variation inequalities for the fejer and poisson kernels in this paper we show that the g th order variation operator for both the fejer and poisson kernels are bounded from l p to l p 1 2 counterexamples are given if g 2

on the littlewood paley stein g function we consider semigroups tt which are contractive on lp m for all p e q q and q e 1 2 we give an example on symmetric spaces of the noncompact type which shows that the littlewood paley stein g function associated to the infinitesimal generator of tt may be unbounded on lq m and on l m we prove that variants of the g function are bounded on these lebesgue spaces in this paper we present some new results concerning the lp boundedness of the littlewood paley stein g function suppose that tt is a co semigroup of operators on lp m where m is a a finite measure space and let a e n the littlewood paley stein function ga associated to the given semigroup is defined by the rule ga f 1 ta ttfi2 dt for all f for which the right hand side makes sense in classical cases where tt is the poisson semigroup acting on lp r or lp t the g function is an important tool in fourier analysis for instance it is frequently used to prove multiplier theorems and pointwise convergence results the reader is referred to 1 1 for a survey of its role in a general context the g function was considered by e m stein he proved that if tt is a symmetric diffusion semigroup then ga satisfies the estimate 1 aplifiip liga f iip bp fllp vf e lp m whenever p e 1 oo a proof of this result can be found in 10 a simpler and more general approach via transference techniques is presented in 2 a third proof is in 4 and 5 it is of an entirely different nature in that it is based on the fact that generators of symmetric diffusion semigroups and even more general semigroups 3 have an hx functional calculus one of the interesting applications of g functions is to functional calculus for infinitesimal generators of co semigroups stein proved a marcinkiewicz type result for those as a corollary of the theory he developed for the g function the relationship between g functions and functional calculus for the infinitesimal generator of tt was investigated in 4 and 5 our treatment here owes received by the editors march 7 1994 originally communiated to the proceedings of the ams by j marshall ash 1991 mathematics subject classification primary 42b25 47a60 secondary 43a85

random martingales and localization of maximal inequalities abstract let x d be a metric measure space for r 0 consider the hardy littlewood maximal operator m r f x def sup r r 1 b x r b x r f d we show that if there is an n 1 such that one has the microdoubling condition b x 1 1 n r b x r for all x x and r 0 then the weak 1 1 norm of m r has the following localization property m r l 1 x l 1 x sup r 0 m r r n r l 1 x l 1 x an immediate consequence is that if x d is ahlfors david n regular then the weak 1 1 norm of m r is n log n generalizing a result of stein and stromberg 1983 47 we show that this bound is sharp by constructing a metric measure space x d that is ahlfors david n regular for which the weak 1 1 norm of m 0 is n log n the localization property of m r is proved by assigning to each f l 1 x a distribution over random martingales for which the associated random doob maximal inequality controls the weak 1 1 inequality for m r

multiple test procedures for identifying the maximum safe dose we consider dose response studies for safety assessment of crop protection compounds and drugs and we offer a hypothesis testing approach for identifying the maximum dose level that is guaranteed to be safe with preassigned confidence the focus is on step down sd multiple test procedures for identifying the maximum safe dose we propose two classes of contrasts among the dose means as test statistics for these procedures pairwise contrasts pc and helmert contrasts hc the first procedure sd2pc consists of a sequence of ordinary t tests and is thus easy to apply but it can have very low power for certain step response functions the second procedure sd1hc does not suffer from this drawback but it requires a weak monotonicity assumption for its mathematical validity the powers of sd2pc and sd1hc are studied via monte carlo simulation we recommend sd2pc for linear response functions and sd1hc for step response functions the procedures are illustrated by applying them to data from an aquatic

