strings mathematical theory and statistical examples strings in the sense of the present paper are sequences of multiarrays with two types of indices tensorial and structural and they are characterized by a transformation law that generalizes those for tensors affine connections and derivatives of scalars the original definition barndorff nielsen proc r soc lond a 406 127 137 1986 is here extended and a systematic study of the mathematics of strings is undertaken in particular a convolutive multiplication of strings is introduced and is used in the discussion of intertwining a type of operation that produces tensors from strings and strings from tensors and connection strings a special kind of string it appears that tensors and connection strings have a role respectively as coordinates and coordinate frames in the calculus of strings a definition of covariant differentiation of strings is proposed and is related to convolutive multiplication and to intertwining the general theory is illustrated by various examples from the context of statistical inference finally a brief comparison is made between strings and the somewhat related concept of extensors

perturbations of elliptic operators in 1 sided chord arc domains part i small and large perturbation for symmetric operators the first and third authors acknowledge financial support from the spanish ministry of economy and competitiveness through the severo ochoa programme for centres of excellence in r d sev 2015 0554 they also acknowledge 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 second author was supported by nsf grant dms 1664047

stability of square root domains associated with elliptic systems of pdes on nonsmooth domains abstract we discuss stability of square root domains for uniformly elliptic partial differential operators l a a in l 2 with mixed boundary conditions on with respect to additive perturbations we consider open bounded and connected sets r n n n 1 that satisfy the interior corkscrew condition and prove stability of square root domains of the operator l a with respect to additive potential perturbations v l p l p n 2 special emphasis is put on the case of uniformly elliptic systems with mixed boundary conditions

hausdorff content and the hardy littlewood maximal operator on metric measure spaces abstract let x d be a complete metric measure space and be a non negative borel regular measure satisfying the doubling condition with some dimensional constant d we prove that the hausdorff content of codimension 0 denoted by h and the hardy littlewood maximal operator m satisfy the strong type inequality x m u p d h c x u p d h 0 u l loc 1 x whenever p max 0 1 d if further satisfies some reverse doubling condition with some other dimensional constant then for the endpoint case p 1 d with 0 d 0 we also obtain the corresponding weak type estimate for h and m the fundamental point in the proofs is to introduce and develop a theory of the dyadic hausdorff content h d which is a choquet capacity comparable to h and has the strong subadditivity property

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

derivative strings differential strings and semi holonomic jets the derivative strings of barndorff nielsen and the differential strings of blaesild mora are considered here from the coordinate free viewpoint it is shown that the derivative strings of given length and degree over a differentiable manifold form a vector bundle associated to a principal bundle of higher order frames and that there is an analogous result for differential strings bundles of derivative strings are identified with vector bundles obtained from 0 truncated versions of ehresmann s semi holonomic jets by dualization and by taking tensor products similarly bundles of differential strings are identified with vector bundles obtained from semiholonomic jets of certain tensor fields

duality of hardy and bmo spaces associated with operators with heat kernel bounds the introduction and development of hardy and bmo spaces on euclidean spaces r in the 1960s and 1970s played an important role in modern harmonic analysis and applications in partial differential equations these spaces were studied extensively in 32 22 18 19 31 and many others an l function f on r is in the hardy spaceh r if the area integral function of the poisson integral e t f satisfies

carleson measure problems for parabolic bergman spaces and homogeneous sobolev spaces abstract let b p r 1 n be the space of solutions to the parabolic equation t u u 0 0 1 having finite l p r 1 n norm we characterize nonnegative radon measures on r 1 n having the property u l q r 1 n u w 1 p r 1 n 1 p q whenever u t x b p r 1 n w 1 p r 1 n meanwhile denoting by v t x the solution of the above equation with cauchy data v 0 x we characterize nonnegative radon measures on r 1 n satisfying v t 2 x l q r 1 n v 0 w p r n 0 n p 1 n q 0 moreover we obtain the decay of v t x an isocapacitary inequality and a trace inequality

classes of carleson type measures generated by capacities we introduce classes of measures in the half space mathbb r n 1 generated by riesz or bessel or besov capacities in mathbb r n and give a geometric characterization of these as carleson type measures

