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

a tree approach to p variation and to integration we consider a real valued path it is possible to associate a tree to this path and we explore the relations between the tree the properties of p variation of the path and integration with respect to the path in particular the fractal dimension of the tree is estimated from the variations of the path and young integrals with respect to the path as well as integrals from the rough paths theory are written as integrals on the tree examples include some stochastic paths such as martingales levy processes and fractional brownian motions for which an estimator of the hurst parameter is given

multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice i homogeneous problems we present a computational approach for the wkb approximation of the wave function of an electron moving in a periodic one dimensional crystal lattice we derive a nonstrictly hyperbolic system for the phase and the intensity where the flux functions originate from the bloch spectrum of the schrodinger operator relying on the framework of the multibranch entropy solutions introduced by brenier and corrias we compute efficiently multiphase solutions using well adapted and simple numerical schemes in this first part we present computational results for vanishing exterior potentials which demonstrate the effectiveness of the proposed method

stability of spectral types for jacobi matrices under decaying random perturbations abstract we study stability of spectral types for semi infinite self adjoint tridiagonal matrices under random decaying perturbations we show that absolutely continuous spectrum associated with bounded eigenfunctions is stable under hilbert schmidt random perturbations we also obtain some results for singular spectral types

a novel mesh regeneration algorithm for 2d fem simulations of flows with moving boundary a novel mesh regeneration algorithm is proposed to maintain the mesh structure during a finite element simulation of flows with moving solid boundary with the current algorithm a new body fitted mesh can be efficiently constructed by solving a set of laplace equations developed to specify the displacements of individual mesh elements these equations are subjected to specific boundary conditions determined by the instantaneous body motion and other flow boundary conditions the proposed mesh regeneration algorithm has been implemented on an arbitrary lagrangian eulerian ale framework that employs an operator splitting technique to solve the navier stokes equations the integrated numerical scheme was validated by the numerical results of four existing problems a flow over a backward facing step a uniform flow over a fixed cylinder the vortex induced vibration of an elastic cylinder in uniformly incident flow and a complementary problem that compares the transient drag coefficient for a cylinder impulsively set into motion to that measured on a fixed cylinder in a starting flow good agreement with the numerical or experimental data in the literature was obtained and new transient flow dynamics was revealed the scheme performance is further examined with respect to the parameter employed in the mesh regeneration algorithm

on a trilinear form related to the carleson theorem abstract demeter and thiele introduced the two dimensional bilinear hilbert transform in demeter and thiele 2010 3 and showed that the carleson operator can be identified in particular instances of the corresponding trilinear form demeter considered the walsh model of one such form in demeter 2012 2 in relation to the discussion of the walsh carleson theorem we prove boundedness of this trilinear form for a single triple of exponents at the boundary of the previously established range for this purpose we adapt the multilinear bellman function technique from kovac 2011 5 to the one and a half dimensional time frequency space

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

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

ell p left mathbb z d right estimates for discrete operators of radon type variational estimates we prove ell p left mathbb z d right bounds for p in 1 infty of r variations r in 2 infty for discrete averaging operators and truncated singular integrals of radon type we shall present a new powerful method which allows us to deal with these operators in a unified way and obtain the range of parameters of p and r which coincide with the ranges of their continuous counterparts

2d nearly orthogonal mesh generation with controls on distortion function a method to control the distortion function of the ryskin and leal rl orthogonal mesh generation system is presented the proposed method considers the effects from not only the local orthogonal condition but also the local smoothness condition the geometry and the mesh size on the distortion function the distortion function is determined by both the scale factors and the averaged scale factors of the constant mesh lines two adjustable parameters are used to control the local balance of the orthogonality and the smoothness the proposed method is successfully applied to several benchmark examples and the natural river channels with complex geometries

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

adaptive mesh redistribution method for domains with complex boundaries an adaptive structured mesh redistribution method asmrm that permits smooth transition from non uniformly distributed boundary points to solution adaptive interior points and enables the resolution of complex flow in the complex boundary region as well as away from the boundary is proposed it is a variant of the traditional variational technique it involves a combination of static and dynamic monitor functions the former for mesh distribution in the vicinity of a complex boundary and the latter for mesh adaption with the evolving solution elsewhere its effectiveness is demonstrated on some example problems and it is then applied to a chevron nozzle the proposed method is shown to be capable of generating a mesh with a good balance of orthogonality and smoothness in the entire domain