combining multiple comparisons and modeling techniques in dose response studies the analysis of data from dose response studies has long been divided according to two major strategies multiple comparison procedures and model based approaches model based approaches assume a functional relationship between the response and the dose taken as a quantitative factor according to a prespecified parametric model the fitted model is then used to estimate an adequate dose to achieve a desired response but the validity of its conclusions will highly depend on the correct choice of the a priori unknown dose response model multiple comparison procedures regard the dose as a qualitative factor and make very few if any assumptions about the underlying dose response model the primary goal is often to identify the minimum effective dose that is statistically significant and produces a relevant biological effect one approach is to evaluate the significance of contrasts between different dose levels while preserving the family wise error rate such procedures are relatively robust but inference is confined to the selection of the target dose among the dose levels under investigation we describe a unified strategy to the analysis of data from dose response studies which combines multiple comparison and modeling techniques we assume the existence of several candidate parametric models and use multiple comparison techniques to choose the one most likely to represent the true underlying dose response curve while preserving the family wise error rate the selected model is then used to provide inference on adequate doses

quasi algebras and multiplication of distributions a self adjoint operatorainl2 defines in a natural way a space of test functions sa and a corresponding space of distributions s a these are considered as quasi algebras and the problem of multiplying distributions is studied in terms of multiplication operators defined on a rigged hilbert space

multiple test procedures for identifying the minimum effective and maximum safe doses of a drug we address the problem of determining the therapeutic window of a drug by finding its minimum effective and maximum safe doses mined and maxsd the mined is the lowest dose that exceeds the mean efficacy of the zero dose by a specified threshold and the maxsd is the highest dose that does not exceed the mean toxicity of the zero dose by a specified threshold step down multiple test procedures are proposed to identify the mined and maxsd assuming a bivariate normal model these procedures control the type i familywise error probability of declaring any ineffective dose as effective or any unsafe dose as safe at a prespecified level a new multivariate t distribution is introduced whose critical points are required to implement the exact normal theory procedures because these critical points depend on the unknown correlation coefficient between the efficacy and safety variables the bonferroni method is proposed as an alternative which amounts to separately testing for efficacy and safety each at t

quantum systems with diagonal singularity their logic and intrinsic irreversibility abstract the work of the brussels austin group on irreversibility over the last years has shown that quantum large poincare systems with diagonal singularity lead to an extension of quantum theory beyond the conventional hilbert space framework and logic we characterize the algebra of observables and the states and the logic of the extended quantum theory of intrinsically irreversible systems with diagonal singularity

a note on multiple testing procedures in dose finding it is shown that in some recently proposed multiple test procedures for dose finding tamhane hochberg and dunnett 1996 biometrics 52 21 37 strong control of the familywise error is only guaranteed if order relations are assumed to hold among the means procedures that can be applied safely without assuming any restrictions on the means can be based on the classical many one comparisons of doses with the zero dose

localization and dimension free estimates for maximal functions abstract in the recent paper a naor t tao random martingales and localization of maximal inequalities j funct anal 259 3 2010 731 779 naor and tao introduce a new class of measures with a so called micro doubling property and present via martingale theory and probability methods a localization theorem for the associated maximal functions as a consequence they obtain a weak type estimate in a general abstract setting for these maximal functions that is reminiscent of the n log n result of stein and stromberg in euclidean spaces the purpose of this work is twofold first we introduce a new localization principle that localizes not only in the time dilation parameter but also in space the proof uses standard covering lemmas and selection processes second we show that a uniform condition for micro doubling in the euclidean spaces provides dimension free estimates for their maximal functions in all l p with p 1 this is done introducing a new technique that allows to differentiate through dimensions

stepwise confidence intervals without multiplicity adjustment for dose response and toxicity studies abstract not all simultaneous inferences need multiplicity adjustment if the sequence of individual inferences is predefined and failure to achieve the desired inference at any step renders subsequent inferences unnecessary then multiplicity adjustment is not needed this can be justified using the closed testing principle to test appropriate hypotheses that are nested in sequence starting with the most restrictive one but what hypotheses are appropriate may not be obvious in some problems we give a fundamentally different confidence set based justification by partitioning the parameter space naturally and using the principle that exactly one member of the partition contains the true parameter in dose response studies designed to show superiority of treatments over a placebo negative control or a drug known to be efficacious active control the confidence set approach generates methods with meaningful guarantee against incorrect decision whereas previous applications of the closed testing app

