cyclic convex bodies and optimization moment problems abstract we deal with two discrete moment problems first deciding when a fixed element of r d is the vector of d first moments for some discrete probability distribution on a given interval a b feasibility moment problem and second maximizing minimizing a given linear combination of moments on the set of discrete probability distributions on a b whose d first moments are given optimization moment problem these problems are linked with the cyclic body which is the union of all cyclic polytopes on a b the cyclic polytopes have been extensively studied and their combinatorial and geometric properties are noteworthy the cyclic body also has interesting geometric properties we totally determine its facial structure and supporting hyperplanes and we construct an external representation by means of linear inequality systems whose coefficients are symmetric polynomials depending on parameters these tools allow us to solve the mentioned moment problems by using linear semi infinite programming and we obtain a representation of non negative polynomials over a b as well

mixed estimates for degenerate multi linear operators associated to simplexes we prove that the degenerate trilinear operator c3 1 1 1 given by the formula c3 1 1 1 f1 f2 f3 x x1 x2 x3f1 x1 f2 x2 f3 x3 e2 ix x1 x2 x3 dx1dx2dx3 satisfies for every p1 p2 p3 such that 2 p1 1 p2 p3 1p1 1p2 1 and 1p2 1p3 3 2 the mixed estimate c3 1 1 1 f1 f2 f3 11p1 1p2 1p3 p1 p2 p3 f 1 p1 f2 p2 f3 p3 for all f1 lp1 r f 1 lp1 r f2 lp2 r and f3 lp3 r mixed estimates for some generalizations of c3 1 1 1 are also established

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

motzkin decomposition of closed convex sets theodore motzkin proved in 1936 that any polyhedral convex set can be expressed as the minkowski sum of a polytope and a polyhedral convex cone this paper provides five characterizations of the larger class of closed convex sets in finite dimensional euclidean spaces which are the sum of a compact convex set with a closed convex cone these characterizations involve different types of representations of closed convex sets as the support functions dual cones and linear systems whose relationships are also analyzed in the paper the obtaining of information about a given closed convex set f and the parametric linear optimization problem with feasible set f from each of its different representations including the motzkin decomposition is also discussed

voronoi cells via linear inequality systems abstract the theory and methods of linear algebra are a useful alternative to those of convex geometry in the framework of voronoi cells and diagrams which constitute basic tools of computational geometry as shown by voigt and weis in 2010 the voronoi cells of a given set of sites t which provide a tesselation of the space called voronoi diagram when t is finite are solution sets of linear inequality systems indexed by t this paper exploits systematically this fact in order to obtain geometrical information on voronoi cells from sets associated with t convex and conical hulls tangent cones and the characteristic cones of their linear representations the particular cases of t being a curve a closed convex set and a discrete set are analyzed in detail we also include conclusions on voronoi diagrams of arbitrary sets

on calderon s conjecture with constants cfi p1 p2 depending only on fi p1 p2 and p p1p2 p1 p2 hold the flrst result of this type is proved in 4 and the purpose of the current paper is to extend the range of exponents p1 and p2 for which 2 is known in particular the case p1 2 p2 1 is solved to the a rmative this was

the intersection local time of the westwater process in this paper self intersection properties of the westwater process are investigated as a result we obtain that the westwater process has an intersection local time bar alpha x 0 s t 1 which is holder continuous with respect tox s t r3 0 1 2 1 2 1 and the hausdorff dimension of the double time set is 1 2 as for brownian motion inr3

representability of convex sets by analytical linear inequality systems the solution sets of analytical linear inequality systems posed in the euclidean space form a transition class between the polyhedral convex sets and the closed convex sets which are representable by means of linear continuous systems the constraint systems of many semi infinite programming problems are analytical and their feasible sets retain geometric properties of the polyhedral sets which are useful in the numerical treatment of such kind of optimization problems the euclidean closed n dimensional balls admit analytical representation if and only if n 3 this paper solves in a negative way the analytical representation problem for a wide class of n dimensional convex sets with n 3 which includes quasi polyhedral sets and smooth convex bodies

on farthest voronoi cells abstract given an arbitrary set t in the euclidean space r n whose elements are called sites and a particular site s the farthest voronoi cell of s denoted by f t s consists of all points which are farther from s than from any other site in this paper we study farthest voronoi cells and diagrams corresponding to arbitrary possibly infinite sets more in particular we characterize for a given arbitrary set t those s t such that f t s is nonempty and study the geometrical properties of f t s in that case we also characterize those sets t whose farthest voronoi diagrams are tesselations of the euclidean space and those sets that can be written as f t s for some t r n and some s t

