a bounded approximation of weakly compact operators abstract we study bounded approximation properties of operators it is shown that for every weakly compact operator t on a banach space x if t is uniformly approximated on each compact subset of x by finite rank operators then t 2 can be approximated in the same topology by finite rank operators with the operator norm less than or equal to t 2

ideals in l l_1 the main result is that there are infinitely many in fact a continuum of closed two sided ideals in the banach algebra l l 1 of bounded linear operators on l 1 0 1 this answers a question from pietsch s 1978 book operator ideals the proof also shows that l c 0 1 contains a continuum of closed ideals finally a duality argument yields that l ell infty has a continuum of closed ideals

λ q processes motivated by some classical notions in harmonic analysis a q processes are introduced in the context of a study of stochastic interdependencies an extension of a classical theorem of salem and zygmund regarding random fourier series is obtained the littlewood exponent of a q processes is estimated and in some archetypical cases computed 0 introduction in 1 we considered stochastic processes with respect to which every deterministic function on 0 1 was stochastically integrable such processes x were normed by iixii sup e e1 ej x tj x tj1 n 0 ej 1 1 n 0 to x e x 0 received by the editors september 20 1988 1980 mathematics subject classification 1985 revision primary 60g05 60g 17 42a6 1 research partially supported by nsf grant dms 8601485 i 1990 american mathematical society 0002 9947 90 1 00 25 per page

on the distance of subspaces of ⁿ_ to _ it is proved that if in is well isomorphic to x e0 y and x either p has small dimension or is a euclidean space then y is well isomorphic to k k dim y the proofs use new forms of the finite dimensional decomposition method it is shown that the constant of equivalence between a normalized kunconditional basic sequence in ipn and a subsequence of the unit vector basis p of in is greatest up to a constant depending on k when the sequence spans a 2 euclidean space

estimates for the square variation of partial sums of fourier series and their rearrangements we investigate the square variation operator v2 which majorizes the partial sum maximal operator on general orthonormal systems ons of size n we prove that the l2 norm of the v2 operator is bounded by o ln n on any ons this result is sharp and refines the classical rademacher menshov theorem we show that this can be improved to o ln n for the trigonometric system which is also sharp we show that for any choice of coefficients this truncation of the trigonometric system can be rearranged so that the l2 norm of the associated v2 operator is o lnln n we also show that for p 2 a bounded ons of size n can be rearranged so that the l2 norm of the vp operator is at most op lnln n uniformly for all choices of coefficients this refines bourgain s work on garsia s conjecture which is equivalent to the v case several other results on operators of this form are also obtained the proofs rely on combinatorial and probabilistic methods

new concentration inequalities in product spaces we introduce three new ways to measure the distance from a point to a subset of a product space and we prove corresponding concentration inequalities each of them allows to control the fluctuation of a new class of random variables

robust norm equivalencies for diffusion problems additive multilevel methods offer an efficient way for the fast solution of large sparse linear systems which arise from a finite element discretization of an elliptic boundary value problem these solution methods are based on multilevel norm equivalencies for the associated bilinear form using a suitable subspace decomposition to obtain a robust iterative scheme it is crucial that the constants in the norm equivalence do not depend or depend only weakly on the ellipticity constants of the problem in this paper we present such a robust norm equivalence for the model problem w u with a scalar diffusion coefficient in c r 2 our estimates involve only very weak information about w and the results are applicable for a large class of diffusion coefficients namely we require to be in the muckenhoupt class a 1 a function class well studied in harmonic analysis the presented multilevel norm equivalencies are a main step towards the realization of an optimal and robust multilevel preconditioner for scalar diffusion problems

necessary conditions for vector valued operator inequalities in harmonic analysis via a random construction we establish necessary conditions for l p ell q inequalities for certain families of operators arising in harmonic analysis in particular we consider dilates of a convolution kernel with compactly supported fourier transform vector maximal functions acting on classes of entire functions of exponential type and a characterization of sobolev spaces by square functions and pointwise moduli of smoothness

regularity of gaussian processes on donne la validite de la conjecture de fernique 1974 sur l existence de mesures de majoration pouvant caracteriser la limitation des processus gaussiens

integer cells in convex sets abstract every convex body k in r n has a coordinate projection pk that contains at least vol 1 6 k cells of the integer lattice p z n provided this volume is at least one our proof of this counterpart of minkowski s theorem is based on an extension of the combinatorial density theorem of sauer shelah and vapnik chervonenkis to z n this leads to a new approach to sections of convex bodies in particular fundamental results of the asymptotic convex geometry such as the volume ratio theorem and milman s duality of the diameters admit natural versions for coordinate sections

an auxiliary space multigrid preconditioner for the weak galerkin method abstract in this paper we construct an auxiliary space multigrid preconditioner for the weak galerkin method for second order diffusion equations discretized on simplicial 2d or 3d meshes the idea of the auxiliary space multigrid preconditioner is to use an auxiliary space as a coarse space in the multigrid algorithm where the discrete problem in the auxiliary space can be easily solved by an existing solver in our construction we conveniently use the h 1 conforming piecewise linear finite element space as an auxiliary space the main technical difficulty is to build the connection between the weak galerkin discrete space and the h 1 conforming piecewise linear finite element space we successfully constructed such an auxiliary space multigrid preconditioner for the weak galerkin method as well as the reduced system of the weak galerkin method involving only the degrees of freedom on edges faces the preconditioned systems are proved to have condition numbers independent of the mesh size numerical experiments further support the theoretical results

on the robustness of the bpx preconditioner with respect to jumps in the coeffiencients we determine the worst case behavior of the standard bpx preconditioner for elliptic problems with arbitrary coefficient jumps along the boundaries of the coarsest partition the counterexamples are also useful for other problems

