uniform rectifiability and harmonic measure ii poisson kernels in l p imply uniform rectifiability the first author was supported by nsf grants dms 0801079 and dms 1101244 the second author was supported by micinn grant mtm2010 16518 and by csic pie 200850i015 the third author was partially supported by grants dms 0901524 career dms 1056965 us nsf sloan research foundation and mtm2010 16232 mtm2009 14694 c02 01 spain

harmonic measure and quantitative connectivity geometric characterization of the l p solvability of the dirichlet problem it is well known that quantitative scale invariant absolute continuity more precisely the weak a infty property of harmonic measure with respect to surface measure on the boundary of an open set omega subset mathbb r n 1 with ahlfors david regular boundary is equivalent to the solvability of the dirichlet problem in omega with data in l p partial omega for some p infty in this paper we give a geometric characterization of the weak a infty property of harmonic measure and hence of solvability of the l p dirichlet problem for some finite p this characterization is obtained under background hypotheses an interior corkscrew condition along with ahlfors david regularity of the boundary that are natural and in a certain sense optimal we provide counter examples in the absence of either of them or even one of the two upper or lower ahlfors david bounds moreover the examples show that the upper and lower ahlfors david bounds are each quantitatively sharp

another look at cesaro sequence spaces abstract we consider the cesaro sequence space ces p as a closed subspace of the infinite l p sum of finite dimensional spaces we easily obtain many known results for example ces p has property of rolewicz uniform opial property and weak uniform normal structure we also consider some generalized cesaro sequence spaces finally we compute the von neumann jordan and james constants of the two dimensional cesaro sequence space ces p 2 when 1 p 2

an analyst s traveling salesman theorem for sets of dimension larger than one in his 1990 inventiones paper p jones characterized subsets of rectifiable curves in the plane via a multiscale sum of beta numbers these beta numbers are geometric quantities measuring how far a given set deviates from a best fitting line at each scale and location jones result is a quantitative way of saying that a curve is rectifiable if and only if it has a tangent at almost every point moreover computing this square sum for a curve returns the length of the curve up to multiplicative constant k okikiolu extended his result from subsets of the plane to subsets of euclidean space g david and s semmes extended the discussion to include sets of integer dimension larger than one under the assumption of ahlfors regularity and using a variant of jones beta numbers this variant has since been used by others to give structure theorems for rectifiable sets and to give upper bounds for the measure of a set in this paper we give a version of p jones theorem for sets of arbitrary integer dimension lying in euclidean space our main result is a lower bound for the d dimensional hausdorff measure of a set in terms of an analogous sum of beta type numbers we also show an upper bound of this type the combination of these results gives a jones theorem for higher dimensional sets while there is no assumption of ahlfors regularity or of a measure on the underlying set there is an assumption of a lower bound on the hausdorff content we adapt david and semmes version of jones beta numbers by redefining them using a choquet integral allowing them to be defined for arbitrary sets and not just sets of locally finite measure a key tool in the proof is g david and t toro s parametrization of reifenberg flat sets with holes

differentiable points of sierpinski like sponges abstract for the sierpinski like sponge in r n we obtain the hausdorff dimension of the set of differentiable points such that there exists an n 1 rectifiable subsurface which has a tangent hyperplane at the point using the techniques such as multi dynamical systems geometric types of slices and restricted frobenius problem furthermore we can use the dimension of the set of differentiable points as a bilipschitz invariant to classify sierpinski like sponges in terms of bilipschitz equivalence

dahlberg s theorem in higher co dimension abstract in 1977 the celebrated theorem of b dahlberg established that the harmonic measure is absolutely continuous with respect to the hausdorff measure on a lipschitz graph of dimension n 1 in r n and later this result has been extended to more general non tangentially accessible domains and beyond in the present paper we prove the first analogue of dahlberg s theorem in higher co dimension on a lipschitz graph of dimension d in r n d n 1 with a small lipschitz constant we construct a linear degenerate elliptic operator l such that the corresponding harmonic measure l is absolutely continuous with respect to the hausdorff measure on more generally we provide sufficient conditions on the matrix of coefficients of l which guarantee the mutual absolute continuity of l and the hausdorff measure

rectifiability and elliptic measures on 1 sided nta domains with ahlfors david regular boundaries let omega subset mathbb r n 1 n geq 2 be 1 sided nta domain aka uniform domain i e a domain which satisfies interior corkscrew and harnack chain conditions and assume that partial omega is n dimensional ahlfors david regular we characterize the rectifiability of partial omega in terms of the absolute continuity of surface measure with respect to harmonic measure we also show that these are equivalent to the fact that partial omega can be covered mathcal h n a e by a countable union of portions of boundaries of bounded chord arc subdomains of omega and to the fact that partial omega possesses exterior corkscrew points in a qualitative way mathcal h n a e our methods apply to harmonic measure and also to elliptic measures associated with real symmetric second order divergence form elliptic operators with locally lipschitz coefficients whose derivatives satisfy a natural qualitative carleson condition

