemannian manifolds with a uniform bound of ricci curvature which include riemannian curvature tensor the ricci curvature and the scalar curvature as fundamental properties of our ricci curvature we show that the ricci curvature coincides with the difference between the hodge laplacian and the connection laplacian and is compatible with gigli s one and lott s ricci measure moreover we prove a lower bound of the ricci curvature is compatible with a reduced riemannian curvature dimension condition we also give a positive answer to lott s question on the behavior of the scalar curvature with respect to the gromov hausdorff topology by using our scalar curvature

higher su n tensor products we extend our recent results on ordinary su n tensor product multiplicities to higher su n tensor products particular emphasis is put on four point couplings where the tensor product of four highest weight modules is considered the number of times the singlet occurs in the decomposition is the associated multiplicity in this framework ordinary tensor products correspond to three point couplings as in that case the four point multiplicity may be expressed explicitly as a multiple sum measuring the discretized volume of a convex polytope this description extends to higher point couplings as well we also address the problem of determining when a higher point coupling exists i e when the associated multiplicity is non vanishing the solution is a set of inequalities in the dynkin labels

n point and higher genus osp 1 2 fusion we study affine osp 1 2 fusion the fusion in osp 1 2 conformal field theory for example higher point and higher genus fusion is discussed the fusion multiplicities are characterized as discretized volumes of certain convex polytopes and are written explicitly as multiple sums measuring those volumes we extend recent methods developed to treat affine su 2 fusion they are based on the concept of generalized berenstein zelevinsky triangles and virtual couplings higher point tensor products of finite dimensional irreducible osp 1 2 representations are also considered the associated multiplicities are computed and written as multiple sums

quasiconformal maps in metric spaces with controlled geometry this paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry the principal message is that such a theory is both relevant and viable the first main issue is the problem of definition which we next describe quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount this requirement makes sense in every metric space given a homeomorphism f from a metric space x to a metric space y then for x x and r 0 set

a differentiable structure for metric measure spaces abstract the main result of this paper is the provision of conditions under which a metric measure space admits a differentiable structure this differentiable structure gives rise to a finite dimensional l cotangent bundle over the given metric measure space and then to a sobolev space h 1 p over the given metric measure space the latter which is reflexive for p 1 this extends results of cheeger geom funct anal 9 1999 3 428 to a wider collection of metric measure spaces

is an orlicz poincare inequality an open ended condition and what does that mean abstract in 16 keith and zhong prove that spaces admitting poincare inequalities also admit a priori stronger poincare inequalities we use their technique with slight adjustments to obtain a similar result in the case of orlicz poincare inequalities we give examples in the plane that show all hypotheses are required

sobolev embedding for m1 p spaces is equivalent to a lower bound of the measure abstract it has been known since 1996 that a lower bound for the measure b x r b r s implies sobolev embedding theorems for sobolev spaces m 1 p defined on metric measure spaces we prove that in fact sobolev embeddings for m 1 p spaces are equivalent to the lower bound of the measure

the equivalence between pointwise hardy inequalities and uniform fatness we prove an equivalence result between the validity of a pointwise hardy inequality in a domain and uniform capacity density of the complement this result is new even in euclidean spaces but our methods apply in general metric spaces as well we also present a new transparent proof for the fact that uniform capacity density implies the classical integral version of the hardy inequality in the setting of metric spaces in addition we consider the relations between the above concepts and certain hausdorff content conditions

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