compact composition operators on weighted hilbert spaces of analytic functions we characterize the compactness of composition operators acting on a large family of hilbert spaces of analytic functions which lie between bergman and dirichlet spaces our characterization is given in terms of generalized nevanlinna counting functions

cone volume measure of general centered convex bodies abstract we show that the cone volume measure of a convex body with centroid at the origin satisfies the subspace concentration condition this extends former results obtained in the discrete as well as in the symmetric case and implies among others a conjectured best possible inequality for the u functional of a convex body

order statistics and concentration of lr norms for log concave vectors abstract we establish upper bounds for tails of order statistics of isotropic log concave vectors and apply them to derive a concentration of l r norms of such vectors

a stability result for mean width of lp centroid bodies we give a different proof of a recent result of klartag b klartag a central limit theorem for convex sets invent math 168 1 2007 91 131 concerning the concentration of the volume of a convex body within a thin euclidean shell and proving a conjecture of anttila ball and perissinaki m anttila k ball i perissinaki the central limit problem for convex bodies trans amer math soc 355 12 2003 4723 4735 it is based on the study of the lp centroid bodies we prove an almost isometric reverse holder inequality for their mean width and a refined form of a stability result

weak convergence of random p mappings and the exploration process of inhomogeneous continuum random trees we study the asymptotics of the p mapping model of random mappings on n as n gets large under a large class of asymptotic regimes for the underlying distribution p we encode these random mappings in random walks which are shown to converge to a functional of the exploration process of inhomogeneous random trees this exploration process being derived aldous miermont pitman 2004 from a bridge with exchangeable increments our setting generalizes previous results by allowing a finite number of attracting points to emerge

the exploration process of inhomogeneous continuum random trees and an extension of jeulin s local time identity we study the inhomogeneous continuum random trees icrt that arise as weak limits of birthday trees we give a description of the exploration process a function defined on 0 1 that encodes the structure of an icrt and also of its width process determining the size of layers in order of height these processes turn out to be transformations of bridges with exchangeable increments which have already appeared in other icrt related topics such as stochastic additive coalescence the results rely on two different constructions of birthday trees from processes with exchangeable increments on weak convergence arguments and on general theory on continuum random trees

on the correlation for kac like models in the convex case the aim of this paper is to stu the behavior asm tends to of a family of measures exp m x dx m on m where m is a potential on m which is a perturbation in a suitable sense of the harmonic potential j x j 2

extremal slabs in the cube and the laplace transform abstract we study the volume of symmetric slabs in the unit cube we show that for t 3 4 the slab parallel to a face has the minimal volume among all symmetric slabs with width t for large width we prove the asymptotic extremality of the slab orthogonal to the main diagonal the proof is based on certain concavity properties of the laplace transform and on several limit theorems from probability the central limit theorem and classical principles of moderate and large deviations finally we extend some of the results to more general classes of bodies

a berry esseen type inequality for convex bodies with an unconditional basis suppose x x1 xn is a random vector distributed uniformly in a convex body k subset mathbb r n we assume the normalization mathbb e x i 2 1 for i 1 n the body k is further required to be invariant under coordinate reflections that is we assume that x1 xn has the same distribution as x1 xn for any choice of signs then we show that mathbb e left x sqrt n right 2 leq c 2 where c 4 is a positive universal constant and is the standard euclidean norm in mathbb r n the estimate is tight up to the value of the constant it leads to a berry esseen type bound in the central limit theorem for unconditional convex bodies

power law estimates for the central limit theorem for convex sets abstract we investigate the rate of convergence in the central limit theorem for convex sets established in b klartag a central limit theorem for convex sets invent math in press 8 we obtain bounds with a power law dependence on the dimension these bounds are asymptotically better than the logarithmic estimates which follow from the original proof of the central limit theorem for convex sets

isotropic constants and mahler volumes abstract this paper contains a number of results related to volumes of projective perturbations of convex bodies and the laplace transform on convex cones first it is shown that a sharp version of bourgain s slicing conjecture implies the mahler conjecture for convex bodies that are not necessarily centrally symmetric second we find that by slightly translating the polar of a centered convex body we may obtain another body with a bounded isotropic constant third we provide a counter example to a conjecture by kuperberg on the distribution of volume in a body and in its polar

singular values and schmidt pairs of composition operators on the hardy space abstract we find the singular values and corresponding schmidt pairs of a compact composition operator c induced by z a z b where a b 1 on the classical hardy space we do so by solving a functional equation that is a generalization of schroder s equation find a function f holomorphic on the open unit disc and a complex number such that g z f z f z where is a holomorphic self map of the open unit disc with an interior fixed point and g is a bounded holomorphic function on the open unit disc in addition we find the spectrum of the weighted composition operator m g c

