new abstract hardy spaces the aim of this paper is to propose an abstract construction of spaces which keep the main properties of the already known hardy spaces h1 we construct spaces through an atomic or molecular decomposition we prove some results about continuity from these spaces into l1 and some results about interpolation between these spaces and the lebesgue spaces we also obtain some results on weighted norm inequalities finally we present partial results in order to understand a characterization of the duals of hardy spaces

learning loopy graphical models with latent variables efficient methods and guarantees the problem of structure estimation in graphical models with latent variables is considered we characterize conditions for tractable graph estimation and develop efficient methods with provable guarantees we consider models where the underlying markov graph is locally tree like and the model is in the regime of correlation decay for the special case of the ising model the number of samples n required for structural consistency of our method scales as n 1 2 min log p where p is the number of variables min is the minimum edge potential is the depth i e distance from a hidden node to the nearest observed nodes and is a parameter which depends on the bounds on node and edge potentials in the ising model necessary conditions for structural consistency under any algorithm are derived and our method nearly matches the lower bound on sample requirements further the proposed method is practical to implement and provides flexibility to control the number of latent variables and the cycle lengths in the output graph

gaussian heat kernel upper bounds via the phragmen lindelof theorem we prove that in presence of l gaussian estimates so called davies gaffney estimates on diagonal upper bounds imply precise off diagonal gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces

factorization in lorentz spaces with an application to centralizers abstract we give an explicit formula to factorize functions in the lorentz spaces l p q some applications to centralizers twisted sums and interpolation theory are included

boundary layers on sobolev besov spaces and poisson s equation for the laplacian in lipschitz domains we study inhomogeneous boundary value problems for the laplacian in arbitrary lipschitz domains with data in sobolev besov spaces as such this is a natural continuation of work in jerison and kenig j funct anal 1995 16 219 where the inhomogeneous dirichlet problem is treated via harmonic measure techniques the novelty of our approach resides in the systematic use of boundary integral methods in this regard the key results are establishing the invertibility of the classical layer potential operators on scales of sobolev besov spaces on lipschitz boundaries for optimal ranges of indices applications tolp based helmholtz type decompositions of vector fields in lipschitz domains are also presented

on maximal regularity of type lp lq under minimal assumptions for elliptic non divergence operators we prove maximal regularity of type l p l q for operators in non divergence form with complex valued measurable coefficients on r n for a certain range of q which depends on dimension and the order of the operators this is done under the sole assumption that they generate an analytic semigroup in l q thus we give for this class of operators and this range of q a positive answer to brezis question whether generation of an analytic semigroup entails maximal l p regularity for other values of q we give several additional assumptions the proof relies on a result on maximal regularity under the assumption of suitable off diagonal bounds due to s blunck and the author which we improve here these off diagonal bounds are obtained by a modification of davies technique suited to cope with operators in non divergence form and they also imply new results on the scale of l q spaces in which a given operator generates an analytic semigroup we thus obtain a completely new approach to maximal l p regularity for operators of this kind in particular for those whose highest order coefficients belong to vmo moreover we obtain extensions of recent results due to kim and krylov for operators with measurable coefficients depending on one coordinate

weak type p p estimates for riesz transforms we consider the problem of l p boundedness of higher order riesz transforms associated to elliptic operators l of order 2m on as an application of the recently solved kato conjecture we show for all this generalizes the result of auscher and tchamitchian restricted to the case d 2m

boundary value problems for the laplacian in convex and semiconvex domains abstract we study the fully inhomogeneous dirichlet problem for the laplacian in bounded convex domains in r n when the size smoothness of both the data and the solution are measured on scales of besov and triebel lizorkin spaces as a preamble we deal with the dirichlet and regularity problems for harmonic functions in convex domains with optimal nontangential maximal function estimates as a corollary sharp estimates for the green potential are obtained in a variety of contexts including local hardy spaces a substantial part of this analysis applies to bounded semiconvex domains i e lipschitz domains satisfying a uniform exterior ball condition