well posedness of quasi geostrophic equations with data in besov q spaces abstract by wavelets and multi resolution analysis we convert the bilinear estimate on tent spaces associated with besov q spaces b p p 1 2 r n to the estimates of the numerical quantities as an application we get the well posedness of the quasi geostrophic equations with initial data in b p p 1 2 r 2

semigroup characterization of besov type morrey spaces and well posedness of generalized navier stokes equations abstract the well posedness of generalized navier stokes equations with initial data in some critical homogeneous besov spaces and in some critical q spaces was known in this paper we establish a wavelet characterization of besov type morrey spaces under the action of semigroup as an application we obtain the well posedness of smooth solution for the generalized navier stokes equations with initial data in some critical homogeneous besov type morrey spaces b p p 1 2 n 1 2 1 1 2 1 2 1 p 2 and n p 2 2 2 n p or 2 p and max n p 2 2 1 2 n p with divergence free these critical homogeneous besov type morrey spaces are larger than corresponding classical besov spaces and cover q spaces

carleson embeddings for sobolev spaces via heat equation abstract let u t x be the solution of the heat equation t x u t x 0 on r 1 n 0 r n subject to u 0 x f x on r n the main goal of this paper is to characterize such a nonnegative measure on r 1 n that f x u t 2 x induces a bounded embedding from the sobolev space w 1 p r n p 1 n into the lebesgue space l q r 1 n q 0

on the analyticity of the semigroup generated by the stokes operator with neumann type boundary conditions on lipschitz subdomains of riemannian manifolds we study the analyticity of the semigroup generated by the stokes operator equipped with neumann type boundary conditions on l p spaces in lipschitz domains our strategy is to regularize this operator by considering the hodge laplacian which has the additional property that it commutes with the leray projection

strong type estimates for homogeneous besov capacities capacitary strong type estimates for homogeneous besov spaces are established moreover the result is used to study embeddings and dualities of function spaces

optimal l p l q estimates for parabolic boundary value problems with inhomogeneous data in this paper we investigate vector valued parabolic initial boundary value problems mathcal a t x d mathcal b j t x d subject to general boundary conditions in domains g in mathbb r n with compact c 2m boundary the top order coefficients of mathcal a are assumed to be continuous we characterize optimal l p l q regularity for the solution of such problems in terms of the data we also prove that the normal ellipticity condition on mathcal a and the lopatinskii shapiro condition on mathcal a mathcal b 1 dots mathcal b m are necessary for these l p l q estimates as a byproduct of the techniques being introduced we obtain new trace and extension results for sobolev spaces of mixed order and a characterization of triebel lizorkin spaces by boundary data

square function non tangential maximal function estimates and the dirichlet problem for non symmetric elliptic operators a we consider divergence form elliptic operators l div a x defined in the half space rn 1 n 2 where the coefficient matrix a x is bounded measurable uniformly elliptic t independent and not necessarily symmetric we establish square function non tangential maximal function estimates for solutions of the homogeneous equation lu 0 and we then combine these estimates with the method of approximability to show that l harmonic measure is absolutely continuous with respect to surface measure i e n dimensional lebesgue measure on the boundary in a scale invariant sense more precisely that it belongs to the class a with respect to surface measure equivalently that the dirichlet problem is solvable with data in lp for some p previously these results had been known only in the case n 1

a john nirenberg type inequality for qα rn abstract the john nirenberg inequality characterizes functions in the space bmo in terms of the decay of the distribution function of their oscillations over a cube in this paper we prove separate necessary and sufficient john nirenberg type inequalities for functions in the space q r n introduced by essen janson peng and xiao who conjectured a version of this inequality our results are a modified version of their conjecture and we give a counterexample to show the necessity for this modification the counterexample also shows that these necessary and sufficient conditions cannot be reconciled

quadratic estimates and functional calculi of perturbed dirac operators we prove quadratic estimates for complex perturbations of dirac type operators and thereby show that such operators have a bounded functional calculus as an application we show that spectral projections of the hodge dirac operator on compact manifolds depend analytically on l changes in the metric we also recover a unified proof of many results in the calderon program including the kato square root problem and the boundedness of the cauchy operator on lipschitz curves and surfaces

