limited range multilinear extrapolation with applications to the bilinear hilbert transform we prove a limited range off diagonal extrapolation theorem that generalizes a number of results in the theory of rubio de francia extrapolation and use this to prove a limited range multilinear extrapolation theorem we give two applications of this result to the bilinear hilbert transform first we give sufficient conditions on a pair of weights w 1 w 2 for the bilinear hilbert transform to satisfy weighted norm inequalities of the form begin aligned bh l p 1 left w 1 p 1 right times l p 2 left w 2 p 2 right longrightarrow l p w p end aligned where w w 1w 2 and frac 1 p frac 1 p 1 frac 1 p 2 frac 3 2 this improves the recent results of culiuc et al by increasing the families of weights for which this inequality holds and by pushing the lower bound on p from 1 down to frac 2 3 the critical index from the unweighted theory of the bilinear hilbert transform second as an easy consequence of our method we obtain that the bilinear hilbert transform satisfies some vector valued inequalities with muckenhoupt weights this reproves and generalizes some of the vector valued estimates obtained by benea and muscalu in the unweighted case we also generalize recent results of carando et al on marcinkiewicz zygmund estimates for multilinear calderon zygmund operators

two parameter version of bourgain s inequality rational frequencies our aim is to establish the first two parameter version of bourgain s maximal logarithmic inequality on l 2 r 2 for the rational frequencies we achieve this by introducing a variant of a two parameter rademacher menschov inequality the method allows us to control an oscillation seminorm as well

complexity of critical functions for hamiltonian systems transition to predominantly chaotic motion in hamiltonian systems with two degrees of freedom is described by a complicated function of frequency which is called the critical function a graph of this function is a fractal set with the local structure which is believed to depend only on the arithmetic nature of the frequency we calculated numerically fractal dimensions of this function for a few typical systems using the method of modular smoothing and an efficient algorithm for computation of the fractal dimensions the dimensions which measure the complexity of the fractal are indeed the same within the error bounds and are equal to the dimension of the exponent of the brjuno function which is a purely arithmetic function

lp estimates for non smooth bilinear littlewood paley square functions on mathbb r in this work we study some non smooth bilinear analogues of linear littlewood paley square functions on the real line we prove boundedness properties in lebesgue spaces for them let us consider the functions phi n satisfying widehat phi n xi bf 1 n n 1 xi and define the bilinear operator s n f g x int f x y g x y phi n y dy these bilinear operators are closely related to the bilinear hilbert transforms then for exponents p q r in 2 infty satisfying frac 1 p frac 1 q frac 1 r we prove that left left sum n in mathbb z left s n f g right 2 right 1 2 right l r mathbb r lesssim f l p mathbb r g l q mathbb r

truncations of multilinear hankel operators we extend to multilinear hankel operators the fact that some truncations of bounded hankel operators are still bounded we prove and use a continuity property of bilinear hilbert transforms on products of lipschitz spaces and hardy spaces

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

vector valued bilinear maximal operator and method of rotations abstract in this paper we obtain a bilinear analogue of fefferman stein s vector valued inequality for classical hardy littlewood maximal function also we prove the boundedness of bilinear hardy littlewood maximal operator from l p 1 r n l p 2 r n l 1 r n where 1 p 1 1 p 2 1 by applying the method of rotations

maximal multilinear operators we establish multilinear l p bounds for a class of maximal multilinear averages of functions of one variable reproving and generalizing the bilinear maximal function bounds of lacey 2000 as an application we obtain almost everywhere convergence results for these averages and in some cases we also obtain almost everywhere convergence for their ergodic counterparts on a dynamical system

modular smoothing and kam tori abstract transformation properties of kam tori under the action of a modular group on frequencies are analysed and used for calculating the kam tori of a hamiltonian system with one and a half degrees of freedom first we derive a relation between the kam torus with a frequency v and the kam torus with the frequency v 1 v 1 v where 1 v is the integer part of 1 v given by a continuous function l1 v in the double limit v m n and k 0 where m n is a rational frequency and k is a perturbation parameter the function l1 v was introduced by buric percival and vivaldi and also gives the transformation properties of the critical parameter as a function of the frequency this function provides a zeroth order modular smoothing approximation to the invariant tori which is better than a perturbation expansion the first order corrections to this approximation can be calculated from a function r1 v that appears to be smooth

uniform estimates for bilinear hilbert transforms and bilinear maximal functions associated to polynomials we study the bilinear hilbert transforms and bilinear maximal functions associated to non flat polynomial curves and obtain uniform l r estimates for r d 1 over d where d is the degree of the polynomial in addition the lower bound of r is sharp up to the end point

semi canonical quantisation of dissipative equations in order to avoid the known problems of canonical quantisation of the dissipative equation mx alpha xn delta v delta x 0 a modified approach is proposed it is based on the classically conserved quantity e mx2 2 v x alpha integral xndx rather than on a hamilton function linear quadratic and cubic damping n 1 2 and 3 are discussed explicitly for n 1 and n 3 normalisable one particle states cannot be stable however for quadratic friction which is classically a reversible process the author shows using the example of the harmonic oscillator that stable normalisable eigenstates can exist