property β of rolewicz and orthogonal convexities of calderon lozanovskiĭ spaces abstract we find criteria for property of rolewicz and orthogonal convexities in calderon lozanovski spaces in order to present characterization for property in the sequence case we introduce a new respective monotonicity property the particular case of orlicz lorentz spaces will also be discussed

three space property for asymptotically uniformly smooth renormings abstract we prove that if y is a closed subspace of a banach space x such that y and x y admit an equivalent asymptotically uniformly smooth norm then x also admits an equivalent asymptotically uniformly smooth norm the proof is based on the use of the szlenk index and yields a few other applications to renorming theory

weyl manifolds and deformation quantization this paper deals with non commutative objects based on the weyl algebra from the differential geometric point of view we propose to extend familiar notions from manifolds to non commutative objects in this algebraic approach the basic object is not a point of a space but certain sections of bundles

logarithms of the derivative of univalent functions in qp spaces abstract for a univalent function d we study the membership of log to the space q p we look for geometric conditions and also a description involving the schwarzian derivative is given

quantifying curvelike structures of measures by using l2 jones quantities we study the curvelike structure of special measures on n in a multiscale fashion more precisely we consider the existence and construction of a sufficiently short curve with a sufficiently large measure our main tool is an l2 variant of jones numbers which measure the scaled deviations of the given measure from a best approximating line at different scales and locations the jones function is formed by adding the squares of the l2 jones numbers at different scales and the same location using a special l2 jones function we construct a sufficiently short curve with a sufficiently large measure the length and measure estimates of the underlying curve are expressed in terms of the size of this jones function 2003 wiley periodicals inc

a fully convex banach space which does not have the banach saks property let k 2 2 be an integer a banach space x is said to be fully k convex kr if for every sequence xn in x lim n o l k ii l x j 1 implies that xn is a cauchy sequence in x it is known s that every uniformly convex space is 2r and that every kr space is reflexive it is also known 3 51 that there are 2r spaces which are not superreflexive milman 10 p 973 has raised the question whether every reflexive space is isomorphic to a 2r space a banach space x is said to have the banach saks property bs if for every bounded sequence x in x there exists a subsequence y of xn such that the cesaro means l n y y of yn is convergent in x it is known that 7 all superreflexive spaces have the bs and that 1 l every banach space with bs is reflexive however there exist e g 1 reflexive banach spaces which do not have the bs it is also known 12 that there are banach spaces with the bs which are not superreflexive since the bs is invariant under isomorphism it is natural to ask whether every kr space has the bs an affirmative answer would yield a negative solution to milman s question the purpose of the paper is to renorm baernstein s space 1 so that it is 2r therefore kr spaces do not necessarily have the bs we shall need the following characterization of kr spaces for the proof when k 2 see s 6 s

k β and k nearly uniformly convex banach spaces vol 162 no 2 1991 k and k nearly uniformly convex banach spaces denka kutzarova different uniform geometrical properties have been defined between the uniform convexity and the reflexivity of banach spaces in the present paper we introduce other properties of this type namely k and k nearly uniform convexity the definitions as well as some of the results presented here are announced in 7 sullivan 24 has defined k uniformly rotund banach spaces which by a suggestion of davis are all superreflexive fan and glicksberg 1 have introduced fully k convex banach spaces let x1 xk 1 be vectors in a banach space x the k dimensional volume enclosed by x1 xk 1 is given by v x1 x2 xk 1

k uniform rotundity is equivalent to k uniform convexity abstract r geremia and f sullivan proved that 2 uniform rotundity is equivalent to 2 uniform convexity this paper extends this result to k uniformly rotund spaces

a implies nta for a class of variable coefficient elliptic operators abstract we consider a certain class of second order variable coefficient divergence form elliptic operators in a uniform domain with ahlfors regular boundary and we show that the a property of the elliptic measure associated to any such operator and its transpose imply that the domain is in fact nta and hence chord arc the converse was already known and follows from work of kenig and pipher 30

an asymptotic property of schachermayer s space under renorming let x be a banach space with closed unit ball b given k n x is said to be k respectively k 1 nearly uniformly convex k 1 nuc if for every 0 there exists 0 1 so that for every x b and every separated sequence xn b there are indices ni ki 1 respectively ni k 1i 1 such that 1 k 1 x ki 1 xni 1 respectively 1 k 1 k 1i 1 xni 1 it is shown that a banach space constructed by schachermayer is 2 but is not isomorphic to any 2 nuc banach space modifying this example we also show that there is a 2 nuc banach space which cannot be equivalently renormed to be 1