characterizations of weighted besov and triebel lizorkin spaces via temperatures abstract hardy littlewood type characterizations via temperatures on a half space of weighted besov and triebel lizorkin spaces studied recently by the author are given in the proof of the sufficient parts of the results a sub mean value property of temperatures is used in an elegant way to get control of certain terms by the hardy maximal function or its vector valued version as a byproduct of the results another characterization of weighted hardy spaces as well as characterizations of the gauss weierstrass integrals on a strip domain of distributions in the above weighted spaces is obtained also

new applications of besov type and triebel lizorkin type spaces abstract in this paper the authors prove that besov morrey spaces are proper subspaces of besov type spaces b p q s r n and that triebel lizorkin morrey spaces are special cases of triebel lizorkin type spaces f p q s r n the authors also establish an equivalent characterization of b p q s r n when 0 1 p these besov type spaces b p q s r n and triebel lizorkin type spaces f p q s r n were recently introduced to connect besov spaces and triebel lizorkin spaces with q spaces moreover for the spaces b p q s r n and f p q s r n the authors investigate their trace properties and the boundedness of the pseudo differential operators with homogeneous symbols in these spaces which generalize the corresponding classical results of jawerth and grafakos torres by taking 0

numerical approximation of the integral fractional laplacian we propose a new nonconforming finite element algorithm to approximate the solution to the elliptic problem involving the fractional laplacian we first derive an integral representation of the bilinear form corresponding to the variational problem the numerical approximation of the action of the corresponding stiffness matrix consists of three steps 1 apply a sinc quadrature scheme to approximate the integral representation by a finite sum where each term involves the solution of an elliptic partial differential equation defined on the entire space 2 truncate each elliptic problem to a bounded domain 3 use the finite element method for the space approximation on each truncated domain the consistency error analysis for the three steps is discussed together with the numerical implementation of the entire algorithm the results of computations are given illustrating the error behavior in terms of the mesh size of the physical domain the domain truncation parameter and the quadrature spacing parameter

vertical versus conical square functions we study the difference between vertical and conical square functions in the abstract and also in the specific case where the square functions come from an elliptic operator trans amer math soc 364 2012 no 10 5469 5489

the kato problem for operators with weighted ellipticity we consider second order operators lw w 1 divawr with ellipticity controlled by a muckemphout a2 weight w we prove that the kato square root estimate l 1 2 w f l2 w kr fkl2 w holds in the weighted space l 2 w

the solution of the kato problem for degenerate elliptic operators with gaussian bounds we prove the kato conjecture for degenerate elliptic operators on r n more precisely we consider the divergence form operator lw w 1 diva where w is a muckenhoupt a2 weight and a is a complex valued n n matrix such that w 1 a is bounded and uniformly elliptic we show that if the heat kernel of the associated semigroup e tlw satisfies gaussian bounds then the weighted kato square root estimate kl 1 2 w f k l2 w k f k l2 w holds

the solution of the kato problem for divergence form elliptic operators with gaussian heat kernel bounds which in turnhas applications to the perturbation theory for certain classes of hyperbolicequations see 16 we remark that 1 3 is equivalent to the oppositeinequality for the square root of the adjoint operator l in 13 14 kato conjectured that an abstract version of 1 3 might hold for regularly accretive operators see 14 18 for the details a counter example to this abstract conjecture was obtained by mcintosh 17 whothen reformulated the conjecture in the following form bearing in mind thatkato s interest in the problem had been motivated by the special case ofelliptic di erential operators conjecture 1 4 the estimate 1 3 holds for lde ned as in 1 2 for anyl

hausdorff besov type and triebel lizorkin type spaces and their applications abstract let s r p 1 and 0 1 p where p denotes the conjugate index of p in this article the authors first introduce the hausdorff besov type space b h p q s r n with q 1 and the hausdorff triebel lizorkin type space f h p q s r n with q 1 via a class of weights defined on the upper plane and then establish some equivalent characterizations of b h p q s r n and f h p q s r n via some classes of weights defined on r n the authors then prove that their dual spaces are besov morrey and triebel lizorkin morrey spaces respectively the relations between these spaces and the known besov triebel lizorkin hausdorff spaces are also studied as an application for p 1 and 0 n the authors obtain the coincidence between the space f h p 2 0 n n p r n and the predual space h p r n of the morrey space l p r n moreover characterizations of b h p q s r n and f h p q s r n via local means and peetre maximal functions as well as the boundedness of riesz potential operators and some singular integrals on b h p q s r n and f h p q s r n are also obtained