potential theory on lipschitz domains in riemannian manifolds sobolev besov space results and the poisson problem abstract we continue a program to develop layer potential techniques for pde on lipschitz domains in riemannian manifolds building on l p and hardy space estimates established in previous papers here we establish sobolev and besov space estimates on solutions to the dirichlet and neumann problems for the laplace operator plus a potential on a lipschitz domain in a riemannian manifold with a metric tensor smooth of class c 1 for some 0 we treat the inhomogeneous problem and extend it to the setting of manifolds results obtained for the constant coefficient laplace operator on a lipschitz domain in euclidean space with the dirichlet boundary condition by d jerison and c kenig

complex interpolation of some quasi banach spaces abstract we interpolate in the complex method some real intermediate quasi banach spaces this enables us for example to get in a unified way complex interpolation of h p spaces 0 p 0 p 1 h p spaces could be the standard ones as well as weighted h p spaces h p spaces on product domains etc

nonlinear centralizers in homology it is shown that every nonlinear centralizer from l p to l q is trivial unless q p this means that if q ne p the only exact sequence of quasi banach l infty modules and homomorphisms 0 rightarrow l q rightarrow z rightarrow l p rightarrow 0 is the trivial one where z l q oplus l p from this it follows that the space of centralizers on l p is essentially independent on p in 0 infty which confirms a conjecture by kalton

the navier slip thin film equation for 3d fluid films existence and uniqueness abstract we consider the thin film equation t h h 2 h 0 in physical space dimensions i e one dimension in time t and two lateral dimensions with h denoting the height of the film in the third spatial dimension which corresponds to the lubrication approximation of the navier stokes equations of a three dimensional viscous thin fluid film with navier slip at the substrate this equation can have a free boundary the contact line moving with finite speed at which we assume a zero contact angle condition complete wetting regime previous results have focused on the 1 1 dimensional version where it has been found that solutions are not smooth as a function of the distance to the free boundary in particular a well posedness and regularity theory is more intricate than for the second order counterpart the porous medium equation or the thin film equation with linear mobility corresponding to darcy dynamics in the hele shaw cell here we prove existence and uniqueness of classical solutions that are perturbations of an asymptotically stable traveling wave profile this leads to control on the free boundary and in particular its velocity

weighted norm inequalities off diagonal estimates and elliptic operators part i general operator theory and weights this is the first part of a series of four articles in this work we are interested in weighted norm estimates we put the emphasis on two results of different nature one is based on a good inequality with two parameters and the other uses calderon zygmund decomposition these results apply well to singular non integral operators and their commutators with bounded mean oscillation functions singular means that they are of order 0 non integral that they do not have an integral representation by a kernel with size estimates even rough so that they may not be bounded on all lp spaces for 1 p 8 pointwise estimates are then replaced by appropriate localized lp lq estimates we obtain weighted lp estimates for a range of p that is different from 1 8 and isolate the right class of weights in particular we prove an extrapolation theorem a la rubio de francia for such a class and thus vector valued estimates

extending sobolev functions with partially vanishing traces from locally e δ domains and applications to mixed boundary problems we prove that given any k n for each open set rn and any closed subset d of such that is locally an e domain near d there exists a linear and bounded extension operator ek d mapping for each p 1 the space wdk p into wdk p rn here with o denoting either or rn the space wdk p o is defined as the completion in the classical sobolev space wk p o of restrictions to o of functions from cc rn whose supports are disjoint from d in turn this result is used to develop a functional analytic theory for the class wdk p including intrinsic characterizations boundary traces and extensions results interpolation theorems among other things which is then employed in the treatment of mixed boundary value problems formulated in locally e domains finally we also prove extension results on the scales of besov and bessel potential spaces on e domains with partially vanishing traces on ahlfors regular sets and explore some of the implications of such extension results

transmission problems and spectral theory for singular integral operators on lipschitz domains we prove the well posedness of the transmission problem for the laplacian across a lipschitz interface with optimal non tangential maximal function estimates for data in lebesgue and hardy spaces on the boundary as a corollary we show that the spectral radius of the adjoint harmonic double layer potential k in lp0 is less than 12 whenever is a bounded convex domain and 1 p 2