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 bernstein approximants and the f variation of a fuzzy random variable abstract in this paper we study different properties of bernstein approximants of a fuzzy random variable as those in relation with the approximation of fuzzy random variables by means of its n bernstein approximant moreover we define and study the concept of variation for a fuzzy random variable

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

the absolutely continuous spectrum of one dimensional schrodinger operators with decaying potentials we investigate one dimensional schrodinger operators with asymptotically small potentials it will follow from our results that if with then is an essential support of the absolutely continuous part of the spectral measure we also prove that if then the spectrum is purely absolutely continuous on these results are optimal

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

frechet differentiability p variation and uniform donsker classes classically a stastistical functional is defined on a space of distribution f on the real line with the supremum norn the values may be real or themselves functions such as the quantile function f 1 nonlinear functionals are studied via their derivatives this paper and a related one dudley 1991a will show that the differentiability properties originally proved by reeds 1976 also treated in fernholz 1983 can be improved substantially

schrodinger operators with many bound states consider the schrodinger operators h d2 dx2 v x we present a method for estimating the potential in terms of the negative eigenvalues of these operators among the applications are inverse lieb thirring inequalities and several sharp results concerning the spectral properties of h

uniqueness and regularity of proper harmonic maps introduction the purpose of this paper is to study some uniqueness and regularity properties of the dirichlet problem at infinity for proper harmonic maps between hyperbolic spaces in general if the metrics of two complete noncompact manifolds mm and n are given locally by ds2 imj gij dxzdxj and dsy en h 3 du du respectively then the energy density function of a c1 map u m n is defined by

a uniform estimate in this article we study bilinear operators whose input consists of two functions on the real line and whose output is again a function on the real line we shall be concerned with a priori estimates for these operators for simplicity we initially require the two input functions to be schwartz functions and allow the output function to be a tempered distribution by abstract generalities such a bilinear operation b can be formally written as

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

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

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

superdiffusive limits for deterministic fast slow dynamical systems we consider deterministic fast slow dynamical systems on mathbb r m times y of the form begin aligned left begin array ll x k 1 n x k n n 1 a big x k n big n 1 alpha b big x k n big v y k y k 1 f y k end array right end aligned where alpha in 1 2 under certain assumptions we prove convergence of the m dimensional process x n t x lfloor nt rfloor n to the solution of the stochastic differential equation begin aligned mathrm d x a x mathrm d t b x diamond mathrm d l alpha end aligned where l alpha is an alpha stable levy process and diamond indicates that the stochastic integral is in the marcus sense in addition we show that our assumptions are satisfied for intermittent maps f of pomeau manneville type

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

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

a class of bilinear multipliers given by littlewood paley decomposition abstract in this paper we study a particular class of bilinear multipliers which are given by littlewood paley decompositions in the first part of the paper we show that if is a bilinear multiplier for p q r 1 p q satisfying the holder s condition 1 p 1 q 1 r and have support inside 0 1 then its periodization j z j is also a bilinear multiplier for the same triplet p q r further we show that for a given triplet p q r of exponents outside the local l 2 range there exists sequence j j z of uniformly bounded bilinear multipliers so that the function j z j is not a bilinear multiplier for the triplet p q r in the second part we describe several results for bilinear multipliers of the type m which are similar to the first part in nature in particular we point out that the results described by p honzik 2014 13 can be generalized to a more general setting

a note on harmonic quasiconformal mappings abstract in this note we show that a harmonic quasiconformal mapping f u i v with respect to the poincare metric of the upper half plane r 2 onto itself such that v x y v y or u x y u x is a conformal mapping

on the singular spectrum of schrodinger operators with decaying potential the relation between hausdorff dimension of the singular spectrum of a schrodinger operator and the decay of its potential has been extensively studied in many papers in this work we address similar questions from a different point of view our approach relies on the study of the so called krein systems for schrodinger operators we show that some bounds on the singular spectrum obtained recently by remling and christ kiselev are optimal

boundary treatment for 2d elliptic mesh generation in complex geometries this paper presents a boundary treatment method for 2d elliptic mesh generation in complex geometries corresponding to neumann dirichlet boundary conditions sliding boundary conditions the proposed method aims at achieving orthogonal and smooth nodal distribution along irregular boundaries in this method a three lined auxiliary mesh is constructed which is composed of the boundary line an auxiliary line generated on its inner side and the reflection line of this auxiliary line the movements of the boundary nodes are determined by solving this auxiliary mesh the boundary nodes are further corrected and adjusted by a second order parabolic interpolation method and a weighting parameter depending on the curvature of the boundary the proposed method was demonstrated through examples and applied to field cases with complex geometries it has been shown that this method is stable and effective in producing high quality meshes