analyticity of layer potentials and l 2 solvability of boundary value problems for divergence form elliptic equations with complex l infty coefficients we consider divergence form elliptic operators of the form l dv a x nabla defined in r n 1 x t in r n times r n geq 2 where the l infty coefficient matrix a is n 1

higher derivatives in geometry and physics two related ways to invariantly study higher derivatives new tensors and derivative strings are compared and the idea of suppression operator or multiplier is introduced it renders the formal expression of the new objects almost as simple as that of ordinary tensors jacobians as a subgroup of the full second order group are identified and the harmonic subgroup leaving laplace s equation invariant is determined the wider principle of general covariance entailed by the full group is considered in particular electrogravitic potentials are defined and the exploration of their field theory begun

q spaces and morrey spaces abstract we characterize functions in morrey space by p carleson measures we then reveal a simple relation between q space and morrey space that is q space can be viewed as a fractional integration of the morrey space therefore many results for morrey space can be translated onto q space for example we show that q space is a dual space by identifying its predual

abelian link invariants and homology we consider the link invariants defined by the quantum chern simons field theory with compact gauge group u 1 in a closed oriented 3 manifold m the relation of the abelian link invariants with the homology group of the complement of the links is discussed we prove that when m is a homology sphere or when a link in a generic manifold m is homologically trivial the associated observables coincide with the observables of the sphere s3 finally we show that the u 1 reshetikhin turaev surgery invariant of the manifold m is not a function of the homology group only nor a function of the homotopy type of m alone

on the quantum invariants for the spherical seifert manifolds we study the witten reshetikhin turaev su 2 invariant for the seifert manifolds s3 gamma where is a finite subgroup of su 2 we show that the wrt invariants can be written in terms of the eichler integral of modular forms with half integral weight and we give an exact asymptotic expansion of the invariants by use of the nearly modular property of the eichler integral we further discuss that those modular forms have a direct connection with the polyhedral group by showing that the invariant polynomials of modular forms satisfy the polyhedral equations associated to

new orlicz hardy spaces associated with divergence form elliptic operators abstract let l be the divergence form elliptic operator with complex bounded measurable coefficients the positive concave function on 0 of strictly critical lower type p 0 1 and t t 1 1 t 1 for t 0 in this paper the authors study the orlicz hardy space h l r n and its dual space bmo l r n where l denotes the adjoint operator of l in l 2 r n several characterizations of h l r n including the molecular characterization the lusin area function characterization and the maximal function characterization are established the carleson measure characterization and the john nirenberg inequality for the space bmo l r n are also given as applications the authors show that the riesz transform l 1 2 and the littlewood paley g function g l map h l r n continuously into l the authors further show that the riesz transform l 1 2 maps h l r n into the classical orlicz hardy space h r n for p n n 1 1 and the corresponding fractional integral l for certain 0 maps h l r n continuously into h l r n where is determined by and and satisfies the same property as all these results are new even when t t p for all t 0 and p 0 1

some results on q p spaces 0lpl1 for p is an element of 0 1 let q p be a proper subspace of bmoa defined by means of a modified garcia norm we give a boundary value criterion for a function to be in q p and a necessary and sufficient condition in terms of p carleson measure for an

decompositions of besov hausdorff and triebel lizorkin hausdorff spaces and their applications abstract let p 1 q 1 s r and 0 1 1 max p q in this paper the authors establish the transform characterizations of besov hausdorff spaces b h p q s r n and triebel lizorkin hausdorff spaces f h p q s r n q 1 as applications the authors then establish their embedding properties which on b h p q s r n is also sharp smooth atomic and molecular decomposition characterizations for suitable moreover using their atomic and molecular decomposition characterizations the authors investigate the trace properties and the boundedness of pseudo differential operators with homogeneous symbols in b h p q s r n and f h p q s r n q 1 which generalize the corresponding classical results on homogeneous besov and triebel lizorkin spaces when p 1 and q 1 by taking 0

fefferman stein decomposition for q spaces and micro local quantities abstract in this paper we study the fefferman stein decomposition of q r n and give an affirmative answer to an open problem posed by essen et al 2000 one of our main methods is to characterize the structure of the predual of q r n by the micro local quantities this result indicates that the norm of the predual space of q r n depends on the micro local structure in a self correlation way

