norm estimates for operators from hp to lq abstract we give upper and lower estimates of the norm of a bounded linear operator from the hardy space h p to l q in terms of the norm of the rows and the columns of its associated matrix in certain vector valued sequence spaces

a lattice renorming theorem and applications to vector valued processes a norm 11 11 on a banach space e is said to be locally uniformly convex if iixnii lxii and iix xli 211xii implies that xn x in norm it is shown that a banach lattice has an order equivalent locally uniformly convex norm if and only if the lattice is order continuous this result is used to reduce convergence theorems for lattice valued positive martingales and submartingales to the scalar case 0 introduction a norm j on a banach space e is said to have the kadec klee property sometimes property h if whenever xn x weakly and llx ll lix ll then x0 x strongly the aim of this paper is to show that if e has an equivalent kadec klee norm then one may obtain convergence theorems for e valued random processes xj whenever j ixjj is a well known real valued convergent process and whenever the limit can be identified in the banach space for instance the classical kadec renorming theorem for separable banach spaces gives the convergence of vector valued martingales uniform amarts and additive processes ergodic theorem since their norms are real valued submartingales amarts and subadditive processes respectively while the identification of the limit requires the radonnikodym property on the space for the first two processes the limit of the third process exists in any banach space if now we consider positive submartingales and subadditive processes valued in a banach lattice one needs that the equivalent kadec klee norm be also a lattice norm in order to conclude that the norms of these processes are real valued submartingales and subadditive processes respectively 1 deals with the existence of such a lattice renorming while in 1i we show how this leads to a unified approach for proving the almost sure convergence of the processes mentioned above i renorming order continuous lattices a norm jj 1 on a banach space e is said to be locally uniformly convex if jjx jj i jjxii and iix xnjj 2j1xii imply that xn x strongly this notion is clearly stronger than the kadec klee property received by the editors october 15 1979 and in revised form february 11 1980 presented to the society san antonio january 1980 ams mos subject classifications 1970 primary 46b99 60g99

l2 estimates for the finite element method for the dirichlet problem with singular data in this paper we consider the approximation by the finite element method of second order elliptic problems on convex domains and homogeneous dirichlet condition on the boundary in these problems the data are borel measures using a quasiuniform mesh of finite elements and polynomials of degree 1 we prove that in two dimensions the convergence is of orderh inl 2 and in three dimensions of orderh 1 2

norm estimates for matrix operators between banach spaces abstract we study norm estimates of matrix operators between banach spaces in particular we prove a general variant of the hardy littlewood result on lower estimate for norms of operators between l p spaces and show applications to calderon lozanovskii sequence lattices we also provide estimates for norms of radially generated operators from abstract hardy spaces to banach sequence lattices likewise we prove the mean convergence theorem in abstract hardy spaces and show general examples of radially generated operators

force traction microscopy an inverse problem with pointwise observations abstract force traction microscopy is an inversion method that allows one to obtain the stress field applied by a living cell on the environment on the basis of a pointwise knowledge of the displacement produced by the cell itself this classical biophysical problem usually addressed in terms of green functions can be alternatively tackled using a variational framework and then a finite elements discretization in such a case a variation of the error functional under suitable regularization is operated in view of its minimization this setting naturally suggests the introduction of a new equation based on the adjoint operator of the elasticity problem in this paper we illustrate the rigorous theory of the two dimensional and three dimensional problems involving in the former case a distributed control and in the latter case a surface control the pointwise observations require one to exploit the theory of elasticity extended to forcing terms that are borel measures

characterizations of kadec klee properties in symmetric spaces of measurable functions we present several characterizations of kadec klee properties in symmetric function spaces on the half line based on the k functional of j peetre in addition to the usual kadec klee property we study those symmetric spaces for which sequential convergence in measure respectively local convergence in measure on the unit sphere coincides with norm convergence

derivation of conservation laws from nonlocal symmetries of differential equations an identity is derived which yields a correspondence between symmetries and conservation laws for self adjoint differential equations this identity does not rely on use of a lagrangian as needed to obtain conservation laws by noether s theorem moreover unlike noether s theorem which can only generate conservation laws from local symmetries the derived identity generates conservation laws from nonlocal as well as local symmetries it is explicitly shown how noether s theorem is extended by the identity conservation laws arising from nonlocal symmetries are obtained for a class of scalar wave equations with variable wave speeds the constants of motion resulting from these nonlocal conservation laws are shown to be linearly independent of all constants of motion resulting from local conservation laws