strong fatou 1 points of blaschke products this paper shows that to every countable set m on the unit circle there corresponds a blaschke product whose set of strong fatou 1 points contains m it also shows that some blaschke products have an uncountable set of strong fatou 1 points

on the hot spots conjecture of j rauch rodrigo banuelos research partially supported by nsf grant dms 9400854 krzysztof burdzy s research partially supported by nsf grant dms 9322689

poincare inequality in mean value for gaussian polytopes let k n g 1 g n be the absolute convex hull of n independent standard gaussian random points in mathbb r n with n n we prove that for any 1 lipschitz function f mathbb r n rightarrow mathbb r the polytope k n satisfies the following poincare inequality in mean value mathbb e omega int limits k n omega left f x frac 1 textup vol n left k n omega right int limits k n omega f y dy right 2 dx leq frac c n mathbb e omega int limits k n omega x 2dx where c 0 is an absolute constant this poincare inequality is the one suggested by a conjecture of kannan lovasz and simonovits for general convex bodies moreover we prove in mean value that the volume of the polytope k n is concentrated in a subexponential way within a thin euclidean shell with the optimal dependence of the dimension n an important tool of the proofs is a representation of the law of g 1 g n conditioned by the event that the convex hull of g 1 g n is a n 1 face of k n as an application we also get an estimate of the number of n 1 faces of the polytope k n valid for every n n

the central limit problem for convex bodies it is shown that every symmetric convex body which satisfies a kind of weak law of large numbers has the property that almost all its marginal distributions are approximately gaussian several quite broad classes of bodies are shown to satisfy the condition

boundary correspondence of nevanlinna counting functions for self maps of the unit disc let be a holomorphic self map of the unit disc d for every a d there is a measure on d sometimes called aleksandrov measure defined by the poisson representation re z z f p z d its singular part measures in a natural way the affinity of for the boundary value a the affinity for values w inside d is provided by the nevanlinna counting function n w of we introduce a natural measure valued refinement m w of n w and establish that the measures d are obtained as boundary values of the refined nevanlinna counting function m more precisely we prove that is the weak limit of m w whenever w converges to a non tangentially outside a small exceptional set e we obtain a sharp estimate for the size of e in the sense of capacity

une theorie combinatoire des series formelles abstract this paper presents a combinatorial theory of formal power series the combinatorial interpretation of formal power series is based on the concept of species of structures a categorical approach is used to formulate it a new proof of cayley s formula for the number of labelled trees is given as well as a new combinatorial proof due to g labelle of lagrange s inversion formula polya s enumeration theory of isomorphism classes of structures is entirely renewed recursive methods for computing cycle index polynomials are described a combinatorial version of the implicit function theorem is stated and proved the paper ends with general considerations on the use of coalgebras in combinatorics

the variance conjecture on projections of the cube abstract we prove that the uniform probability measure on every n k dimensional projection of the n dimensional unit cube verifies the variance conjecture with an absolute constant c var x 2 c sup s n 1 e x 2 e x 2 provided that 1 k n we also prove that if 1 k n 2 3 log n 1 3 the conjecture is true for the family of uniform probabilities on its projections on random n k dimensional subspaces

affine images of isotropic measures necessary and sufficient conditions are given in order for a borel measure on the euclidean sphere to have an affine image that is isotropic a sharp reverse affine isoperimetric inequality for borel measures on the sphere is presented this leads to sharp reverse affine isoperimetric inequalities for convex bodies 2015 international press of boston inc all rights reserved

the b conjecture for the gaussian measure of dilates of symmetric convex sets and related problems we show that given a symmetric convex set k rd the function t etk is log concave on r where denotes the standard d dimensional gaussian measure we also comment on the extension of this property to unconditional log concave measures and sets and on the complex case

some examples of composition operators and their approximation numbers on the hardy space of the bi disk we give examples of composition operators c phi on h 2 d 2 showing that the condition phi infty 1 is not sufficient for their approximation numbers a n c phi to satisfy lim n to infty a n c phi 1 sqrt n 1 contrary to the 1 dimensional case we also give a situation where this implication holds we make a link with the monge amp ere capacity of the image of phi

invariances in variance estimates we provide variants and improvements of the brascamp lieb variance inequality which takes into account the invariance properties of the underlying measure this is applied to spectral gap estimates for log concave measures with many symmetries and to non interacting conservative spin systems

