l p estimates for a singular integral operator motivated by calderxc3xb3n s second commutator abstract we prove a wide range of l p estimates for a trilinear singular integral operator motivated by dropping one average in calderon s second commutator for comparison by dropping two averages in calderon s second commutator one faces the trilinear hilbert transform the novelty in this paper is that in order to avoid difficulty of the level of the trilinear hilbert transform we choose to view the symbol of the operator as a non standard symbol the methods used come from time frequency analysis but must be adapted to the fact that our symbol is non standard

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

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

nonconventional ergodic averages and nilmanifolds we study the l2 convergence of two types of ergodic averages the first is the average of a product of functions evaluated at return times along arithmetic progressions such as the expressions appearing in furstenberg fs proof of szemer ledi fs theorem the second average is taken along cubes whose sizes tend to for each average we show that it is sufficient to prove the convergence for special systems the characteristic factors we build these factors in a general way independent of the type of the average to each of these factors we associate a natural group of transformations and give them the structure of a nilmanifold from the second convergence result we derive a combinatorial interpretation for the arithmetic structure inside a set of integers of positive upper density

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

a multidimensional szemeredi theorem for hardy sequences of different growth abstract we prove a variant of the multidimensional polynomial szemeredi theorem of bergelson and leibman where one replaces polynomial sequences with other sparse sequences defined by functions that belong to some hardy field and satisfy certain growth conditions we do this by studying the limiting behavior of the corresponding multiple ergodic averages and obtaining a simple limit formula a consequence of this formula in topological dynamics shows denseness of certain orbits when the iterates are restricted to suitably chosen sparse subsequences another consequence is that every syndetic set of integers contains certain nonshift invariant patterns and every finite coloring of n with each color class a syndetic set contains certain polychromatic patterns results very particular to our non polynomial setup

from harmonic analysis to arithmetic combinatorics arithmetic combinatorics or additive combinatorics is a fast developing area of research combining elements of number theory combinatorics harmonic analysis and ergodic theory its arguably best known result and the one that brought it to global prominence is the proof by ben green and terence tao of the long standing conjecture that primes contain arbitrarily long arithmetic progressions there are many accounts and expositions of the green tao theorem including the articles by kra 119 and tao 182 in the bulletin the purpose of the present article is to survey a broader highly interconnected network of questions and results built over the decades and spanning several areas of mathematics of which the green tao theorem is a famous descendant an old geometric problem lies at the heart of key conjectures in harmonic analysis a major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles an unexpected argument points harmonic analysts towards additive number theory with consequences that could have hardly been anticipated we will not try to give a comprehensive survey of harmonic analysis combinatorics or additive number theory we will not even be able to do full justice to our specific areas of focus instead referring the reader to the more complete expositions and surveys listed in section 7 our goal here is to emphasize the connections between these areas we will thus concentrate on relatively few problems chosen as much for their importance to their fields as for their links to each other the article is written from the point of view of an analyst who in the course of her work was gradually introduced to the questions discussed here and found them fascinating we hope that the reader will enjoy a taste of this experience

the galerkin method for the kdv equation using a new basis of smooth piecewise cubic polynomials abstract in this paper we introduce the galerkin method using a new basis of smooth piecewise cubic polynomials for the one dimensional nonlinear kdv equation the stability analysis shows that the present method is unconditionally stable in l 2 norm numerical experiments show that the proposed method preserves the conservation laws and is effective for simulating the motions and the interactions of solitary waves

three level fourier spectral approximations for fluid flow with low mach number fourier spectral approximation combined with a second order time differencing technique for fluid flow with low mach number is considered in this paper its generalized stability and convergence are strictly proven

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

rudin shapiro sequences along squares we estimate exponential sums of the form sigma n x f n 2 e nu n for a large class of digital functions f and nu is an element of r we deduce from these estimates the distribution along squares of this class of digital functions which includes the rudin shapiro sequence and some of its generalizations

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

the fourier pseudospectral method with a restrain operator for the korteweg de vries equation abstract in this paper we develop a fourier pseudospectral method with a restrain operator for the korteweg de vries equation we prove the generalized stability of the schemes and give convergence estimations depending on the smoothness of the solution of the p d e

estimates for n widths of multiplier operators of multiple walsh series abstract estimates for kolmogorov and gelfand n widths of multiplier operators of multiple walsh series are obtained upper and lower bounds are established for n widths of general multiplier operators these results are applied to get upper and lower bounds for n widths of specific multiplier operators which generate sets of finitely and infinitely differentiable functions in the dyadic sense it is shown that these estimates have order sharp in various important cases