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

2d orthogonal grid generation with boundary point distribution control a numerical method for 2d orthogonal grid generation with control of the boundary point distribution is presented the method is based on the solution of a system of partial differential equations the grid cell aspect ratio the so called distortion function is calculated from its definition equation in the entire domain the method allows the specification of the boundary point distribution in all the boundaries and also the distance of the first grid node to the boundary the method is successfully applied to several geometries including geometries with grid singularities and grids around airfoils and cross sections of ship sterns

first kdv integrals and absolutely continuous spectrum for 1 d schrodinger operator we consider 1 d schrodinger operators on l 2 r with slowly decaying potentials under some conditions on the potential related to the first integrals of the kdv equation we prove that the a c spectrum of the operator coincides with the positive semiaxis and the singular spectrum is unstable examples show that for special classes of sparse potentials these results can not be improved

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

harmonicity of the inverse of a harmonic diffeomorphism abstract in this paper we obtain a necessary and sufficient condition for the harmonicity of the inverse of a harmonic diffeomorphism as an application we give explicit representations of reversible logharmonic diffeomorphisms as another application we connect the harmonicity of the inverse of a hyperbolic harmonic quasiconformal diffeomorphism with the schoen conjecture

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

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

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

orthogonal mapping in two dimensions abstract a method for the generation of orthogonal boundary fitted curvilinear coordinates for arbitrary simply and doubly connected domains is developed on the basis of the theory of quasi conformal mappings of quadrilaterals and of previous work by ryskin and leal the method has useful applications in orthogonal grid generation in two dimensional and axi symmetric domains and in the extension of rapid elliptic solvers and spectral methods to complex geometries a new technique for the calculation of the conformal module of quadrilaterals is also presented

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

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

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

imbedded singular continuous spectrum for schrodinger operators 1 1 hv v x and some self adjoint boundary condition at zero the operator 1 1 describes a charged particle such as an electron in the electric field v x when v x is decaying quickly one expects the spectral and dynamical properties of hy to remain close to those of the free operator ho recall that with the operator 1 1 one can associate in a canonical way a spectral measure p see for example 6 2 which contains much information about the quantum system the classical result in this direction going back to the beginning of the century is that if v g l1 the spectral measure on the positive semi axis remains purely absolutely continuous in one dimension an absolutely continuous spectrum corresponds to the ballistic rate of propagation of the particle a natural question to ask is what are the critical rates of decay at which some changes in the spectral and dynamical properties of hy may happen as far as v x has any decay at all the essential spectrum coincides with 0 oo but the quality of the spectrum and dynamics may change in 1928 wigner and von neumann 27 showed that there exist potentials v x satisfying v x jsuch that hy has positive imbedded eigenvalue e 1 this is a purely quantum resonance phenomenon since v can be chosen arbitrarily small and the bound state originates from long range correlations in the potential rather than the usual confining effect naboko 17 and simon 25 provided constructions which show that much more drastic changes are possible if the potential decays arbitrarily slower than coulomb namely for any positive function h x which grows at infinity there exist potentials v such that v x yjgand hy has a dense set of positive eigenvalues due to some recent results 3 21 it is known that the absolutely continuous spectrum on the positive semi axis is preserved for v o c l x l 2 e and in fact for v g l2 8 more precisely for such potentials the absolutely continuous part of the spectral measure iac gives positive weight to any subset of 0 oo of positive lebesgue measure therefore the naboko

quasi banach valued inequalities via the helicoidal method abstract we extend the helicoidal method from 1 to the quasi banach context proving in this way multiple banach and quasi banach vector valued inequalities for paraproducts and for the bilinear hilbert transform bht as an immediate application we obtain mixed norm estimates for in the whole range of lebesgue exponents one of the novelties in the quasi banach framework that is when 0 r 1 which we expect to be useful in other contexts as well is the linearization of the operator k t f k g k r 1 r achieved by dualizing its weak l p quasinorms through l r see proposition 8 another important role is played by the sharp evaluation of the operatorial norm t i 0 f 1 f g 1 g 1 h r which is obtained by dualizing the weak l p quasinorms through l with r in the banach case the linearization of the operator and the sharp estimates for the localized operatorial norm can be both achieved through the classical generalized restricted type l 1 dualization