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

a high order characteristics upwind fv method for incompressible flow and heat transfer simulation on unstructured grids abstract this paper presents a new unstructured grid upwind finite volume algorithm for accurate numerical simulation of incompressible flows and convection heat transfer on unstructured grids it is an upwind method at both the differential equation level and discretized equation level based on the method of characteristics this is made possible with the introduction of chorin s j comput phys 2 1967 12 26 artificial compressibility formulation to the incompressible flow equations flow variables are calculated along characteristics and their initial values are interpolated based on the signs of the corresponding characteristic speed in addition an upwind biased interpolation method of third order accuracy is used for interpolating flow variables on unstructured grids with these inherent upwinding techniques for evaluating convection fluxes at control volume surfaces no artificial viscosity is required and the method is accurate and less sensitive to the grid orientation thus suitable for unstructured grid application the use of the finite volume method combined with unstructured grids makes the scheme very flexible in dealing with very complex boundary geometries while remaining to be fully conservative at the cell as well as global level the discretized equations are solved by an explicit multistage runge kutta time stepping scheme which is found to be efficient in terms of cpu and memory overheads a computer code has been developed which uses the numerical methods presented and a highly efficient edge based data structure for flux calculation and data storage a number of well documented test cases including 2d laminar flow in a channel with a backward facing step forced and natural convection flows have been calculated in order to validate the code and evaluate the performance of the numerical algorithm good agreement between the computed and measured results have been obtained for all the cases to demonstrate the accuracy of the third order scheme flow in another channel with a backward facing step studied by gartling int j numer meth fluids 1990 11 is simulated with both the third order and first order schemes it is shown that the third order scheme can produce a velocity profile across the channel that has perfect agreement with that of gartling local citation whilst the first order scheme cannot even predict the separation near the upper wall the proposed numerical methods are also used to calculate 3d incompressible inviscid flows and results show that the method is accurate and robust for general 3d simulation the 3d navier stokes code is later used to study a lid driven cavity flow at a reynolds number of 400 the unstructured grid used has 97 336 nodes and 455 625 tetrahedral elements with strong grid clustering near the walls the convergence rate is found to be satisfactory which shows that the method is also robust for 3d navier stokes simulation the results obtained agree well with those reported by jiang et al comput meth appl mech eng 114 109 1994 results obtained using the second order central scheme with artificial viscosity are also presented for comparison it is observed that the third order characteristics scheme is more accurate than the latter thus confirming its third order accuracy

lower upper implicit schemes with multiple grids for the euler equations on montre qu une methode de surrelaxation successive pour la resolution du schema implicite non factorise est equivalente a une forme modifiee du schema implicite inferieur superieur

a theory of complex interpolation for families of banach spaces abstract a detailed development is given of a theory of complex interpolation for families of banach spaces which extends the well known theory for pairs of spaces

robust estimation of latent tree graphical models inferring hidden states with inexact parameters latent tree graphical models are widely used in computational biology signal and image processing and network tomography here we design a new efficient estimation procedure for latent tree models including gaussian and discrete reversible models that significantly improves on previous sample requirement bounds our techniques are based on a new hidden state estimator that is robust to inaccuracies in estimated parameters more precisely we prove that latent tree models can be estimated with high probability in the so called kesten stigum regime with o log2n samples where n is the number of nodes

interpolation of fredholm operators we prove novel results on interpolation of fredholm operators including an abstract factorization theorem the main result of this paper provides sufficient conditions on the parameters 0 1 and q 1 under which an operator a is a fredholm operator from the real interpolation space x0 x1 q to y0 y1 q for a given operator a x0 x1 y0 y1 between compatible pairs of banach spaces such that its restrictions to the endpoint spaces are fredholm operators these conditions are expressed in terms of the corresponding indices generated by the k functional of elements from the kernel of the operator a in the interpolation sum x0 x1 if in addition a is invertible operator on endpoint spaces then these conditions are also necessary we apply these results to present a solution of the variant of a long standing problem of lions magenes for the real interpolation method we also discuss some applications to the spectral theory of operators as well as to perturbation of the hardy operator by identity on weighted lp spaces