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

on the cauchy problem for the evolution p laplacian equations with gradient term and source abstract in this paper the authors study the equation u t div d u p 2 d u u q 1 u d u l in r n with p 2 we first prove that for 1 l p 1 the solution exists at least for a short time then for p 2 l p 1 the existence and nonexistence of global in time solutions are studied in various situations

a non singular dynamical system without maximal ergodic inequality abstract it is shown that there is a non singular dynamical system for which the maximal ergodic inequality does not hold we further discuss the connection between a non singular dynamical systems and the pointwise convergence of the furstenberg ergodic averages

fully generic sequences and a multiple term return times theorem the return times theorem of bourgain furstenberg katznelson and ornstein states theorem bfko given any measure preserving transformation of a probability space and there is a subset of full measure so that for any and second dynamical system and there is a subset of full measure so that for any we will prove here a multiple term version of this result first proposed by i assani this will be accomplished by recasting the result completely as a fact about infinite names in where is a compact metrizable space we define what it means for such an infinite name to be fully generic for some shift invariant measure the core result here is theorem 1 which verifies that if is fully generic in some then the collection of names for which the pair is fully generic in will have measure one with respect to any shift invariant measure the multiple term return times theorem is an obvious induction on this statement by replacing the role played by orthogonality in the proofs in bfko and r2 by a disjointness result on joinings the argument avoids the need to identify distinguished factor algebras for the higher term averages

optimal error estimates of the legendre petrov galerkin method for the korteweg de vries equation in this paper the legendre petrov galerkin method for the korteweg de vries equation with nonperiodic boundary conditions is analyzed the nonlinear term is computed with the legendre spectral method and some pseudospectral methods respectively optimal error estimates in l2 norm are obtained for both semidiscrete and fully discrete schemes the method is also applicable to some 2m 1 th order differential equations

intersective polynomials and the polynomial szemeredi theorem abstract let p p 1 p r q n 1 n m be a family of polynomials such that p i z m z i 1 r we say that the family p has the psz property if for any set e z with d e lim sup n m e m n 1 n m 0 there exist infinitely many n z m such that e contains a polynomial progression of the form a a p 1 n a p r n we prove that a polynomial family p p 1 p r has the psz property if and only if the polynomials p 1 p r are jointly intersective meaning that for any k n there exists n z m such that the integers p 1 n p r n are all divisible by k to obtain this result we give a new ergodic proof of the polynomial szemeredi theorem based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds we also obtain as a corollary the following generalization of the polynomial van der waerden theorem if p 1 p r q n are jointly intersective integral polynomials then for any finite partition z i 1 k e i of z there exist i 1 k and a n e i such that a a p 1 n a p r n e i

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

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

the fourier pseudospectral method with a restrain operator for the rlw equation in this paper we develop a new fourier pseudospectral method with a restrain operator for the modified korteweg de vries mkdv equation we prove the generalized stability of the scheme from which the convergence follows

multifractal analysis of some multiple ergodic averages in linear cookie cutter dynamical systems in this paper we study the multiple ergodic averages of a locally constant real valued function in linear cookie cutter dynamical systems the multifractal spectrum of these multiple ergodic averages is completely determined

pointwise convergence of some multiple ergodic averages abstract we show that for every ergodic system x t 1 t d with commuting transformations the average 1 n d 1 0 n 1 n d n 1 0 n n 1 f 1 t 1 n j 1 d t j n j x f 2 t 2 n j 1 d t j n j x f d t d n j 1 d t j n j x converges for a e x x as n if x is distal we prove that the average 1 n n 0 n 1 f 1 t 1 n x f 2 t 2 n x f d t d n x converges for a e x x as n we also establish the pointwise convergence of averages along cubical configurations arising from a system with commuting transformations our methods combine the existence of sated and magic extensions introduced by austin and host respectively with ideas on topological models by huang shao and ye

multiple recurrence and nilsequences aiming at a simultaneous extension of khintchine s and furstenberg s recurrence theorems we address the question if for a measure preserving system x mathcal x mu t and a set a in mathcal x of positive measure the set of integers n such that mu a cap t n a cap t 2n a cap ldots cap t kn a mu a k 1 epsilon is syndetic the size of this set surprisingly enough depends on the length k 1 of the arithmetic progression under consideration in an ergodic system for k 2 and k 3 this set is syndetic while for k 4 it is not the main tool is a decomposition result for the multicorrelation sequence int f x f t n x f t 2n x ldots f t kn x d mu x where k and n are positive integers and f is a bounded measurable function we also derive combinatorial consequences of these results for example showing that for a set of integers e with upper banach density d e 0 and for all e 0 the set big n in mathbb z colon d big e cap e n cap e 2n cap e 3n big d e 4 epsilon big is syndetic