strings tensorial combinants and bartlett adjustments the transformation laws for tensors for affine connections and for derivatives of scalars may be subsumed under a general concept termed a string the intertwining of certain types of strings leads to new strings thus intertwining of a symmetric scalar string and a symmetric connection string yields a sequence of symmetric tensors some statistical instances of strings are given and these are of use for representing various asymptotic expansions in invariant terms as an example of this invariant decompositions of bartlett adjustments are briefly discussed

a new class of function spaces connecting triebel lizorkin spaces and q spaces abstract let s r 0 p 1 and q 1 in this paper we introduce a new class of function spaces f p q s r n which unify and generalize the triebel lizorkin spaces with both p 1 and p and q spaces by establishing the carleson measure characterization of q space we then determine the relationship between triebel lizorkin spaces and q spaces which answers a question posed by dafni and xiao in g dafni j xiao some new tent spaces and duality theorem for fractional carleson measures and q r n j funct anal 208 2004 377 422 moreover via the hausdorff capacity we introduce a new class of tent spaces f t p q s r n 1 and determine their dual spaces f w p q s q r n where s r p q 1 max p q 1 0 q max p q and t denotes the conjugate index of t 1 as an application of this we further introduce certain hardy hausdorff spaces f h p q s r n and prove that the dual space of f h p q s r n is just f p q s q r n when p q 1

derivative strings contravariant aspect a concept termed strings which generalizes those of derivatives of scalars tensors and affine connections was introduced and studied in two previous papers by the authors the results obtained were essentially of a covariant nature in this paper a corresponding contravariant theory is developed and combined with the previous results a fairly complete theoretical framework is thereby established in particular the previously defined operations of intertwining and of covariant differen tiation for strings are generalized to arbitrary strings of mixed covariant contravariant type and further properties of these operations are derived to distinguish strings in the sense of this and the two foregoing papers from various other concepts that are also termed strings and that are now of great interest in physics and cosmology we shall occasionally refer to the present concept as derivative strings as has already been done in the title

three manifold invariants and their relation with the fundamental group we consider the 3 manifold invariant i m which is defined by means of the chern simons quantum field theory and which coincides with the reshetikhin turaev invariant we present some arguments and numerical results supporting the conjecture that for nonvanishing i m the absolute value i m only depends on the fundamental group 1 m of the manifold m for lens spaces the conjecture is proved when the gauge group is su 2 in the case in which the gauge group is su 3 we present numerical computations confirming the conjecture

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

l2 solvability of boundary value problems for divergence form parabolic equations with complex coefficients abstract we consider parabolic operators of the form t l l div a x t in r n 2 x t x x n 1 t r n r r x n 1 0 n 1 we assume that a is a n 1 n 1 dimensional matrix which is bounded measurable uniformly elliptic and complex and we assume in addition that the entries of a are independent of the spatial coordinate x n 1 as well as of the time coordinate t for such operators we prove that the boundedness and invertibility of the corresponding layer potential operators are stable on l 2 r n 1 c l 2 r n 2 c under complex l perturbations of the coefficient matrix subsequently using this general result we establish solvability of the dirichlet neumann and regularity problems for t l by way of layer potentials and with data in l 2 assuming that the coefficient matrix is a small complex perturbation of either a constant matrix or of a real and symmetric matrix

coordinate systems and taylor series in statistics a geometric theory of taylor series is presented and shown to reduce to the choice of a set of coordinates at each point of a family of probability distributions the so called coordinate string any family admits a naturally defined coordinate string which generalizes the natural coordinates of exponential families this concept gives a coordinate free description of the mobius derivatives of mccullagh cox and the connection strings of barndorff nielsen

