on certain maximal functions and approach regions if f e l ir and u x y is the poisson integral of j a classical theorem of fatou 2 asserts that u has nontangential limits almost everywhere on r littlewood 5 and later zygmund 9 showed that this conclusion is false if nontangential approach is replaced by approach along translates of a curve which approaches the boundary tangentially in this paper we find a necessary and sufficient condition on an approach region for the associated maximal function to be appropriately bounded it then turns out that there are many approach regions which are not contained in any nontangential region but for which the conclusions of fatou s theorem remain true suppose q c ir has the property that whenever xi u e b and whenever x x y y it follows that x v e q then the maximal function associated to such a set is bounded if and only if the cross sectional measure at height y is bounded by a constant times y we also study regions where the cross sectional area is larger than y we introduce a modified maximal function which is again bounded this result leads to a simple proof of the boundedness of certain tangential maximal functions of poisson integrals of potentials studied in 161 the plan of this paper is as follows in section 1 we motivate our main ideas by discussing two particular examples the method of proof here is somewhat different from the proofs of the more general results in section 2 we obtain the necessary and sufficient conditions for fatou s theorem in section 3 we study more general approach regions and the appropriately modified maximal functions in section 4 we show how this maximal function controls the tangential boundary behavior of poisson integrals of potentials

tangential boundary limits and exceptional sets for holomorphic functions in dirichlet type spaces according to the well known theorem of fatou the harmonic extension u to the upper half space r x of a function f in lp rn 1 1 in this paper we study the analogous question in the unit ball b of c recall that the correct analogue of fatou s theorem for b is the theorem of kor myi k stating that invariant harmonic extensions poisson szeg6 integrals of functions in lp s 1 p oo have admissible limits at almost every point of the unit sphere s in particular this is true for holomorphic functions in the hardy space hp b as is well known korfinyi s admissible regions allow parabolic tangential approach along the complex tangential directions these admissible regions are best possible as far as the order of tangency to the boundary is concerned even if only bounded holomorphic functions are considered as shown in h s however they are not best possible in a stronger sense su 1 by analogy with the real variable situation it seems reasonable to expect kor myi s regions not to be optimal for holomorphic functions whose admissible boundary values are in some sense bessel type potentials of lp s functions thus we consider holomorphic functions in the unit ball which are given by fractional cauchy integrals of lp s functions these are obtained by modifying the exponent in the denominator of the cauchy kernel of b a procedure suggested by the

the importance of the selberg integral it has been remarked that a fair measure of the impact of atle selberg s work is the number of mathematical terms that bear his name one of these is the selberg integral an n dimensional generalization of the euler beta integral we trace its sudden rise to prominence initiated by a question to selberg from enrico bombieri more than thirty years after its initial publication in quick succession the selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the macdonald conjectures it further initiated the study of q analogues which in turn enriched the macdonald conjectures we review these developments and proceed to exhibit the sustained prominence of the selberg integral as evidenced by its central role in random matrix theory calogero sutherland quantum many body systems knizhnik zamolodchikov equations and multivariable orthogonal polynomial theory

first layer formulas for characters of sl n c some problems concerning the decomposition of certain characters of sl n c are studied from a combinatorial point of view the specific characters considered include those of the exterior and symmetric algebras of the adjoint representation and the euler characteristic of hanlon s so called macdonald complex a general recursion is given for computing the irreducible decomposition of these characters the recursion is explicitly solved for the first layer representations which are the irreducible representations corresponding to partitions of n in the case of the exterior algebra this settles a conjecture of gupta and hanlon a further application of the recursion is used to give a family of formal laurent series identities that generalize the equal parameter q dyson theorem

a new approach to constant term identities and selberg type integrals abstract selberg type integrals that can be turned into constant term identities for laurent polynomials arise naturally in conjunction with random matrix models in statistical mechanics built on a recent idea of karasev and petrov we develop a general interpolation based method that is powerful enough to establish many such identities in a simple manner the main consequence is the proof of a conjecture of forrester related to the calogero sutherland model in fact we prove a more general theorem which includes aomoto s constant term identity at the same time we also demonstrate the relevance of the method in additive combinatorics

