optimal domain for the hardy operator abstract we study the optimal domain for the hardy operator considered with values in a rearrangement invariant space in particular this domain can be represented as the space of integrable functions with respect to a vector measure defined on a ring a precise description is given for the case of the minimal lorentz spaces

choquet type l 1 spaces of a vector capacity abstract given a set function with values in a banach space x we construct an integration theory for scalar functions with respect to by using duality on x and choquet scalar integrals our construction extends the classical bartle dunford schwartz integration for vector measures since just the minimal necessary conditions on are required several l 1 spaces of integrable functions associated to appear in such a way that the integration map can be defined in them we study the properties of these spaces and how they are related we show that the behavior of the l 1 spaces and the integration map can be improved in the case when x is an order continuous banach lattice providing new tools for non linear operator theory and information sciences

completeness ofl1 spaces for measures with values in complex vector spaces we provide a simple and direct proof of the completeness of the space of c valued integrable functions with respect to a vector measure taking its values in a complex frechet space a similar result is established via a quite different argument for closed measures taking their values in more general sequentially complete locally convex hausdorff spaces over c

bounds on the joint and generalized spectral radius of the hadamard geometric mean of bounded sets of positive kernel operators abstract let 1 m be bounded sets of positive kernel operators on a banach function space l we prove that for the generalized spectral radius and the joint spectral radius the inequalities 1 1 m m 1 m 1 2 m 1 m 1 1 m m 1 m 1 2 m 1 m hold where 1 1 m m 1 m denotes the hadamard schur geometric mean of the sets 1 m

algebra structure for lp of a vector measure abstract in this paper we present new results about the space l p for being a vector measure defined in the borel algebra of a compact abelian group g and satisfying certain property concerning translation of simple functions namely we show that l p is a translation invariant space which can be endowed with an algebra structure via usual convolution product we apply these results to the optimal domain of the fourier transform and convolution operators

unified analysis of secondary traits in case control association studies it has been repeatedly shown that in case control association studies analysis of a secondary trait that ignores the original sampling scheme can produce highly biased risk estimates although a number of approaches have been proposed to properly analyze secondary traits most approaches fail to reproduce the marginal logistic model assumed for the original case control trait and or do not allow for interaction between secondary trait and genotype marker on primary disease risk in addition the flexible handling of covariates remains challenging we present a general retrospective likelihood framework to perform association testing for both binary and continuous secondary traits which respects marginal models and incorporates the interaction term we provide a computational algorithm based on a reparameterized approximate profile likelihood for obtaining the maximum likelihood ml estimate and its standard error for the genetic effect on secondary traits in the presence of covariates for completen

the effect of sample design on principal component analysis abstract most sample surveys are multivariate and many lend themselves to multivariate methods of analysis the most usual mode of such analysis is a standard statistical package such as bmdp or spss in which the multivariate analyses are based on the underlying assumption that the data are generated as independent observations from a common probability distribution this assumption ignores the sample selection procedure involved in the survey which leads to the following basic questions what effects can the sample design have on methods of multivariate analysis how should such effects be taken into account this article considers the case of principal component analysis and in particular the point estimation of the eigenvalues and eigenvectors of a covariance matrix it is assumed that the selection of the sample depends on the population values of auxiliary variables as for example in stratified sampling the conventional estimators based on the assumption of simple random sampling are compar

optimal sobolev imbeddings involving rearrangement invariant quasinorms let m and n be positive integers with n 2 and 1 m n 1 we study rearrangement invariant quasinorms r and d on functions f 0 1 view the mathml source such that to each bounded domain in view the mathml sourcen with lebesgue measure there corresponds c c 0 for which one has the sobolev imbedding inequality r u t c d mu t u cm0 involving the nonincreasing rearrangements of u and a certain mth order gradient of u when m 1 we deal in fact with a closely related imbedding inequality of talenti in which d need not be rearrangement invariant r u t c d d dt x view the mathml sourcen u x u t u x dx u c10 in both cases we are especially interested in when the quasinorms are optimal in the sense that r cannot be replaced by an essentially larger quasinorm and d cannot be replaced by an essentially smaller one our results yield best possible refinements of such limiting sobolev inequalities as those of trudinger strichartz hansson brezis and wainger

rearrangement invariant spaces of functions on lca groups in this paper the rearrangement invariant hulls and kernels of the classes f f lp and f f l1loc on a noncompact nondiscrete locally compact abelian group are characterized the spacel1loc s of tempered distributions serves as the extended domain of the fourier transform instead of the classical domainl1 l2 we also introduce and study a new function spacep0which arises as the rearrangement invariant hull of some of the classes under consideration

factorizing operators on banach function spaces through spaces of multiplication operators abstract in order to extend the theory of optimal domains for continuous operators on a banach function space x over a finite measure we consider operators t satisfying other type of inequalities than the one given by the continuity which occur in several well known factorization theorems for instance pisier factorization theorem through lorentz spaces p th power factorable operators we prove that such a t factorizes through a space of multiplication operators which can be understood in a certain sense as the optimal domain for t our extended optimal domain technique does not need necessarily the equivalence between and the measure defined by the operator t and by using rings is allowed to be infinite classical and new examples and applications of our results are also given including some new results on the hardy operator and a factorization theorem through hilbert spaces

boundedness of classical operators on rearrangement invariant spaces abstract we study the behaviour on rearrangement invariant r i spaces of such classical operators of interest in harmonic analysis as the hardy littlewood maximal operator including the fractional version the hilbert and stieltjes transforms and the riesz potential the focus is on sharpness questions and we present characterisations of the optimal domain or range partner spaces when the range domain is fixed when an r i partner space exists at all a complete characterisation of the situation is given we illustrate the results with a variety of examples of sharp particular results involving customary function spaces