exact solutions for wave equations of two layered media with smooth transition the wave equation c2 x uxx utt 0 is solved for wave speeds c x corresponding to two layered media with smooth transition from layer to layer the wave speed c x has four free parameters to fit a given medium solutions are constructed from invariant solutions of a related system of first order partial differential equations that admit a four parameter symmetry group these solutions are superposed to solve general initial value problems for data with compact support the computation of the superposition coefficients uses elementary fourier analysis solutions are illustrated for various initial conditions

nonlocally related systems linearization and nonlocal symmetries for the nonlinear wave equation abstract the nonlinear wave equation u t t c 2 u u x x arises in various physical applications ames et al w f ames r j lohner e adams group properties of u t t f u u x x int j nonlin mech 16 1981 439 447 did the complete group classification for its admitted point symmetries with respect to the wave speed function c u and as a consequence constructed explicit invariant solutions for some specific cases by considering conservation laws for arbitrary c u we find a tree of nonlocally related systems and subsystems which include related linear systems through hodograph transformations we use existing work on such related linear systems to extend the known symmetry classification in w f ames r j lohner e adams group properties of u t t f u u x x int j nonlin mech 16 1981 439 447 to include nonlocal symmetries moreover we find sets of c u for which such nonlinear wave equations admit further nonlocal symmetries and hence significantly further extend the group classification of the nonlinear wave equation

weak strong continuity of multilinear mappings and the pelczynski pitt theorem abstract in generalizing a result of pelczynski the sequential weak to norm continuity ofn linear mappings between certain banach spaces including spaces of type and cotype will be studied in particular it is shown that everyn linear continuous mapping from lp1 lpninto lqis compact if and only if i 1n 1 pi

control of an elliptic problem with pointwise state constraints this paper deals with a quadratic control problem for elliptic equations with pointwise state constraints existence and uniqueness of the solution is proved optimality conditions are given and regularity of the optimal solution is investigated

equivalence constants for certain matrix norms we obtain optimal equivalence constants for certain families of matrix norms our results generalize inequalities of goldberg linear and multilinear algebra 21 1987 173

an interpolation approach to hardy littlewood inequalities for norms of operators on sequence spaces we revisit estimates of hardy and littlewood for norms of operators on sequence spaces and exploit recent advances in interpolation theory to provide relatively simple proofs

on geometric structure of symmetric spaces abstract in this article we discuss local approach to strict k monotonicity and local uniform rotundity in symmetric spaces we prove several general results on local structure of symmetric spaces e showing relation between strict monotonicity and strict k monotonicity and the kadec klee property for global convergence in measure we also present the full criteria for points of upper k monotonicity in lorentz spaces p w for degenerated weight function w next we characterize local uniform rotundity in symmetric spaces e proving several correspondences between x e a point of local uniform rotundity and its decreasing rearrangement x and absolute value x finally we apply these results to find complete criteria for local uniform rotundity of lorentz spaces p w

optimal exponents for hardy littlewood inequalities for m linear operators the hardy littlewood inequalities on lplp spaces provide optimal exponents for some classes of inequalities for bilinear forms on lplp spaces in this paper we investigate in detail the exponents involved in hardy littlewood type inequalities and provide several optimal results that were not achieved by the previous approaches our first main result asserts that for q1 qm 0q1 qm 0 and an infinite dimensional banach space y attaining its cotype cot ycot y if 1p1 1pm 1cot y then the following assertions are equivalent a there is a constant cp1 pmy 1 such that j1 1 j2 1 jm 1 a ej1 ejm qm qm 1qm q1q2 1q1 cp1 pmy a for all continuous m linear operators a lp1 lpm y a lp1 lpm y b the exponents q1 qmq1 qm satisfy q1 m cot yp1 pm q2 m 1 cot yp2 pm qm 1 cot ypm where for k 1 m k 1 m m k 1 cot ypk pm cot y1 1pk 1pm cot y as an application of the above result we generalize one of the classical hardy littlewood inequalities for bilinear forms to the m linear setting our result is sharp in a very strong sense the constants and exponents are optimal even if we consider mixed sums