absolute continuity of harmonic measure for domains with lower regular boundaries abstract we study absolute continuity of harmonic measure with respect to surface measure on domains that have large complements we show that if r d 1 is ahlfors d regular and splits r d 1 into two nta domains then h d on this result is a natural generalization of a result of wu in 49 we also prove that almost every point in is a cone point if is a lipschitz graph combining these results and a result from 8 we characterize sets of absolute continuity with finite h d measure if d 1 for domains with large complements both in terms of the cone point condition and in terms of the rectifiable structure of the boundary even in the plane this extends the results of mcmillan in 38 and pommerenke in 43 which were only known for simply connected planar domains finally we also show our first result holds for elliptic measure associated with real second order divergence form elliptic operators with a mild assumption on the gradient of the matrix

uniform rectifiability carleson measure estimates and approximation of harmonic functions the first author was supported by nsf grant dms 1361701 the second author was supported in part by mineco grant mtm2010 16518 icmat severo ochoa project sev 2011 0087 he also acknowledges that the research leading to these results has received funding from the european research council under the european union s seventh framework programme fp7 2007 2013 erc agreement no 615112 hapdegmt the third author was supported by the alfred p sloan fellowship the nsf career award dms 1056004 the nsf inspire award dms 1344235 and the nsf materials research science and engineering center seed grant dmr 0212302

a comparison between star products on regular orbits of compact lie groups in this paper an algebraic and a differential star product defined on a regular coadjoint orbit of a compact semisimple group are compared it has been proved that there is an injective algebra homomorphism between the algebra of polynomials with the algebraic star product and the algebra of differential functions with the differential star product structure

compactly uniformly convex spaces and property β of rolewicz abstract we study different facets of property of rolewicz we remark that the notions of compact uniform convexity and property are isometrically equivalent and present new examples of spaces with that property an observation is made that the property can be formulated in terms of graphs and an estimate of the modulus is also given

on the k uniform rotund and the fully convex banach spaces let x be a real banach space if x is uniformly convex then it is known that x is 2 uniformly rotund s and fully 2 convex 2 furthermore if a banach space x is either k uniformly rotund or fully k convex for some k then x is reflexive in this paper we show that if x is strictly convex and kuniformly rotund then x is fully k 1 convex however there exists a superreflexive space which is fully 2 convex but is not 2 uniformly rotund and for each k 2 there exists a strictly convex space which is k uniformly rotund but is not fully k convex thus for each k 2 there exist fully kconvex banach space which is not fully k 1 convex for x x2 xk in x let

rectifiable sets and the traveling salesman problem let k c c be a bounded set in this paper we shall give a simple necessary and sufficient condit ion for k to lie in a rectifiable curve we say that a set is a rectifiable curve if it is the image of a finite interval under a lipschitz mapping recall that for a connected set f c c f is a rectifiable curve not necessarily simple if and only if l f where l denotes one dimensional hausdorff measure this classical result follows from the fact that on any finite graph there is a tour which covers the entire graph and which crosses each edge but not necessarily each vertex at most twice if k is a finite set then we are essentially reduced to the classical traveling salesman problem tsp compute the length of the shortest hami l ton ian cycle which hits all points of k this is the same up to a constant multiple as asking for the inf imum of l f where f is a curve k c f such a f is called a spanning tree in tsp theory for infinite sets k we cannot hope in general to have k be a subset of a jordan curve what we should therefore look at is connected sets which conta in k let fmi n be the shortest minimal spanning tree then we cannot possibly solve our problem for sets k of infinite cardinality if we cannot find f i f c o fmin for any finite set k here and throughout the paper c co c1 c o etc denote various universal constants while there are several algorithms for computing l fml these algorithms work for finite graphs satisfying the triangle inequality and do not use the euclidean properties of k see 13 for an excellent discussion of some of these algorithms therefore these methods cannot solve our problem for general infinite k we present a method which is a minor modification of a well known algorithm farthest insert ion see 13 which yields a f with i f c o l fmi the farthest insert ion algorithm has been extensively studied with large numerical calculations on computers and is experimentally good in the sense that the f produced satisfy i f c o l f i for all examples which have

on tangential star products for the coadjoint poisson structure we derive necessary conditions on a lie algebra from the existence of a star product on a neighbourhood of the origin in the dual of the lie algebra for the coadjoint poisson structure which is both differential and tangential to all the coadjoint orbits in particular we show that when the lie algebra is semisimple there are no differential and tangential star products on any neighbourhood of the origin in the dual of its lie algebra