weighted tent spaces with whitney averages factorization interpolation and duality in this paper we introduce a new scale of tent spaces which covers the weighted tent spaces of coifman meyer stein and of hofmann mayboroda mcintosh and some other tent spaces considered by dahlberg kenig pipher and auscher axelsson in studying boundary value problems for elliptic systems the strong factorizations within our tent spaces with applications to quasi banach complex interpolation and to multiplier duality theory are then established this way we unify and extend the corresponding results obtained by coifman meyer stein cohn verbitsky and hytonen rosen

weighted admissibility and wellposedness of linear systems in banach spaces we study linear control systems in infinite dimensional banach spaces governed by analytic semigroups for p in 1 infty and alpha in mathbb r we introduce the notion of l p admissibility of type alpha for unbounded observation and control operators generalizing earlier work by le merdy j london math soc 2 67 2003 pp 715 738 and haak and le merdy houston j math 31 2005 pp 1153 1167 we give conditions under which l p admissibility of type alpha is characterized by boundedness conditions which are similar to those in the well known weiss conjecture we also study l p wellposedness of type alpha for the full system here we use recent ideas due to pruss and simonett arch math basel 82 2004 pp 415 431 our results are illustrated by a controlled heat equation with boundary control and boundary observation where we take lebesgue and besov spaces as state space this extends the considerations in c i byrnes e j dynam control systems 8 2002 pp 341 370 to non hilbertian settings and to p neq2

latent variable graphical model selection via convex optimization suppose we observe samples of a subset of a collection of random variables no additional information is provided about the number of latent variables nor of the relationship between the latent and observed variables is it possible to discover the number of latent components and to learn a statistical model over the entire collection of variables we address this question in the setting in which the latent and observed variables are jointly gaussian with the conditional statistics of the observed variables conditioned on the latent variables being specified by a graphical model as a first step we give natural conditions under which such latent variable gaussian graphical models are identifiable given marginal statistics of only the observed variables essentially these conditions require that the conditional graphical model among the observed variables is sparse while the effect of the latent variables is spread out over most of the observed variables next we propose a tractable convex program based on regularized maximum likelihood for model selection in this latent variable setting the regularizer uses both the ell 1 norm and the nuclear norm our modeling framework can be viewed as a combination of dimensionality reduction to identify latent variables and graphical modeling to capture remaining statistical structure not attributable to the latent variables and it consistently estimates both the number of latent components and the conditional graphical model structure among the observed variables these results are applicable in the high dimensional setting in which the number of latent observed variables grows with the number of samples of the observed variables the geometric properties of the algebraic varieties of sparse matrices and of low rank matrices play an important role in our analysis

spectral theory and complex interpolation abstract some relations between the spectral theory for linear operators and the complex theory of interpolation for families of banach spaces are studied an invertibility theorem is proven and size estimates for the spectra of operators on the interpolation spaces are given

notes on limits of sobolev spaces and the continuity of interpolation scales we extend lemmas by bourgain brezis mironescu 2001 and maz ya shaposhnikova 2002 on limits of sobolev spaces to the setting of interpolation scales this is achieved by means of establishing the continuity of real and complex interpolation scales at the end points a connection to extrapolation theory is developed and a new application to limits of sobolev scales is obtained we also give a new approach to the problem of how to recognize constant functions via sobolev conditions

on the lp theory of c0 semigroups associated with second order elliptic operators ii abstract we study positive c 0 semigroups on l p associated with second order uniformly elliptic divergence type operators with singular lower order terms subject to a wide class of boundary conditions we obtain an interval p min p max in the l p scale where these semigroups can be defined including the case 2 p min p max we present an example showing that the result is optimal we also show that the semigroups are analytic with angles of analyticity and spectra of the generators independent of p for the whole range of p where the semigroups are defined

spectral multiplier theorems via h infty calculus and r bounds we prove spectral multiplier theorems for hormander classes mathcal h alpha p for 0 sectorial operators a on banach spaces assuming a bounded h infty sigma sigma calculus for some sigma in 0 pi and norm and certain r bounds on one of the following families of operators the semigroup e za on mathbb c the wave operators e isa for s in mathbb r the resolvent lambda a 1 on mathbb c backslash mathbb r the imaginary powers a it for t in mathbb r or the bochner riesz means 1 a u alpha for u 0 in contrast to the existing literature we neither assume that a operates on an l p scale nor that a is self adjoint on a hilbert space furthermore we replace generalized gaussian or poisson bounds and maximal estimates by the weaker notion of r bounds which allow for a unified approach to spectral multiplier theorems in a more general setting in this setting our results are close to being optimal moreover we can give a characterization of the r bounded mathcal h alpha 1 calculus in terms of r boundedness of bochner riesz means