the partial sum process of orthogonal expansions as geometric rough process with fourier series as an example an improvement of menshov rademacher theorem abstract the partial sum process of orthogonal expansion n 0 c n u n is a geometric 2 rough process for any orthonormal system u n n 0 in l 2 and any sequence of numbers c n satisfying n 0 log 2 n 1 2 c n 2 since being a geometric 2 rough process implies the existence of a limit function up to a null set our theorem could be treated as an improvement of menshov rademacher theorem for fourier series the condition can be strengthened to n 0 log 2 n 1 c n 2 which is equivalent to f u f v 2 sin u v 2 d u d v with f the limit function

the bilinear hilbert transform in umd spaces we prove l p bounds for the bilinear hilbert transform acting on functions valued in intermediate umd spaces such bounds were previously unknown for umd spaces that are not banach lattices our proof relies on bounds on embeddings from bochner spaces l p mathbb r x into outer lebesgue spaces on the time frequency scale space mathbb r 3

locally polyhedral linear inequality systems abstract linear systems of an arbitrary number of inequalities provide external representations for the closed convex sets in the euclidean space in particular the locally polyhedral systems introduced in this paper are the natural linear representation for quasipolyhedral sets those subsets of the euclidean space whose nonempty intersections with polytopes are polytopes for these systems the geometrical properties of the solution set are investigated and their extreme points and edges are characterized the class of locally polyhedral systems includes the quasipolyhedral systems introduced by marchi puente and vera de serio in order to generalize the weyl property of finite linear inequality systems

maximal functions associated to filtrations abstract let t be a bounded linear or sublinear operator from lp y to lq x a maximal operator t f x supj t f yj x is associated to any sequence of subsets yj of y under the hypotheses that q p and the sets yj are nested we prove that t is also bounded classical theorems of menshov and zygmund are obtained as corollaries multilinear generalizations of this theorem are also established these results are motivated by applications to the spectral analysis of schrodinger operators

wkb and spectral analysis of one dimensional schrodinger operators with slowly varying potentials consider a schrodinger operator on l 2 of the line or of a half line with appropriate boundary conditions if the potential tends to zero and is a finite sum of terms each of which has a derivative of some order in l 1 l p for some exponent p 2 then an essential support of the the absolutely continuous spectrum equals almost every generalized eigenfunction is bounded and satisfies certain wkb type asymptotics at infinity if moreover these derivatives belong to l p with respect to a weight x with 0 then the hausdorff dimension of the singular component of the spectral measure is strictly less than one

generalizations of the carleson hunt theorem i the classical singularity case in this article we prove l p estimates for a general maximal operator which extend both the classical coifman meyer and carleson hunt theorems in harmonic analysis

on calderon s conjecture for the bilinear hilbert transform abstract we show that the bilinear hilbert transform defined by maps lp lq into lr for 1 p q 1 p 1 q 1 r and 2 3 r

variational estimates for the bilinear iterated fourier integral abstract we prove pointwise variational l p bounds for a bilinear fourier integral operator in a large but not necessarily sharp range of exponents this result is a joint strengthening of the corresponding bounds for the classical carleson operator the bilinear hilbert transform the variation norm carleson operator and the bi carleson operator terry lyon s rough path theory allows for extension of our result to multilinear estimates we consider our result a proof of concept for a wider array of similar estimates with possible applications to ordinary differential equations

quasipolyhedral sets in linear semiinfinite inequality systems abstract this paper provides an extension to linear semiinfinite systems of a well known property of finite linear inequality systems the so called weyl property which characterizes the extreme points of the solution set as those solution points such that the gradient vectors of the active constraints form a complete set a class of linear semiinfinite systems which satisfy this property is identified the p systems it is also shown that any p system contains an equivalent minimal subsystem

a theory of linear inequality systems abstract this paper provides results on the boundedness the dimension the boundary and the relative boundary of the feasible set of possibly infinite linear inequality systems defined on a finite dimensional space it analyzes the redundancy the minimality and the finite reduction of such systems as well as the relation between any system and its associated homogeneous one

wkb asymptotic behavior of almost all generalized eigenfunctions for one dimensional schrodinger operators with slowly decaying potentials we prove the wkb asymptotic behavior of solutions of the differential equation d2u dx2 v x u eu for a e e a where v v1 v2 v1 lp r and v2 is bounded from above with a lim supx v x while v 2 x lp r 1 p 2 these results imply that schrodinger operators with such potentials have absolutely continuous spectrum on a we also establish wkb asymptotic behavior of solutions for some energy dependent potentials

l p estimates for the biest ii the fourier case we prove l p estimates theorem 1 2 for the biest a trilinear multiplier operator with singular symbol the methods used are based on the treatment of the walsh analogue of the biest in the prequel 13 of this paper but with additional technicalities due to the fact that in the fourier model one cannot obtain perfect localization in both space and frequency

lp estimates for the bilinear hilbert transform for the bilinear hilbert transform given by we announce the inequality h fg p3 kp1 p2 f p1 g p2 provided 2 p1 p2 1 p3 1 p1 1 p2 and 1 p3 2