solid extensions of the cesaro operator on the hardy space h2 d abstract we introduce and study the largest banach space of analytic functions on the unit disc which is solid for the coefficient wise order and to which the classical cesaro operator c h 2 h 2 can be continuously extended while still maintaining its values in h 2 properties of this banach space h c e s 2 as well as a characterization of individual analytic functions which belong to h c e s 2 are presented in addition both the multiplier space of h c e s 2 and the spectrum of c h c e s 2 h c e s 2 are determined

salem sets in local fields the fourier restriction phenomenon and the hausdorff young inequality abstract we establish the existence of salem sets in the ring of integers of any local field and study the fourier restriction phenomenon on such sets optimal extension of the hausdorff young inequality initially attained for the torus by g mockenhaupt and w ricker is also established in the local field setting

logistic regression in case control studies the effect of using independent as dependent variables in case control studies cases are sampled separately from controls in such studies the primary analysis concerns the estimation of the effect of covariables on being a case or a control to explore causal pathways further secondary analysis could concern the relationships among the covariables in this paper the validity of such secondary analysis is addressed in particular the use of multiple logistic regression in case control studies where the dependent variable is not the case control indicator is explored it is shown that only under very restrictive conditions will sample regression coefficients correctly estimate their true value in many situations it may be valid to regress one covariable on others in the control group but not in the case group or the combined sample this principle is illustrated by a study of sexually transmitted disease in kenya

optimal rearrangement invariant range for hardy type operators we characterize in the context of rearrangement invariant spaces the optimal range space for a class of monotone operators related to the hardy operator the connection between optimal range and optimal domain for these operators is carefully analyzed

abstract cesaro spaces duality we study abstract cesaro spaces cx which may be regarded as generalizations of cesaro sequence spaces cesp and cesaro function spaces cesp i on i 0 1 or i 0 and also as the description of optimal domain from which cesaro operator acts to x we find the dual of such spaces in a very general situation what is however even more important we do it in the simplest possible way our proofs are more elementary than the known ones for cesp and cesp i this is the point how our paper should be seen i e not as a generalization of known results but rather like grasping and exhibiting the general nature of the problem which is not so easily visible in previous publications our results show also an interesting phenomenon that there is a big difference between duality in the cases of finite and infinite intervals

compactness properties of sobolev imbeddings for rearrangement invariant norms compactness properties of sobolev imbeddings are studied within the context of rearrangement invariant norms attention is focused on the extremal situation namely when the imbedding is considered as defined on its optimal sobolev domain with the range space fixed the techniques are based on recent results which reduce the question of boundedness of the imbedding to boundedness of an associated kernel operator of just one variable

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

local approach to order continuity in cesaro function spaces abstract the goal of this paper is to present a complete characterization of points of order continuity in abstract cesaro function spaces cx for x being a symmetric function space under some additional assumptions mentioned result takes the form c x a c x a we also find simple equivalent condition for this equality which in the case of i 0 1 comes to x l furthermore we prove that x is order continuous if and only if cx is under assumption that the cesaro operator is bounded on x this result is applied to particular spaces namely cesaro orlicz function spaces cesaro lorentz function spaces and cesaro marcinkiewicz function spaces to get criteria for oc points

the integration theory of banach space valued measures and the tonelli fubini theorems ii pettis integration in this part of the paper we turn to the concepts of gelfand and pettis integrability and integration of scalar valued functions with respect to a banach space valued measure the problem before us and our method for solving it have been outlined in section 1 of part i of this paper in what follows we adhere to the notation of part i and continue the trend set in that part all references with numbers such as 1 2 2 3 and a5 are to the items in sections 1 and 2 and appendix a in part i

a unified framework for studying parameter identifiability and estimation in biased sampling designs summary based on the odds ratio representation of a joint density we propose a unified framework to study parameter identifiability in biased sampling designs it is shown that most of these designs encountered in practice can be reformulated within the proposed framework and as a result the question of parameter identifiability can be largely clarified estimation of the identifiable parameters is considered and traditional results on the equivalence of the prospective and retrospective likelihoods are extended information contained in data on certain identifiable parameters is often very limited such parameters can be poorly estimated by the likelihood approach with practically attainable sample sizes which can substantially affect the estimates of parameters of primary interest a partially penalized likelihood approach is proposed to address this simulation results suggest that the proposed approach has good performance

semiparametrically efficient estimation in quantile regression of secondary analysis analysing secondary outcomes is a common practice for case control studies traditional secondary analysis employs either completely parametric models or conditional mean regression models to link the secondary outcome to covariates in many situations quantile regression models complement mean based analyses and provide alternative new insights on the associations of interest for example biomedical outcomes are often highly asymmetric and median regression is more useful in describing the central behaviour than mean regressions there are also cases where the research interest is to study the high or low quantiles of a population as they are more likely to be at risk we approach the secondary quantile regression problem from a semiparametric perspective allowing the covariate distribution to be completely unspecified we derive a class of consistent semiparametric estimators and identify the efficient member the asymptotic properties of the resulting estimators are established simulation results and a real data analysis are provided to demonstrate the superior performance of our approach with a comparison with the only existing approach so far in the literature

estimating marginal causal effects in a secondary analysis of case control data when an initial case control study is performed data can be used in a secondary analysis to evaluate the effect of the case defining event on later outcomes in this paper we study the example in

explicit salem sets in r2 abstract we construct explicit i e non random examples of salem sets in r 2 of dimension s for every 0 s 2 in particular we give the first explicit examples of salem sets in r 2 of dimension 0 s 1 this extends a theorem of kaufman