navier stokes equations on lipschitz domains in riemannian manifolds the navier stokes equations are a system of nonlinear evolution equations modeling the flow of a viscous incompressible fluid one ingredient in the analysis of this system is the stationary linear system known as the stokes system a boundary value problem bvp that will be described in detail in the next section layer potential methods in smoothly bounded domains in euclidean space have proven useful in the analysis of the stokes system starting with work of odqvist and lichtenstein and including work of solonnikov and many others see the discussion in chapter iii of 10 and in 17 for the case of flow in regions with smooth boundary a treatment based on the modern language of pseudodifferential operators can be found in 18 in 1988 e fabes c kenig and g verchota 6 extended this classical layer potential approach to cover lipschitz domains in euclidean space in 6 the main result concerning the constant coefficient stokes system on lipschitz domains with connected boundary in euclidean space is the treatment of the l dirichlet boundary value problem and its regular version to achieve this the authors solve certain auxiliary neumann type problems and then exploit the duality between these and the original bvp s at the level of boundary integral operators p deuring and w von wahl 4 made use of the analysis in 6 to demonstrate the short time existence of solutions to the navier stokes equations in bounded lipschitz domains in threedimensional euclidean space it was necessary in 4 to include the hypothesis that the boundary be connected the hypothesis that the boundary be connected pervaded much work on the application of layer potentials to analysis on lipschitz domains it was certainly natural to speculate that this restriction was an artifact of the methods used and not partly supported by nsf grant dms 9870018 partially supported by nsf grant dms 9877077 1991mathematics subject classification primary 35q30 76d05 35j25 secondary 42b20 45e05

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

interpolation between hp spaces the complex method the corollary follows immediately from theorem 1 and fefferman s theorem see 111 that bmo is the dual space of h before proceeding to further results we make some comments on the history of theorem 1 unfortunately the theorem h lm e lp l p 1 0 has been folklore for many years and is stated e g in 2 221 this is due to a small but inopportune typographical error in 111 it is stated on page 157 of that paper that h lp lq l q 1 8 e p 1 p co

duality and interpolation of anisotropic triebel lizorkin spaces we study properties of anisotropic triebel lizorkin spaces associated with general expansive dilations and doubling measures on mathbb r n using wavelet transforms this paper is a continuation of bownik in j geom anal 2007 to appear trans am math soc 358 1469 1510 2006 where we generalized the isotropic methods of dyadic varphi transforms of frazier and jawerth j funct anal 93 34 170 1990 to non isotropic settings by working at the level of sequence spaces we identify the duals of anisotropic triebel lizorkin spaces we also obtain several real and complex interpolation results for these spaces

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

abstract capacitary estimates and the completeness and separability of certain classes of non locally convex topological vector spaces abstract we are concerned with establishing completeness and separability criteria for large classes of topological vector spaces which are typically non locally convex including lebesgue like spaces lorentz spaces orlicz spaces mixed normed spaces tent spaces and discrete triebel lizorkin and besov spaces for vector spaces of measurable functions we also derive pointwise convergence results our approach relies on abstract capacitary estimates and works in certain cases of interest even in the absence of a background measure space and or of a vector space structure

stability results on interpolation scales of quasi banach spaces and applications we investigate the stability of fredholm properties on interpolation scales of quasi banach spaces this analysis is motivated by problems arising in pde s and several applications are presented

a solution to the problem of l p maximal regularity we give a negative solution to the problem of the l p maximal regularity on various classes of banach spaces including l q spaces with 1 q neq 2 infty