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

dispersive estimate for the 1d schrodinger equation with a steplike potential abstract we prove a sharp dispersive estimate p a c u t x c t 1 2 u 0 l 1 r for the one dimensional schrodinger equation i u t u x x v x u v 0 x u 0 where 1 x 2 v l 1 r and v 0 is a step function real valued and constant on the positive and negative real axes

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

bounded solutions and absolute continuity of sturm liouville operators this work is a contribution to the theory of the absolutely continuous spectrum of one dimensional schriidinger operators i e sturm liouville operators generated by differential expressions of the form ru u qu in proofs of absolute continuity for such operators and also for the closely related dirac systems always asymptotic properties of solutions of zu lu for real or complex values of 1 have played an important role see for example 18 19 8 4 9 lo in these works and a number of others particular classes of potentials q were treated using properties of solutions in the course of the proof of absolute continuity somewhat more recently the connection between asymptotic properties of solutions and absolute continuity was given a theory i e results of the following form were proved let the solutions of zu lu satisfy some specific asymptotic property for a varying in some subset of r no or almost no a priori assumptions on q then operators generated by z are absolutely continuous in this set a first result of this type was given by carmona 3 who proved that local uniform boundedness of solutions together with their derivatives in some open a set implies pure absolute continuity in this set here we will mostly rely on the concept of non or racy of solutions which was shown by gilbert and pearson 7 12 61 to be a powerful tool in the investigation of the continuous spectrum of sturm liouville operators in section 2 we give some consequences of their results whi ch are useful in proofs of absolute continuity these consequences range from non existence of imbedded eigenvalues and the existence of absolutely continuous parts to pure absolute continuity and the existence of transient 210 0022 247x 92 5 00

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

bilinear oscillatory integrals and boundedness for new bilinear multipliers abstract we consider bilinear oscillatory integrals i e pseudo product operators whose symbol involves an oscillating factor lebesgue space inequalities are established which give decay as the oscillation becomes stronger this extends the well known linear theory of oscillatory integral in some directions the proof relies on a combination of time frequency analysis of coifman meyer type with stationary and non stationary phase estimates as a consequence of this analysis we obtain lebesgue estimates for new bilinear multipliers defined by non smooth symbols

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

stability of singular spectral types under decaying perturbations we look at invariance of a e boundary condition spectral behavior under perturbations w of half line continuum or discrete schrodinger operators we extend the results of del rio simon stolz from compactly supported w s to suitable short range w we also discuss invariance of the local hausdorff dimension of spectral measures under such perturbations

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

the hormander multiplier theorem for multilinear operators in this paper we provide a version of the mikhlin hormander multiplier theorem for multilinear operators in the case where the target space is l p for pe 1 this extends a recent result of tomita 15 who proved an analogous result for p 1

on the linear combinations of harmonic univalent mappings abstract in this paper we derive several sufficient conditions of the linear combinations of harmonic univalent mappings to be univalent and convex in the direction of the real axis furthermore some illustrative examples and imagine domains of the linear combinations satisfying the desired conditions are enumerated

modified prufer and efgp transforms and the spectral analysis of one dimensional schrodinger operators using control of the growth of the transfer matrices wediscuss the spectral analysis of continuum and discrete half line schrodinger operators with slowly decaying potentials among our results we show if where w has compact support and then h has purely a c resp purely s c spectrum on 0 if for n 1 2 a n potentials where a n are independent identically distributed random variables with e a n 0 e a 2 n 1 and 2 we find singular continuous spectrum with explicitly computable fractional hausdorff dimension

on the absolutely continuous spectrum of stark operators the stability of the absolutely continuous spectrum of the one dimensional stark operator under perturbations of the potential is discussed the focus is on proving this stability under minimal assumptions on smoothness of the perturbation a general criterion is presented together with some applications these include the case of periodic perturbations where we show that any perturbation v l 1 h 1 2 preserves the a c spectrum

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

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 absolutely continuous spectrum of one dimensional schrodinger operators with square summable potentials for continuous and discrete one dimensional schrodinger operators with square summable potentials the absolutely continuous part of the spectrum is essentially supported by 0 and 2 2 respectively this fact is proved by considering a priori estimates for the transmission coefficient

jump estimates for operators associated with polynomials abstract in this study we prove the jump inequalities for singular integrals and averages of radon type with rough kernels