wiener s lemma along primes and other subsequences abstract inspired by subsequential ergodic theorems we study the validity of wiener s lemma and the extremal behavior of a measure on the unit circle via the behavior of its fourier coefficients k n along subsequences k n we focus on arithmetic subsequences such as polynomials primes and polynomials of primes and also discuss connections to rigidity sequences return times sequences and strongly sweeping out sequences as well as measures on r we also present consequences for orbits of operators and of c 0 semigroups on hilbert and banach spaces extending the results of goldstein 34 and goldstein nagy 36 the results are complemented by some open questions and indication of interesting research directions

noncommutative multi parameter wiener wintner type ergodic theorem abstract in this paper we establish a multi parameter version of bellow and losert s wiener wintner type ergodic theorem for dynamical systems not necessarily commutative more precisely we introduce a weight class d which is shown to strictly include the multi parameter bounded besicovitch weight class thus including the set d 1 k 1 d k d k 1 k d n d 1 d t d then we prove a multi parameter bellow and losert s wiener wintner type ergodic theorem for the class d and for a noncommutative trace preserving dynamical system m t m being a von neumann algebra restricted to d we also prove a noncommutative multi parameter analogue of bourgain s uniform wiener wintner ergodic theorem the noncommutativity and the multi parameter characters induce some difficulties in the proofs for instance our argument of proving the uniform convergence for a dense subset turns out to be quite different from the classical case since the pointwise argument does not work in the noncommutative setting also to obtain the uniform convergence in the largest spaces we need a maximal inequality between the orlicz spaces but it cannot be deduced by using classical extrapolation argument directly junge and xu s noncommutative maximal inequalities with the optimal order together with the atomic decomposition of orlicz spaces play the essential role in overcoming the second difficulty

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

holder estimates for nonlinear degenerate parabolic sytems on etablit des estimations de holder pour le gradient d espace u des solutions locales faibles des systemes paraboliques degeneres on etablit la continuite de holder des solutions non negatives d une equation parabolique degeneree

the quartile operator and pointwise convergence of walsh series the bilinear hilbert transform is given by h f g x p v f x t g x t t it satisfies estimates of the type h f g r c s t f s g t in this paper we prove such estimates for a discrete model of the bilinear hilbert transform involving the walsh fourier transform we also reprove the well known fact that the walsh fourier series of a function in lp 0 1 with 1 p converges pointwise almost everywhere the purpose of this exposition is to clarify the connection between these two results and to present an easy approach to recent methods of time frequency analysis

on the cauchy problem for the singular parabolic equations with gradient term abstract the cauchy problem for a singular parabolic equation with gradient term of the form u t div d u p 2 d u d u q is studied this paper both extends the existence of solutions of the porous media equation with the same gradient term to this cauchy problem for the case q p 1 and obtains the existence of solutions on the assumptions that measures as initial data for the case 0 q p 1 however as q p 1 the optimality of the stipulated assumptions to existence of solutions is still open

iterated trilinear fourier integrals with arbitrary symbols abstract we prove l p estimates for trilinear multiplier operators with singular symbols these operators arise in the study of iterated trilinear fourier integrals which are trilinear variants of the bilinear hilbert transform specifically we consider trilinear operators determined by multipliers that are products of two functions m 1 1 2 and m 2 2 3 such that the singular set of m 1 lies in the hyperplane 1 2 and that of m 2 lies in the hyperplane 2 3 while previous work 8 requires that the multipliers satisfy 1 2 2 3 our results allow for the case of the arbitrary multipliers which have common singularities

multifractal analysis of some multiple ergodic averages in this paper we study the multiple ergodic averages on the symbolic space m 0 1 m 1 n where m 2 l 2 q 2 are integers we give a complete solution to the problem of multifractal analysis of the limit of the above multiple ergodic averages actually we develop a non invariant and non linear version of thermodynamic formalism that is of its own interest we study a large class of measures called telescopic product measures the special case of telescopic product measures defined by the fixed points of some non linear transfer operators plays a crucial role in studying the level sets of the limit which are not shift invariant these measures share many properties with gibbs measures in the classical thermodynamic formalism our work also concerns variational principle pressure function and legendre transform in this new setting less

uniform bounds for the bilinear hilbert transforms i it is shown that the bilinear hilbert transforms h f g x p v r f x t g x t dt t map l p1 r l p2 r l p r uniformly in the real parameters when 2 p 1 p 2 oo and 1 p p 1 p 2 p 1 p 2 2 combining this result with the main result in 9 we deduce that the operators h 1 map l 2 r x l r l 2 r uniformly in the real parameter 0 1 this completes a program initiated by a calderon