learning latent tree graphical models we study the problem of learning a latent tree graphical model where samples are available only from a subset of variables we propose two consistent and computationally efficient algorithms for learning minimal latent trees that is trees without any redundant hidden nodes unlike many existing methods the observed nodes or variables are not constrained to be leaf nodes our algorithms can be applied to both discrete and gaussian random variables and our learned models are such that all the observed and latent variables have the same domain state space our first algorithm recursive grouping builds the latent tree recursively by identifying sibling groups using so called information distances one of the main contributions of this work is our second algorithm which we refer to as clgrouping clgrouping starts with a pre processing procedure in which a tree over the observed variables is constructed this global step groups the observed nodes that are likely to be close to each other in the true latent tree thereby guiding subsequent recursive grouping or equivalent procedures such as neighbor joining on much smaller subsets of variables this results in more accurate and efficient learning of latent trees we also present regularized versions of our algorithms that learn latent tree approximations of arbitrary distributions we compare the proposed algorithms to other methods by performing extensive numerical experiments on various latent tree graphical models such as hidden markov models and star graphs in addition we demonstrate the applicability of our methods on real world data sets by modeling the dependency structure of monthly stock returns in the s p index and of the words in the 20 newsgroups data set

learning hidden variable networks the information bottleneck approach a central challenge in learning probabilistic graphical models is dealing with domains that involve hidden variables the common approach for learning model parameters in such domains is the expectation maximization em algorithm this algorithm however can easily get trapped in sub optimal local maxima learning the model structure is even more challenging the structural em algorithm can adapt the structure in the presence of hidden variables but usually performs poorly without prior knowledge about the cardinality and location of the hidden variables in this work we present a general approach for learning bayesian networks with hidden variables that overcomes these problems the approach builds on the information bottleneck framework of tishby et al 1999 we start by proving formal correspondence between the information bottleneck objective and the standard parametric em functional we then use this correspondence to construct a learning algorithm that combines an information theoretic smoothing term with a continuation procedure intuitively the algorithm bypasses local maxima and achieves superior solutions by following a continuous path from a solution of an easy and smooth target function to a solution of the desired likelihood function as we show our algorithmic framework allows learning of the parameters as well as the structure of a network in addition it also allows us to introduce new hidden variables during model selection and learn their cardinality we demonstrate the performance of our procedure on several challenging real life data sets

weighted norm inequalities off diagonal estimates and elliptic operators part iii harmonic analysis of elliptic operators this is the third part of a series of four articles on weighted norm in equalities o diagonal estimates and elliptic operators for l in some class of elliptic operators we study weighted norm l p inequalities for singular non integral opera tors arising from l those are the operators l for bounded holomorphic functions the riesz transforms rl 1 2 or 1 2 l 1 2 and its inverse l 1 2 1 2 some quadratic functionals gl and gl of littlewood paley stein type and also some vector valued inequalities such as the ones involved for maximal l p regularity for each we obtain sharp or nearly sharp ranges of p using the general theory for bound edness of part i and the o diagonal estimates of part ii we also obtain commutator results with bmo functions

sharp hodge decompositions maxwell s equations and vector poisson problems on nonsmooth three dimensional riemannian manifolds we solve three basic potential theoretic problems hodge decompositions for vector fields poisson problems for the hodge laplacian and inhomogeneous maxwell equations in arbitrary lipschitz subdomains of a smooth compact three dimensional riemannian manifold in each case we derive sharp estimates on sobolev besov scales and establish integral representation formulas for the solution the proofs rely on tools from harmonic analysis and algebraic topology such as calderon zygmund theory and de rham theory

erratum to perturbation and interpolation theorems for the h calculus with applications to differential operators we prove comparison theorems for the h calculus that allow to transfer the property of having a bounded h calculus from one sectorial operator to another the basic technical ingredient are suitable square function estimates these comparison results provide a new approach to perturbation theorems for the h calculus in a variety of situations suitable for applications our square function estimates also give rise to a new interpolation method the rademacher interpolation we show that a bounded h calculus is characterized by interpolation of the domains of fractional powers with respect to rademacher interpolation this leads to comparison and perturbation results for operators defined in interpolation scales such as the l p scale we apply the results to give new proofs on the h calculus for elliptic differential operators including schrodinger operators and perturbed boundary conditions as new results we prove that elliptic boundary value problems with bounded uniformly coefficients have a bounded h calculus in certain sobolev spaces and that the stokes operator on bounded domains with c 1 1 has a bounded h calculus in the helmholtz scale l p p 1