interpolation theory and refinement of nested hilbert spaces using the method of quadratic interpolation between hilbert spaces we show that a nested hilbert space with countable index set admits arbitrarily many proper refinements by nested hilbert spaces

subdyadic square functions and applications to weighted harmonic analysis abstract through the study of novel variants of the classical littlewood paley stein g functions we obtain pointwise estimates for broad classes of highly singular fourier multipliers on r d satisfying regularity hypotheses adapted to fine subdyadic scales in particular this allows us to efficiently bound such multipliers by geometrically defined maximal operators via general weighted l 2 inequalities in the spirit of a well known conjecture of stein our framework applies to solution operators for dispersive pde such as the time dependent free schrodinger equation and other highly oscillatory convolution operators that fall well beyond the scope of the calderon zygmund theory

characterization of sobolev spaces on the sphere abstract we introduce a new characterization of the sobolev spaces h on the unit sphere s d 1 where the smoothness index is any positive real number and d 2 this characterization is in terms of a square function and it avoids the notion of differentiability the result is fully based on zonal fourier multipliers special functions and the subtle use of classical analysis tools

a two stage stepwise estimation procedure this article proposes a two stage simultaneous confidence procedure for the comparisons of k pairs of population means without using multiplicity adjustment of more than two populations the proposed procedure can be broadly applied to parametric or nonparametric models it is robust and versatile because its derivation only utilizes a partitioning approach in conjunction with a bivariate adjustment without any assumption on the underlying distribution to elucidate the application the proposed procedure is intertwined with the estimation of the therapeutic window of a drug it provides confidence limits for the efficacy and the toxicity of the effective doses highest ineffective dose safe doses and lowest unsafe dose simultaneously such estimation information facilitates follow up studies in clinical trials as an illustrative example the new procedure is applied to analyze a data set on molecular cancer therapeutics regarding the apoptotic killing effects of different chemical compounds on two leukemia cell lines

the best constant in the davis inequality for the expectation of the martingale square function a method is introduced for the simultaneous study of the square function and the maximal function of a martingale that can yield sharp norm inequalities between the two one application is that the expectation of the square function of a martingale is not greater than 3 times the expectation of the maximal function this gives the best constant for one side of the davis two sided inequality the martingale may take its values in any real or complex hilbert space the elementary discrete time case leads quickly to the analogous results for local martingales m indexed by 0 some earlier inequalities are also improved and closely related the levy martingale is embedded in a large family of submartingales

the weak type 1 1 bounds for the maximal function associated to cubes grow to infinity with the dimension let md be the centered hardy littlewood maximal function associated to cubes in r d with lebesgue measure and let cd denote the lowest constant appearing in the weak type 1 1 inequality satised by md we show that cd 1 as d 1 thus answering for the case of cubes a longstanding

field algebras do not leave field domains invariant it is proved that von neumann algebras associated to op algebra p d cannot leave the domaind ofp invariant if they are type i or type iii factors or finite direct sums of such factors hence it follows that in quantum field theory global and local von neumann field algebras in typical cases do not leave invariant the definition domain of wightman fields