new besov type spaces and triebel lizorkin type spaces including q spaces let s tau in mathbb r and q in 0 infty we introduce besov type spaces dot b s tau p q mathbb r n for p in 0 infty and triebel lizorkin type spaces dot f s tau p q mathbb r n rm for p in 0 infty which unify and generalize the besov spaces triebel lizorkin spaces and q spaces we then establish the varphi transform characterization of these new spaces in the sense of frazier and jawerth using the varphi transform characterization of dot b s tau p q mathbb r n rm and dot f s tau p q mathbb r n we obtain their embedding and lifting properties moreover for appropriate we also establish the smooth atomic and molecular decomposition characterizations of dot b s tau p q mathbb r n rm and dot f s tau p q mathbb r n for s in mathbb r p in 1 infty q in 1 infty and tau in 0 frac 1 max p q via the hausdorff capacity we introduce certain hardy hausdorff spaces b dot h s tau p q mathbb r n and prove that the dual space of b dot h s tau p q mathbb r n is just dot b s tau p q mathbb r n where t denotes the conjugate index of t in 1 infty

homogeneous endpoint besov space embeddings by hausdorff capacity and heat equation two embeddings of a homogeneous endpoint besov space are established via the hausdorff capacity and the heat equation meanwhile a co capacity formula and a trace inequality are derived from the besov space

square roots of elliptic second order divergence operators on strongly lipschitz domains l p theory we study l p estimates for square roots of second order elliptic non necessarily selfadjoint operators in divergence form l mathrm div a nabla on lipschitz domains subject to dirichlet or to neumann boundary conditions pursuing our work 4 where we considered operators on mathbb r n we obtain among other things l 1 2 f p le c nabla f p for all 1 p infty if l is real symmetric and the domain bounded which is new for 1 p 2 we also obtain similar results for perturbations of constant coefficients operators our methods rely on a singular integral representation calderon zygmund theory and quadratic estimates a feature of this study is the use of a commutator between the resolvent of the laplacian dirichlet and neumann and partial derivatives which carries the geometry of the boundary

necessary and sufficient conditions for absolute continuity of elliptic harmonic measure our purpose in this paper is to give a necessary and sufficient condition on the modulus of continuity of the coefficients of an elliptic operator in divergence form in order that the corresponding poisson kernel for the dirichlet problem exist this condition on the coefficients is global continuity together with the additional property that the modulus of continuity along some nontangential direction at each boundary point be bounded uniformly in these points and directions by a function 7 t satisfying the dini type condition

hardy and bmo spaces associated to divergence form elliptic operators consider a second order divergence form elliptic operator l with complex bounded measurable coefficients in general operators based on l such as the riesz transform or square function may lie beyond the scope of the calderon zygmund theory they need not be bounded in the classical hardy bmo and even some l p spaces in this work we develop a theory of hardy and bmo spaces associated to l which includes in particular a molecular decomposition maximal and square function characterizations duality of hardy and bmo spaces and a john nirenberg inequality

well posedness and regularity of generalized navier stokes equations in some critical q spaces we study the well posedness and regularity of the generalized navier stokes equations with initial data in a new critical space q 1 rn q rn n 12 1 which is larger than some known critical homogeneous besov spaces here q rn is a space defined as the set of all measurable functions with sup l i 2 1 n i i f x f y 2 x y n 2 1 dxdy where the supremum is taken over all cubes i with edge length l i and edges parallel to the coordinate axes in rn in order to study the well posedness and regularity we give a carleson measure characterization of q rn by investigating a new type of tent spaces and an atomic decomposition of the predual for q rn in addition our regularity results apply to the incompressible navier stokes equations with initial data in q 1 1 rn

square function estimates and the kato problem for second order parabolic operators in rn 1 in 4 the kato square root problem for second order complex uniformly elliptic operators of the form l div a del f with only bounded and measurable coefficients was solved the solution is a

invariants and likelihood ratio statistics because the likelihood ratio statistic is invariant under reparameterization it is possible to make a large sample expansion of the statistic itself and of its expectation in terms of invariants in particular the bartlett adjustment factor can be expressed in terms of invariant combinations of cumulants of the first two log likelihood derivatives such expansions are given first for a scalar parameter and then for vector parameters geometrical interpretation is given where possible and some special cases discussed

characterizations of besov type and triebel lizorkin type spaces via maximal functions and local means abstract let s r in this paper the authors first establish the maximal function characterizations of the besov type space b p q s r n with p q 0 and 0 the triebel lizorkin type space f p q s r n with p 0 q 0 and 0 the besov hausdorff space b h p q s r n with p 1 q 1 and 0 1 max p q and the triebel lizorkin hausdorff space f h p q s r n with p q 1 and 0 1 max p q where t denotes the conjugate index of t 1 using this characterization the authors further obtain the local mean characterizations of these function spaces via functions satisfying the tauberian condition and establish a fourier multiplier theorem on these spaces all these results generalize the existing classical results on besov and triebel lizorkin spaces by taking 0 and are also new even for q spaces and hardy hausdorff spaces