a nilpotent roth theorem let t and s be invertible measure preserving transformations of a probability measure space x we prove that if the group generated by t and s is nilpotent then lim n ras infty frac 1 n sum n 1 n u t n x v s n x exists in l 2 norm for any u v l x we also show that for a with a 0 one has lim n ras infty frac 1 n sum n 1 n mu a cap t n a cap s n a 0 by the way of contrast we bring examples showing that if measure preserving transformations t s generate a solvable group then i the above limits do not have to exist ii the double recurrence property fails that is for some a a 0 one may have a t n a s n a 0 for all n finally we show that when t and s generate a nilpotent group of class c lim n ras infty frac 1 n sum n 1 n u t n x v s n x int ud mu int vd mu in l 2 x for all u v l x if and only if t s is ergodic on x x and the group generated by t 1 s t 2 s 2 t c s c acts ergodically on x

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

a pseudospectral scheme for the numerical calculation of shocks abstract a pseudospectral numerical scheme is developed in order to calculate the propagation of a shock wave we use a two step time differencing method and a chebyshev transform method to compute the space derivative such a scheme has been used to reduce the oscillations due to gibb s phenomenon this numerical method is applied to the solution of the burgers equation without viscosity term the accuracy of the numerical solutions is compared to the one given by two finite difference methods

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

integrable discretization of time and its application on the fourier pseudospectral method to the korteweg de vries equation this paper presents a new integrable discretization of the korteweg de vries kdv equation different from other discrete analogues we discretize the variable time and obtain an integrable differential difference system this system has the original kdv equation as a standard limit when the step size tends to zero the main idea is based on hirota s bilinear method and backlund transformation and can be applied to other integrable systems by applying the fourier pseudospectral method to the space variable we derive a new numerical scheme for the kdv equation numerical results are found to agree with the exact solution and the first five conservation quantities are preserved quite well

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

legendre and chebyshev spectral approximations of burgers equation we give error estimates for both legendre and chebyshev spectral approximations of the steady state burgers problem u xx lambda uu x f 0 in i 1 1 u 1 u 1 0 to do that we prove some abstract approximation properties of orthogonal projection operators in some weighted sobolev spacesh s i for both legendre and chebyshev weights

a spectral difference method for two dimensional viscous flow abstract we propose a spectral difference method for the 2 dimensional vorticity equation with a periodic boundary condition in one direction the solution satisfies a semidiscrete conservation law and thus better numerical results are obtained we also prove stability and convergence

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

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

a legendre petrov galerkin and chebyshev collocation method for third order differential equations a legendre petrov galerkin lpg method for the third order differential equation is developed by choosing appropriate base functions the method can be implemented efficiently also this new approach enables us to derive an optimal rate of convergence in l2 norm the method is applied to some nonlinear problems such as the korteweg de vries kdv equation with the chebyshev collocation treatment for the nonlinear term it is a legendre petrov galerkin and chebyshev collocation lpg cc method numerical experiments are given to confirm the theoretical result

spectral and pseudo spectral approximations of the navier stokes equations galerkin and collocation approximations by trigonometric polynomials to the stationary navier stokes equations with periodic boundary conditions are analyzed stability results and optimal rates of convergence for both velocity and pressure in sobolev spaces are proven essential tools to carry out this analysis are some abstract results concerning general polynomial approximations to nonsingular solutions of nonlinear problems brezzi rappag raviart numer math 36 1980 pp 1 25 maday quarteroni rairo numer anal 1980

combined finite element and pseudospectral method for the two dimensional evolutionary navier stokes equations a combined finite element and pseudospectral scheme is proposed for solving the two dimensional evolutionary navier stokes equations the artificial compressibility and filtering techniques are used it is also shown that the inf sup condition holds naturally for this combined method the generalized stability and optimal convergence rate in l 2 norm are proved some numerical results are presented