multiple test procedures for dose finding the problem of identifying the lowest dose level for which the mean response differs from that at the zero dose level is considered a general framework for stepwise testing procedures that use contrasts among the dose level means is proposed using this framework several new procedures are derived these and some existing procedures including that of williams 1971 biometrics 27 103 117 1972 biometrics 28 519 531 are compared analytically and by an extensive simulation study for the normal theory balanced one way layout case it is pointed out that the procedures based on the so called step and basin contrasts proposed by ruberg 1989 journal of american statistical association 84 816 822 have excessively high type i familywise error rates fwes and hence they should not be used some findings of the simulation study are as follows for monotone dose mean configurations williams procedure and two step down test procedures based on helmert and linear contrasts offer the best performance for nonmonotone dose mean configurations the performance of williams procedure does degrade somewhat but the other two procedures are still the best for more complex designs a simple step down test procedure that uses any alpha level tests not necessarily t tests to compare each dose level with the zero dose level controls the fwe and is the only alternative available but its power is rather low especially under nonmonotone configurations step up procedures are generally dominated by step down procedures when the same contrasts are used although the differences are not great

partial algebras of matrices and operators the set m of infinite matrices and the set q of squarable matrices are considered as partial algebras the connection between q and two partial algebras of closed operators is studied conditions for a matrix representation in von neumann s sense of a family of closed operators are given

equivalence with respect to a control stepwise tests summary the problem of selecting those treatments which are equivalent or bioequivalent to a control treatment is investigated by using single step step down and step up multipletesting procedures for each of the tests considered the null hypothesis is that of nonequivalence and the hypotheses are rejected in either a single step or a stepwise fashion while controlling the familywise error rate the single step procedure is based on expanded confidence intervals as discussed by bofinger the step down procedure is based on the method of naik and the step up procedure on the method of dunnett and tamhane almost all the critical constants used in the procedures are based on tables already published

partial algebras and volterra convolution of distribution kernels the set of distribution kernels is considered as a partial algebra when the multiplication of two kernels is defined as their volterra convolution the partial algebra of distribution kernels corresponding to a certain partial algebra of closable operators in l2 is studied

decision theory results for one sided multiple comparison procedures a resurgence of interest in multiple hypothesis testing has occurred in the last decade motivated by studies in genomics microarrays dna sequencing drug screening clinical trials bioassays education and psychology statisticians have been devoting considerable research energy in an effort to properly analyze multiple endpoint data in response to new applications new criteria and new methodology many ad hoc procedures have emerged the classical requirement has been to use procedures which control the strong familywise error rate fwe at some predetermined level a that is the probability of any false rejection of a true null hypothesis should be less than or equal to a finding desirable and powerful multiple test procedures is difficult under this requirement one of the more recent ideas is concerned with controlling the false discovery rate fdr that is the expected proportion of rejected hypotheses which are in fact true many multiple test procedures do control the fdr a much earlier approach to multiple testing was formulated by lehmann ann math statist 23 1952 541 552 and 28 1957 1 25 lehmann s approach is decision theoretic and he treats the multiple endpoints problem as a 2 k finite action problem when there are k endpoints this approach is appealing since unlike the fwe and fdr criteria the finite action approach pays attention to false acceptances as well as false rejections in this paper we view the multiple endpoints problem as a 2 k finite action problem we study the popular procedures single step step down and step up from the point of view of admissibility bayes and limit of bayes properties for our model which is a prototypical one and our loss function we are able to demonstrate the following results under some fairly general conditions to be specified i the single step procedure is admissible ii a sequence of prior distributions is given for which the step down procedure is a limit of a sequence of bayes procedures iii for a vector risk function where each component is the risk for an individual testing problem various admissibility and inadmissibility results are obtained in a companion paper cohen and sackrowitz ann statist 33 2005 145 158 we are able to give a characterization of bayes procedures and their limits the characterization yields a complete class and the additional useful result that the step up procedure is inadmissible the inadmissibility of step up is demonstrated there for a more stringent loss function additional decision theoretic type results are also obtained in this paper

intrinsic irreversibility of quantum systems with diagonal singularity the work of the brussels austin group on irreversibility over the last years has shown that quantum large poincare systems with diagonal singularity lead to an extension of quantum theory beyond the conventional hilbert space framework and logic we characterize the algebra of observables the states and the logic of the extended quantum theory of intrinsically irreversible systems with diagonal singularity we illustrate the general ideas for the friedrichs model