some new function spaces and their applications to harmonic analysis abstract in this paper a family of spaces is introduced which seems well adapted for the study of a variety of questions related to harmonic analysis and its applications these spaces are the tent spaces they provide the natural setting for the study of such things as maximal functions the relevant space here is t p and also square functions where the space t2p is relevant as such these spaces lead to unifications and simplifications of some basic techniques in harmonic analysis thus they are closely related to lp and hardy spaces important parts of whose theory become corollaries of the description of tent spaces also as proc conf harmonic analysis cortona lect notes in math vol 992 springer verlag berlin new york 1983 already indicated where these spaces first appeared explicitly the tent spaces can be used to simplify some of the results related to the cauchy integral on lipschitz curves and multilinear analysis in retrospect one can recognize that various ideas important for tent spaces had been used if only implicitly for quite some time here one should mention carleson s inequality its simplifications and extensions the theory of hardy spaces and atomic decompositions

nonsymmetric macdonald polynomials and demazure characters we establish a connection between a specialization of the nonsymmetric macdonald polynomials and the demazure characters of the corresponding affine kac moody algebra this allows us to obtain a representation theoretical interpretation of the coefficients of the expansion of the specialized symmetric macdonald polynomials in the basis formed by the irreducible characters of the associated finite lie algebra

constant term identities extending the q dyson theorem andrews 1 has conjectured that the constant term in a certain product is equal to a q multinomial coefficient this conjecture is a q analogue of dyson s conjecture 5 and has been proved combinatorically by zeilberger and bressoud 15 in this paper we give a combinatorial proof of a master theorem that the constant term in a similar product computed over the edges of a nontransitive tournament is zero many constant terms are evaluated as consequences of this master theorem including andrews q dyson theorem in two ways one of which is a q analogue of good s 6 recursive proof

limit distributions of u statistics resampled by symmetric stable laws if y i and v i are independent random sequences such thaty i are i i d random variables belonging to the normal domain of attraction of a symmetric stable law 0 2 andv i are i i d random variables then the limit distributions of u statistics n 1 alpha sum limits 1 leqq i t ldots i d leqq n y i 1 ldots y i d f v i 1 ldots v i d coincide with the probability laws of multiple stochastic integralsx d f f t 1 t d dx t d with respect to a symmetric stable processx t

multiple integration with respect to poisson and levy processes necessary and sufficient conditions are given for the existence of a multiple stochastic integral of the form fdx1 dxd where x1 xd are components of a positive or symmetric pure jump type levy process in d conditions are also given for a sequence of integrals of this type to converge in probability to zero or infinity or to be tight all arguments proceed via reduction to the special case of poisson integrals

constant term identities and poincare polynomials in 1982 macdonald published his now famous constant term conjectures for classical root systems this paper begins with the almost trivial observation that macdonald s constant term identities admit an extra set of free parameters thereby linking them to poincare polynomials we then exploit these extra degrees of freedom in the case of type a to give the first proof of kadell s orthogonality conjecture a symmetric function generalisation of the q dyson conjecture or zeilberger bressoud theorem key ingredients in our proof of kadell s orthogonality conjecture are mul tivariable lagrange interpolation the scalar product for demazure characters and 0 1 matrices 2015 american mathematical society

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

boundary tangential convergence on spaces of homogeneous type we study tangential convergence of convolutions with approximate identities of functions defined on a homogeneous type space and having a certain regularity our results contain those already known for the euclidean case and give new ones for stratified nilpotent lie groups and for solutions of the dirichlet problem on lipschitz domains

random multilinear forms on etudie la convergence des formes multilineaires f n n k x n 1 x n k dans des variables aleatoires independantes symetriques

invariance principles for sums of banach space valued random elements and empirical processes almost sure and probability invariance principles are established for sums of independent not necessarily measurable random elements with values in a not necessarily separable banach space it is then shown that empirical processes readily fit into this general framework thus we bypass the problems of measurability and topology characteristic for the previous theory of weak convergence of empirical processes

on maximal functions and poisson szego integrals we study a class of maximal functions of hardy littlewood type defined on spaces of homogeneous type and we give necessary and sufficient conditions for the corresponding maximal operators to be of weak type 1 1 as a consequence we show that poisson szego integrals of lp functions possess certain boundary limits which are not implied by koranyi s theorem introduction the classical theorem of fatou is concerned with the existence of nontangential boundary limits of the poisson integral of a function defined on rn or on the unit circle such limits exist almost everywhere if is in lp rn 1 0 if is locally integrable on rn the maximal function mn is defined by ain io sup j ii