the strong p variation of martingales and orthogonal series let 1 p and letx x n n 0 be a sequence of scalars the strongp variation ofx denoted byw p x is defined as w p x sup left left x 0 p sum limits k 1 infty x n k x n k 1 p right 1 p right where the supremum runs over all increasing sequences of integers 0 n 0 n 1 n 2 let 1 p 2 and letm m n n 0 be a martingale inl p our main results are as follows if sigma mathbb e m n m n 1 p infty thenw p m is finite a s and we have mathbb e w p m p leqq c mathbb e m 0 p sum limits n geqq 1 mathbb e m n m n 1 p for some constantc depending only onp on the other hand let n be an arbitrary orthonormal system of functions inl 2 considerx x n n 0 inl 2 and lets n 0 x i i ands s n n 0 we prove that if x n p 1 p 2 thenw p s t for a e t and w p s 2 c x n p 1 p for some constantc each of these results is an extension of a result proved by bretagnolle for sums of independent mean zero r v s the casep 2 in also discussed our proofs use the real interpolation method of lions peetre they admit extensions in the banach space valued case provided suitable assumptions are imposed on the banach space

is tame open abstract is tame open no answer so far one may pose the tame open conjecture tame is open but how to support it no effective way to date in this note the rank of a wild algebra is introduced the wild rank conjecture which implies the tame open conjecture is formulated the wild rank conjecture is improved to the basic wild rank conjecture a covering criterion on the rank of a basic wild algebra is given which can be effectively applied to verify the basic wild rank conjecture for concrete algebras it makes all conjectures much reliable

wild two point algebras abstract the representation type of two point algebras is determined it is also shown that a two point algebra is wild if and only if it is controlled wild moreover it turns out that a minimal wild two point algebra is strongly minimal wild if and only if it is not strictly wild

an operator valued t1 theory for symmetric czos abstract we provide a natural bmo criterion for the l 2 boundedness of calderon zygmund operators with operator valued kernels satisfying a symmetric property our arguments involve both classical and quantum probability theory in the appendix we give a proof of the l 2 boundedness of the commutators r j b whenever b belongs to the bourgain s vector valued bmo space where r j is the j th riesz transform a common ingredient is the operator valued haar multiplier studied by blasco and pott

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

jump inequalities via real interpolation jump inequalities are the r 2 endpoint of lepingle s inequality for r variation of martingales extending earlier work by pisier and xu probab theory relat fields 77 4 497 514 1988 we interpret these inequalities in terms of banach spaces which are real interpolation spaces this interpretation is used to prove endpoint jump estimates for vector valued martingales and doubly stochastic operators as well as to pass via sampling from mathbb r d to mathbb z d for jump estimates for fourier multipliers

the p variation of partial sum processes and the empirical process the p variation of a function f is the supremum of the sums of the p th powers of absolute increments of f over nonoverlapping intervals let f be a continuous probability distribution function dudley has shown that the p variation of the empirical process is bounded in probability as n to infty if and only if p 2 and for 1 leq p leq 2 the p variation of the empirical process is at least n 1 p 2 and is at most of the order n 1 p 2 log log n p 2 in probability in this paper we prove that the exact order of the 2 variation of the empirical process is log log n in probability and for 1 leq p 2 the p variation of the empirical process is of exact order n 1 p 2 in expectation and almost surely let s j x 1 x 2 dots x j then the p variation of the partial sum process for x 1 x 2 dots x n is defined as that of f on 0 n where f t s j for j 1 t leq j j 1 2 dots n bretagnolle has shown that the expectation of the p variation for independent centered random variables x i with bounded p th moments is of order n for 1 leq p 2 we prove that for p 2 the 2 variation of the partial sum process of i i d centered nonconstant random variables with finite 2 delta moment for some delta 0 is of exact order n log log n in probability

vector valued q variation for ergodic averages and analytic semigroups abstract in this paper we establish umd lattice valued variational inequalities for ergodic averages of contractively regular operators and analytic semigroups these results generalize on the one hand some scalar valued variational inequalities in ergodic theory on the other hand xu s very recent result on umd lattice valued maximal inequality as an application we deduce the jump estimates and obtain quantitative information on the rate of the pointwise convergence for semigroups acting on umd lattice valued functions