inhomogeneous continuum random trees and the entrance boundary of the additive coalescent regard an element of the set of ranked discrete distributions x1 x2 x1 x2 0 i x i 1 as a fragmentation of unit mass into clusters of masses x i the additive coalescent is the valued markov process in which pairs of clusters of masses x i x j merge into a cluster of mass x i x j at rate x i x j aldous and pitman 1998 showed that a version of this process starting from time with infinitesimally small clusters can be constructed from the brownian continuum random tree of aldous 1991 1993 by poisson splitting along the skeleton of the tree in this paper it is shown that the general such process may be constructed analogously from a new family of inhomogeneous continuum random trees

a central limit theorem for convex sets we show that there exists a sequence varepsilon n searrow0 for which the following holds let k n be a compact convex set with a non empty interior let x be a random vector that is distributed uniformly in k then there exist a unit vector in n t0 and 0 such that sup a subset mathbb r left textit prob langle x theta rangle in a frac 1 sqrt 2 pi sigma int ae frac t t 0 2 2 sigma 2 dt right leq varepsilon n qquad ast where the supremum runs over all measurable sets a and where denotes the usual scalar product in n furthermore under the additional assumptions that the expectation of x is zero and that the covariance matrix of x is the identity matrix we may assert that most unit vectors satisfy with t0 0 and 1 corresponding principles also hold for multi dimensional marginal distributions of convex sets

a spectral radius type formula for approximation numbers of composition operators abstract for approximation numbers a n c of composition operators c on weighted analytic hilbert spaces including the hardy bergman and dirichlet cases with symbol of uniform norm lim n a n c 1 n e 1 cap d where cap d is the green capacity of d in d this formula holds also for h p with 1 p

on the role of convexity in isoperimetry spectral gap and concentration we show that for convex domains in euclidean space cheeger s isoperimetric inequality spectral gap of the neumann laplacian exponential concentration of lipschitz functions and the a priori weakest requirement that lipschitz functions have arbitrarily slow uniform tail decay are all quantitatively equivalent to within universal constants independent of the dimension this substantially extends previous results of maz ya cheeger gromov milman buser and ledoux as an application we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume up to constants and under maps which are on average lipschitz we also provide a new characterization up to constants of the spectral gap of a convex domain as one over the square of the average distance from the worst subset having half the measure of the domain in addition we easily recover and extend many previously known lower bounds on the spectral gap of convex domains due to payne weinberger li yau kannan lovasz simonovits bobkov and sodin the proof involves estimates on the diffusion semi group following bakry ledoux and a result from riemannian geometry on the concavity of the isoperimetric profile our results extend to the more general setting of riemannian manifolds with density which satisfy the cd 0 curvature dimension condition of bakry emery

nevanlinna counting function and carleson function of analytic maps we show that the maximal nevanlinna counting function and the carleson function of analytic self maps of the unit disk are equivalent up to constants

compact composition operators on the dirichlet space and capacity of sets of contact points we prove several results about composition operators on the dirichlet space d for every compact set k d of logarithmic capacity capk 0 there exists a schur function both in the disk algebra a d and in d such that the composition operator c is in all schatten classes sp d p 0 and for which k eit eit 1 eit eit 1 for every bounded composition operator c on d and every d the logarithmic capacity of eit eit is 0 every compact composition operator c on d is compact on b 2 and on h 2 in particular c is in every schatten class sp p 0 both on h2 and on b2 there exists a schur function such that c is compact on h 2 but which is not even bounded on d there exists a schur function such that c is compact on d but in no schatten class sp d

the hot spots conjecture for domains with two axes of symmetry consider a convex planar domain with two axes of symmetry we show that the maximum and minimum of a neumann eigenfunction with lowest nonzero eigenvalue occur at points on the boundary only we deduce j rauch s hot spots conjecture in the following form if the initial temperature distribution is not orthogonal to the first nonzero eigenspace then the point at which the temperature achieves its maximum tends to the boundary in fact the maximum point reaches the boundary in finite time if the boundary has positive curvature results of this type have already been proved by bafiuelos and burdzy bb using the heat equation and probabilistic methods to deform initial conditions to eigenfunctions we introduce here a new technique based on deformation of the domain an advantage of our method is that it works even in the case of multiple eigenvalues on the way toward our results we prove monotonicity properties for neumann eigenfunctions for symmetric domains that need not be convex and deduce a sharp comparison of eigenvalues with the dirichlet problem of independent interest