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

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

the bilinear maximal functions map into l p for 2 3 p 1 the bilinear maximal operator defined below maps lp x lq into lr provided 1 0 2t t in particular mfg is integrableif f and g are square integrable answering a conjecture posed by alberto calder6n

double recurrence and almost sure convergence on repond de facon affirmative a une question posee par furstenberg sur la convergence presque sure de moyennes ergodiques de la forme 1 n n t n f t mn g avec 1 n ou n est un entier fixe t une transformation ergodique et preservant la mesure sur un espace de probabilite et f et g appartenant a l la somme ci dessus est un cas particulier d une recurrence multiple a laquelle peuvent etre facilement generalises les resultats obtenus ici pour la recurrence double

on boundedness of discrete multilinear singular integral operators let m be a measurable locally bounded function defined in r2 let 1 p1 q1 p2 q2 such that pi 1 implies qi let also 0 p3 q3 and 1 p 1 p1 1 p2 1 p3 we prove the following transference result the operator cm f g x r rf g m e2 ix d d initially defined for integrable functions with compact fourier support extends to a bounded bilinear operator from lp1 q1 r lp2 q2 r into lp3 q3 r if and only if the family of operators dm t p a b n t1p 1212 1212p q m t t e2 in d d initially defined for finite sequences a ak1 k1 z b bk2 k2 z where p k1 zak1e 2 ik1 and q k2 zbk2e 2 ik2 extend to bounded bilinear operators from lp1 q1 z lp2 q2 z into lp3 q3 z with norm bounded by uniform constant for all t 0 we apply this result to prove boundedness of the discrete bilinear hilbert transforms and other related discrete multilinear singular integrals including the endpoints

modular smoothing of action the central quantity in the theory of transport for hamiltonian systems and in particular the area preserving twist maps is the action of rotational periodic orbits usually this is a complicated discontinuous function of two arguments some perturbation parameter k and a rational rotation number denoted by we applied the idea of modular smoothing to this complicated fractal our main result is that all the information contained in the fractal can be retrieved from a set of continuous or smooth functions of one variable

on a biparameter maximal multilinear operator abstract it is well known that estimates for maximal operators and questions of pointwise convergence are strongly connected in recent years convergence properties of so called non conventional ergodic averages have been studied by a number of authors including assani austin host kra tao etc in particular much is known regarding convergence in l 2 of these averages but little is known about pointwise convergence in this spirit we consider the pointwise convergence of a particular ergodic average and study the corresponding maximal trilinear operator over r thanks to a transference principle lacey in 15 and demeter tao and thiele in 6 have studied maximal multilinear operators previously however the maximal operator we develop has a novel biparameter structure which has not been previously encountered and cannot be estimated using their techniques we will carve this biparameter maximal multilinear operator using a certain taylor series and produce non trivial holder type estimates for one of the two main terms by treating it as a singular integral the symbol of which has singular set given by two intersecting planes similarly to that of the biest operator studied by muscalu tao and thiele in 24 and 25

discrete maximal functions in higher dimensions and applications to ergodic theory we establish a higher dimensional counterpart of bourgain s pointwise ergodic theorem along an arbitrary integer valued polynomial mapping we achieve this by proving variational estimates v r on l p spaces for all 1

endpoint bounds for the bilinear hilbert transform we study the behavior of the bilinear hilbert transform bht at the boundary of the known boundedness regionh a sample of our results is the estimate bht f1 f2 f3 c f1 3 4 f2 3 4 f3 1 2 log log ee f3 min f1 f2 valid for all tuples of sets f j r of finite measure and functions f j such that f j 1f j j 1 2 3 with the additional restriction that f3 be supported on a major subset f 3 of f3 that depends on f j j 1 2 3 the use of subindicator functions in this fashion is standard in the given context see 25 the double logarithmic term improves over the single logarithmic term obtained in 1 whether the double logarithmic term can be removed entirely as is the case for the quartile operator discussed in 7 remains open we employ our endpoint results to describe the blow up rate of weak type and strong type estimates for bht as the tuple approaches the boundary ofh we also discuss bounds on lorentz orlicz spaces near l 2 3 improving on results of 5 the main technical novelty in our article is an enhanced version of the multi frequency calderon zygmund decomposition of 26

chaotic boundary of a hamiltonial map abstract the half plane maps are a class of complex hamiltonial maps whose invariant curves at most fixed irrational frequencies can be obtained as a convergent taylor series expansion for these maps the boundary between regular ordered motion on invariant curves and irregular chaotic motion is given by the radius of convergence of the series the successive terms of the series oscillate wildly due to the presence of small divisors methods are presented for taming the series based on the conversion of the convergenceexponentc ln c into the integral of a continuous but nondifferentiable lambda function whose graph whows a similarity structure on small scales self similarity properties are illustrated for the chaotic boundary function c where v is the 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