hyperbolic beta integrals abstract hyperbolic beta integrals are analogues of euler s beta integral in which the role of euler s gamma function is taken over by ruijsenaars hyperbolic gamma function they may be viewed as q q bibasic analogues of the beta integral in which the two bases q and q are interrelated by modular inversion and they entail q analogues of the beta integral for q 1 the integrals under consideration are the hyperbolic analogues of the ramanujan integral the askey wilson integral and the nassrallah rahman integral we show that the hyperbolic nassrallah rahman integral is a formal limit case of spiridonov s elliptic nassrallah rahman integral

weighted jump and variational inequalities for rough operators abstract in this paper we systematically study weighted jump and variational inequalities for rough operators more precisely we show some weighted jump and variational inequalities for the families t t e e 0 of truncated singular integrals and m m t t 0 of averaging operators with rough kernels which are defined respectively by t e f x y e y y n f x y d y and m t f x 1 t n y t y f x y d y where the kernel belongs to l q s n 1 for q 1

controlled wild algebras the controlled wild algebra is introduced a covering criterion for a finite dimensional algebra to be controlled wild is given and this criterion is applied to the algebras with radical square zero algebras with zero relations local algebras and finite p group algebras 2000 mathematical subject classification 16g20 16g60

vector valued littlewood paley stein theory for semigroups abstract we develop a generalized littlewood paley theory for semigroups acting on l p spaces of functions with values in uniformly convex or smooth banach spaces we characterize in the vector valued setting the validity of the one sided inequalities concerning the generalized littlewood paley stein g function associated with a subordinated poisson symmetric diffusion semigroup by the martingale cotype and type properties of the underlying banach space we show that in the case of the usual poisson semigroup and the poisson semigroup subordinated to the ornstein uhlenbeck semigroup on r n this general theory becomes more satisfactory and easier to be handled in virtue of the theory of vector valued calderon zygmund singular integral operators

weighted variation inequalities for differential operators and singular integrals abstract we prove weighted strong q variation inequalities with 2 q for differential and singular integral operators for the first family of operators the weights used can be either sawyer s one sided a p weights or muckenhoupt s a p weights according to whether the differential operators in consideration are one sided or symmetric we use only muckenhoupt s a p weights for the second family all these inequalities hold equally in the vector valued case that is for functions with values in l for 1 as application we show variation inequalities for mean bounded positive invertible operators on l p with positive inverses

vector valued q variation for differential operators and semigroups i in this paper we establish mathcal b valued variational inequalities for differential operators ergodic averages and symmetric diffusion semigroups under the condition that banach space mathcal b has martingale cotype property these results generalize on the one hand pisier and xu s result on the variational inequalities for mathcal b valued martingales on the other hand many classical variational inequalities in harmonic analysis and ergodic theory moreover we show that rademacher cotype q is necessary for the mathcal b valued q variational inequalities as applications of the variational inequalities we deduce the jump estimates and obtain quantitative information on the rate of convergence it turns out the rate of convergence depends upon the geometry of the banach space under consideration while cowling and leinert have shown that convergence holds for all separable banach spaces

oscillation and variation for singular integrals in higher dimensions in this paper we continue our investigations of square function inequalities in harmonic analysis here we investigate oscillation and variation inequalities for singular integral operators in dimensions d 1 our estimates give quantitative information on the speed of convergence of truncations of a singular integral operator including upcrossing and a jump inequalities

hecke algebras of classical type and their representation type the purpose of this article is to determine the representation type for all of the hecke algebras of classical type to do this we combine methods from our previous work which is used to obtain information on their gabriel quivers and recent advances in the theory of finite dimensional algebras the principal computation is for hecke algebras of type b with two parameters then we show that the representation type of hecke algebras is governed by their poincare polynomials

pointwise convergence for semigroups in vector valued l p spaces suppose that tt t 0 is a symmetric diffusion semigroup on l2 x and denote by widetilde t t t geq0 its tensor product extension to the bochner space l p x mathcal b where mathcal b belongs to a certain broad class of umd spaces we prove a vector valued version of the hopf dunford schwartz ergodic theorem and show that this extends to a maximal theorem for analytic continuations of widetilde t t t geq 0 on l p x mathcal b as an application we show that such continuations exhibit pointwise convergence

noncommutative maximal ergodic theorems this paper is devoted to the study of various maximal ergodic theorems in noncommutative spaces in particular we prove the noncommutative analogue of the classical dunford schwartz maximal ergodic inequality for positive contractions on and the analogue of stein s maximal inequality for symmetric positive contractions we also obtain the corresponding individual ergodic theorems we apply these results to a family of natural examples which frequently appear in von neumann algebra theory and in quantum probability