the a property of elliptic measures for operators with coefficients supported in whitney type cubes abstract it is proved the preservation of the a property with respect to surface measure of harmonic measure associated to second order divergence form operators with bounded measurable coefficients when the discrepancy between the main coefficients of the operators is supported on whitney type rectangles of the unit ball of r n in particular neither the discrepancy between the coefficients satisfies a carleson measure condition nor the coefficients of the operators involved are radially independent as in previous well known works as an application of the perturbation result we provide a class of operators with coefficients supported on whitney type rectangles for which the a property holds

generalized poincare embeddings and weighted hardy operator on qpα q spaces abstract the poincare embeddings w 1 n r n bmo r n and the john nirenberg inequality in bmo r n are classical results in modern analysis and partial differential equations this paper studies the generalized poincare embeddings and the john nirenberg inequality in q p q r n spaces furthermore for the weighted hardy operator u and its adjoint v this paper gives sufficient and necessary conditions on weight function to ensure the boundedness of them on q p q r n spaces

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

likelihood and observed geometries a partir du point de vue de la conditionnalite on etablit un parallele observe a la theorie de chentsov efron amari etc en utilisant l information observee et un tenseur d asymetrie observee

littlewood paley characterizations of triebel lizorkin morrey spaces via ball averages abstract let 1 p u and q 1 in this article the authors establish equivalent characterizations of the triebel lizorkin morrey space e u q p r n with smoothness order 0 2 in terms of the lusin area function and the g function generated by the difference between f x and its ball average b 2 j f x 1 b x 2 j b x 2 j f y d y j 1 2 x r n where for any j 1 2 and x r n b x 2 j y r n y x 2 j as applications the authors obtain several characterizations of e u p r n via pointwise inequalities involving ball averages

some new tent spaces and duality theorems for fractional carleson measures and qα rn several duality questions for fractional carleson measures and the spaces q rn are resolved using a new type of tent spaces these tent spaces are defined in terms of choquet integrals with respect to hausdorff capacity a predual for q rn is then defined as a space of distributions containing the hardy space h1 and an atomic decomposition is proved

some new function spaces and their applications to harmonic analysis abstract in this paper a family of spaces is introduced which seems well adapted for the study of a variety of questions related to harmonic analysis and its applications these spaces are the tent spaces they provide the natural setting for the study of such things as maximal functions the relevant space here is t p and also square functions where the space t2p is relevant as such these spaces lead to unifications and simplifications of some basic techniques in harmonic analysis thus they are closely related to lp and hardy spaces important parts of whose theory become corollaries of the description of tent spaces also as proc conf harmonic analysis cortona lect notes in math vol 992 springer verlag berlin new york 1983 already indicated where these spaces first appeared explicitly the tent spaces can be used to simplify some of the results related to the cauchy integral on lipschitz curves and multilinear analysis in retrospect one can recognize that various ideas important for tent spaces had been used if only implicitly for quite some time here one should mention carleson s inequality its simplifications and extensions the theory of hardy spaces and atomic decompositions

besov type spaces with variable smoothness and integrability abstract in this article the authors introduce besov type spaces with variable smoothness and integrability the authors then establish their characterizations respectively in terms of transforms in the sense of frazier and jawerth smooth atoms or peetre maximal functions as well as a sobolev type embedding as an application of their atomic characterization the authors obtain a trace theorem of these variable besov type spaces

generalized navier stokes equations with initial data in local q type spaces abstract in this paper we establish a link between leray mollified solutions of the three dimensional generalized navier stokes equations and mild solutions for initial data in the adherence of the test functions for the norm of q loc 1 r 3 this result applies to the usual incompressible navier stokes equations

infereni on full or partial parameters based on the standardized signed log likelihood ratio on etudie des tests et des limites de confiance bases sur la statistique du rapport de log vraisemblance signee en ajustant cette statistique de telle sorte qu a un ordre superieur d approximation la distribution normale standard soit valable