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

approximation of zonoids by zonotopes on etudie une propriete d approximation des corps convexes de r n qui sont des limites de sommes de segments

restriction and kakeya phenomena for finite fields the restriction and kakeya problems in euclidean space have received much attention in the last few decades and they are related to many problems in harmonic analysis partial differential equations pdes and number theory in this paper we initiate the study of these problems on finite fields in many cases the euclidean arguments carry over easily to the finite setting and are in fact somewhat cleaner but there are some new phenomena in the finite case which deserve closer study

lacunary sets and function spaces with finite cotype abstract we investigate translation invariant subspaces of the space of uniformly convergent fourier series and orlicz spaces with finite cotype in the case of orlicz spaces this leads to some new characterizations of p rider sets

a recovery based a posteriori error estimator for h curl interface problems abstract this paper introduces a new recovery based a posteriori error estimator for the lowest order nedelec finite element approximation to the h curl interface problem the error estimator is analyzed by establishing both the reliability and the efficiency bounds and is supported by numerical results under certain assumptions it is proved that the reliability and efficiency constants are independent of the jumps of the coefficients

characterizations of hankel multipliers we give characterizations of radial fourier multipliers as acting on radial l p functions 1 p 2d d 1 in terms of lebesgue space norms for fourier localized pieces of the convolution kernel this is a special case of corresponding results for general hankel multipliers besides l p l q bounds we also characte rize weak type inequalities and intermediate inequalities involving lorentz spaces applications include results on interpolation of multiplier spaces

schwarz methods of neumann neumann type for three dimensional elliptic finite element problems several domain decomposition methods of neumann neumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by finite elements we will only consider problems in three dimensions several new variants of the basic algorithm are introduced in a schwarz method framework that provides tools which have already proven very useful in the design and analysis of other domain decomposition and multi level methods the neumann neumann algorithms have several advantages over other domain decomposition methods the subregions which define the subproblems only share the boundary degrees of freedom with their neighbors the subregions can also be of quite arbitrary shape and many of the major components of the preconditioner can be constructed from subprograms available in standard finite element program libraries in its original form however the algorithm lacks a mechanism for global transportation of information and its performance therefore suffers when the number of subregions increases in the new variants of the algorithms considered in this paper the preconditioners include global components of low rank to overcome this difficulty bounds are established for the condition number of the iteration operator which are independent of the number of subregions and depend only polylogarithmically on the number of degrees of freedom of individual local subproblems results are also given for problems with arbitrarily large jumps in the coefficients across the interfaces separating the subregions 1995 john wiley sons inc

the square variation of rearranged fourier series we prove that there exists a rearrangement of the first n elements of the trigonometric system such that the l 2 norm of the square variation operator is at most o epsilon log 9 22 epsilon n this is an improvement over o log 1 2 n from the canonical ordering

on the hardy littlewood majorant problem for random sets abstract we study the classical hardy littlewood majorant problem for trigonometric polynomials we show that the constant in the majorant inequality grows at most like an arbitrary small power of the degree provided the spectrum is chosen at random we also give an example of a deterministic set where the majorant property fails i e the constant grows like a fixed small power in the degree

λ p spaces abstract we say that a closed subspace h of the space l p l p 0 1 1 p is a p space if in h convergence in l p norm is equivalent to convergence in measure mainly we focus on the problem when the unit ball b h f h f p 1 of a p space h has equi absolutely continuous norms in l p moreover assuming that a rearrangement invariant space x is embedded into l p 1 p and that the inclusion operator i x l p fails to be strictly singular we are interested in what we can say about the properties of subspaces on which the norms of x and l p are equivalent we reveal their essential dependence on the value of p which resembles the difference in the classical theorems of bourgain and bachelis ebenstein on p sets

the strong approximation property and the weak bounded approximation property abstract we show that the strong approximation property strong ap respectively strong cap and the weak bounded approximation property respectively weak bcap are equivalent for every banach space this gives a negative answer to oja s conjecture as a consequence we show that each of the spaces c 0 and l 1 has a subspace which has the ap but fails to have the strong ap

corrigendum and addendum to the strong approximation property and the weak bounded approximation property j funct anal 266 2014 5439 5447 abstract we correct a mistake in the proof of 2 theorem 1 1 and a proof of the main result in 2 is presented without depending on 2 theorem 1 1