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

multiple ergodic theorems for arithmetic sets we establish results with an arithmetic flavor that generalize the polyno mial multidimensional szemeredi theorem and related multiple recurrence and conver gence results in ergodic theory for instance we show that in all these statements we can restrict the implicit parameter n to those integers that have an even number of dis tinct prime factors or satisfy any other congruence condition in order to obtain these refinements we study the limiting behavior of some closely related multiple ergodic averages with weights given by appropriately chosen multiplicative functions these averages are then analysed using a recent structural result for bounded multiplicative functions proved by the authors

spectral methods for a nonlinear initial value problem involving pseudo differential operators spectral methods fourier methods for approximating the solution of a nonlinear initial value problem involving pseudo differential operators are defined and analysed a semidiscrete approximation to the nonlinear equation based on an l 2 projection is described the semidiscrete l 2 approximation is shown to be a priori stable and convergent under sufficient decay and smoothness assumptions on the initial data it is shown that the semidiscrete method converges with infinite order that is higher order decay and smoothness assumptions imply higher order error bounds spectral schemes based on spacial collocation are also discussed

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

on solutions of some doubly nonlinear degenerate parabolic equations with absorption abstract the existence uniqueness regularity and behavior of solutions to the initial boundary value problem for the equation u t i 1 n x i u x i p 2 u x i u 0 with zero dirichlet boundary condition and with nonnegative initial condition are studied in a bounded domain in r n where 0 and p 1

cauchy problem for doubly degenerate parabolic equation with gradient source abstract we consider the cauchy problem for a doubly degenerate parabolic equation with a source term depending solely on the gradient a major result of this paper is to establish the local existence of solutions for this equation under optimal stipulated assumptions on the initial data

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

fourier chebyshev pseudospectral method for two dimensional vorticity equation a fourier chebyshev pseudospectral scheme is proposed for two dimensional unsteady vorticity equation the generalized stability and convergence are proved strictly the numerical results are presented

higher order regionally proximal equivalence relations for general minimal group actions abstract we introduce higher order regionally proximal relations suitable for an arbitrary acting group for minimal abelian group actions these relations coincide with the ones introduced by host kra and maass our main result is that these relations are equivalence relations whenever the action is minimal this was known for abelian actions by a result of shao and ye we also show that these relations lift through extensions between minimal systems answering a question by tao given a minimal system we prove that the regionally proximal equivalence relation of order d corresponds to the maximal dynamical antolin camarena szegedy nilspace factor of order at most d in particular the regionally proximal equivalence relation of order one corresponds to the maximal abelian group factor finally by using a result of gutman manners and varju under some restrictions on the acting group it follows that the regionally proximal equivalence relation of order d corresponds to the maximal pronilfactor of order at most d a factor which is an inverse limit of nilsystems of order at most d

singular parabolic equations with measures as initial data abstract in this paper we study the cauchy problem for the singular evolution p laplacian equations with gradient term and source on the assumption of measures as initial conditions for the supercritical case q p 1 p n we obtain that for every nonnegative solution there exists a nonnegative radon measure as initial trace and has some local regularity

bifurcation diagrams of a p laplacian dirichlet problem with allee effect and an application to a diffusive logistic equation with predation abstract we study bifurcation diagrams of positive solutions for the p laplacian dirichlet problem p u x f u 0 1 x 1 u 1 u 1 0 f u u p 1 g u where p 1 p y y p 2 y p u is the one dimensional p laplacian 0 is a bifurcation parameter and g is of allee effect type assuming one suitable condition on g we prove that on the u plane the bifurcation diagram consists of exactly one continuous curve with exactly one turning point where the curve turns to the right hence the problem has at most two positive solutions for each 0 more precisely we are able to prove the exact multiplicity of positive solutions we give an application to a p laplacian diffusive logistic equation with predation of holling type ii functional response to this logistic equation with multiparameters more precisely we give a complete description of the structure of the bifurcation diagrams

l p estimates for the biest i the walsh case we prove lp estimates theorem 1 8 for the walsh model of the biest a trilinear multiplier with singular symbol the corresponding estimates for the fourier model will be obtained in the sequel 11 biest of this paper

on the cauchy problem for the evolution p laplacian equations with gradient term and source and measures as initial data abstract in this paper we study the cauchy problem for the evolution p laplacian equations with gradient term and source on the assumption that measures as initial conditions

classification of blowup solutions for a parabolic p laplacian equation with nonlinear gradient terms abstract we mainly consider the following degenerate p 2 parabolic equation u t p u u m u q with homogeneous dirichlet boundary condition in a bounded domain r n before studying the properties of the solutions of this equation we first establish the local in time existence of its weak solutions then in different ranges of reaction exponents we give the complete classification of blowup results including l blowup and gradient blowup moreover under the subcritical condition of q p 1 m p 1 we can also prove that the solution is global in time

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

on the two dimensional bilinear hilbert transform we investigate the bilinear hilbert transform in the plane and the pointwise convergence of bilinear averages in ergodic theory arising from bf z 2 actions our techniques combine novel one and a half dimensional phase space analysis with more standard one dimensional theory