a distance between orbits that controls commutator estimates and invertibility of operators abstract a distance between orbit spaces generated by a single element is introduced and it is shown that if an operator is invertible in one orbit it is also invertible in nearby orbits thus proving a version of shneiberg s theorem for orbital methods the same machinery is used to extend the celebrated rochberg weiss commutator theorem to the setting of orbital methods it is shown that these results apply to the real and complex methods of interpolation by proving that these methods can be suitably obtained as orbits of a single element

the stationary navier stokes system in nonsmooth manifolds the poisson problem in lipschitz and c 1 domains we consider the linearized version of the stationary navier stokes equations on a subdomain of a smooth compact riemannian manifold m the emphasis is on regularity the boundary of is assumed to be only c1 and even lipschitz and the data are selected from appropriate sobolev besov scales our approach relies on the method of boundary integral equations suitably adapted to the variable coefficient setting we are considering here applications to the stationary nonlinear navier stokes equations in this context are also discussed

the poisson problem with mixed boundary conditions in sobolev and besov spaces in non smooth domains we introduce certain sobolev besov spaces which are particularly well adapted for measuring the smoothness of data and solutions of mixed boundary value problems in lipschitz domains in particular these are used to obtain sharp well posedness results for the poisson problem for the laplacian with mixed boundary conditions on bounded lipschitz domains which satisfy a suitable geometric condition introduced by r brown in 1994 in this context we obtain results which generalize those by d jerison and c kenig 1995 as well as e fabes o mendez and m mitrea 1998 applications to hodge theory and the regularity of green operators are also presented

potential theory on lipschitz domains in riemannian manifolds the case of dini metric tensors we study the applicability of the method of layer potentials in the treatment of boundary value problems for the laplace beltrami operator on lipschitz sub domains of riemannian manifolds in the case when the metric tensor g jk dx j desk has low regularity under the assumption that g jk x g jk y c x y where the modulus of continuity satisfies a dini type condition we prove the well posedness of the classical dirichlet and neumann problems with l p boundary data for sharp ranges of p s and with optimal nontangential maximal function estimates

fredholm toeplitz operators with vmo symbols and the duality of generalized fock spaces with small exponents we characterize fredholmness of toeplitz operators acting on generalized fock spaces of the n dimensional complex space for symbols in the space of vanishing mean oscillation vmo our results extend the recent characterizations for toeplitz operators on standard weighted fock spaces to the setting of generalized weight functions and also allow for unbounded symbols in vmo for the first time another novelty is the treatment of small exponents 0 p 1 which to our knowledge has not been seen previously in the study of the fredholm properties of toeplitz operators on any function spaces we accomplish this by describing the dual of the generalized fock spaces with small exponents

hardy spaces and the neumann problem in l p for laplace s equation in lipschitz domains on donne des resultats optimaux pour la resolubilite du probleme de neumann dans des domaines de lipschitz a donnees dans l p on obtient des resultats du point extreme correspondant pour des espaces de hardy

the kato square root problem for mixed boundary conditions we consider the negative laplacian subject to mixed boundary conditions on a bounded domain we prove under very general geometric assumptions that slightly above the critical exponent 12 its fractional power domains still coincide with suitable sobolev spaces of optimal regularity in combination with a reduction theorem recently obtained by the authors this solves the kato square root problem for elliptic second order operators and systems in divergence form under the same geometric assumptions thereby we answer a question posed by j l lions in 1962 30

weighted norm inequalities off diagonal estimates and elliptic operators part iv riesz transforms on manifolds and weights this is the fourth article of our series here we study weighted norm inequalities for the riesz transform of the laplace beltrami operator on riemannian manifolds and of subelliptic sum of squares on lie groups under the doubling volume property and gaussian upper bounds

necessary and sufficient conditions for invertibility of operators in spaces of real interpolation let a be a bounded linear operator from a couple x 0 x 1 to a couple y 0 y 1 such that the restrictions of a on the end spaces x 0 and x 1 have bounded inverses defined on y 0 and y 1 respec