on the interpolation of injective or projective tensor products of banach spaces abstract we prove a general result on the factorization of matrix valued analytic functions we deduce from it that if e 0 e 1 and f 0 f 1 are interpolation pairs with dense intersections then under some conditions on the spaces e i f i we have e 0 bo f 0 e 1 bo f 1 e 0 e 1 bo f 0 f 1 0 we find also conditions on the spaces e i f i so that the following holds e 0 bo f 0 e 1 bo f 1 e 0 e 1 bo f 0 f 1 0 some applications of these results are also considered

new classes of symmetries for partial differential equations new classes of symmetries for partial differential equations are introduced by writing a given partial differential equation s in a conserved form a related system t with potentials as additional dependent variables is obtained the lie group of point transformations admitted by t induces a symmetry group of s new symmetries may be obtained for s that are neither point nor lie backlund symmetries they are determined by a completely algorithmic procedure significant new symmetries are found for the wave equation with a variable wave speed and the nonlinear diffusion equation

inverse scattering from arbitrary two dimensional objects in stratified environments via a green s operator an important problem in geophysics medical imaging and nondestructive imaging today is the construction of a practical accurate and efficient means of imaging geophysical anomalies tumours or material defects in layered media this paper discusses such a method the use of a stratified green s function for the solution of the forward problem is detailed this forward problem is then incorporated into an efficient and accurate inversion algorithm based on optimization the method is nonperturbative unlike diffraction tomography which relies on linearization to make the problem tractable in the inversion a pair of lippmann schwinger like integral equations are solved simultaneously via the galerkin procedure for the unknown total internal fields and speed distribution the computational burden is high but made manageable by utilizing biconjugate gradients fast fourier transforms and sinc basis functions to speed up the solution of the forward problem the size and contrasts for which the me

on invariance properties of the wave equation a complete group classification is given of both the wave equation c2 x uxx utt 0 i and its equivalent system vt ux c2 x vx ut ii when the wave speed c x const equations i and ii admit either a two or four parameter group for the exceptional case c x ax b 2 equation i admits an infinite group equations i and ii do not always admit the same group for a given c x the group for i can have more parameters or fewer parameters than that for ii moreover the groups can be different with the same number of parameters separately for i and ii all possible c x that admit a four parameter group are found explicitly the corresponding invariant similarity solutions are considered some of these wave speeds have realistic physical properties c x varies monotonically from one positive constant to another positive constant as x goes from to

on mapping linear partial differential equations to constant coefficient equations a constructive algorithm is developed to determine whether or not a given linear p d e can be mapped into a linear p d e with constant coefficients the algorithm is based on analyzing the infinitesimals of the lie group of transformations leaving invariant the given p d e as consequences in two dimensions necessary and sufficient conditions are given for mapping 1 a parabolic p d e into the heat equation 2 a hyperbolic p d e into the wave equation 3 an elliptic p d e into laplace s equation or the helmholtz equation the corresponding mappings are given explicitly

hardy littlewood inequalities for norms of positive operators on sequence spaces abstract we consider estimates of hardy and littlewood for norms of operators on sequence spaces and we apply a factorization result of maurey to obtain improved estimates and simplified proofs for the special case of a positive operator

well posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks we prove a regularity result for the anisotropic linear elasticity equation p u rm div left boldmath mathsf c cdot nabla u right f with mixed displacement and traction boundary conditions on a curved polyhedral domain omega subset mathbb r 3 in weighted sobolev spaces mathcal k m 1 a 1 omega for which the weight is given by the distance to the set of edges in particular we show that there is no loss of mathcal k m a regularity our curved polyhedral domains are allowed to have cracks we establish a well posedness result when there are no neighboring traction boundary conditions and a 0 that depends on p on the boundary conditions and on the domain our results extend to other strongly elliptic systems and higher dimensions

boundary control of semilinear elliptic equations with pointwise state constraints this paper is concerned with state constrained optimal control problems of semilinear elliptic equations the control being on the boundary optimality conditions are derived and regularity of the optimal solution is investigated

equivalence constants for matrix norms a problem of goldberg abstract we investigate the equivalence constants for the l p coefficient norms and l q operator norms 1 p q of complex m n matrices and provide estimates which are either best possible or close to best possible