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

multilinear calderon zygmund theory abstract a systematic treatment of multilinear calderon zygmund operators is presented the theory developed includes strong type and endpoint weak type estimates interpolation a multilinear t 1 theorem and a variety of results regarding multilinear multiplier operators

axiom a maps are dense in the space of unimodal maps in the ck topology in this paper we prove c k structural stability conjecture for unimodal maps in other words we shall prove that axiom a maps are dense in the space of c k unimodal maps in the c k topology here k can be 1 2 1 1 the structural stability conjecture the structural stability conjecture was and remains one of the most interesting and important open problems in the theory of dynamical systems this conjecture states that a dynamical system is structurally stable if and only if it satisfies axiom a and the transversality condition in this paper we prove this conjecture in the simplest nontrivial case in the case of smooth unimodal maps these are maps of an interval with just one critical turning point to be more specific let us recall the definition of axiom a maps definition 1 1 let x be an interval we say that a c k map f x satisfies the axiom a conditions if f has finitely many hyperbolic periodic attractors the set f x f i sh yperbolic where f i sa union of the

nonuniversal critical behaviour in planar ising models with extended radially symmetric defects using conformal mapping and finite size scaling the authors determine the local magnetic exponent near a defect characterized by couplings of the form j r j 1 a r in accordance with phenomenological considerations they find that it depends continuously on the amplitude a both for a defect in the interior and on the boundary

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

conformal invariance and universality in finite size scaling the universal relation between critical exponents and the amplitude of the correlation length divergence as a function of finite size at the critical point of two dimensional systems is shown to be a consequence of conformal invariance both periodic and free boundary conditions are considered

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

finite size scaling and the two dimensional xy model shows that the finite size scaling assumption and particularly a conjectured relation involving the exponent eta are valid in the low temperature phase of the 2d xy model order by order in a perturbative expansion of the two point correlation function

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

variations of hodge structure legendre submanifolds and accessibility variations of hodge structure of weight two are integral manifolds for a distribution in the tangent bundle of a period domain this distribution has dimension h2 0h 1 and is nonintegrable for h2 0 1 in this case it is known that the dimension of an integral manifold does not exceed 1h2oh 2 here we give a new proof based on an analogy between griffiths horizontal differential system of algebraic geometry and the contact system of classical mechanics we show also that any two points in such a domain can be joined by a horizontal curve which is piecewise holomorphic

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

semi infinite two dimensional ising model with marginally inhomogeneous couplings and conformal invariance the authors consider a semi infinite two dimensional ising model with nearest neighbour couplings that deviate from the bulk critical couplings by al 2 where l is the distance from the surface the surface critical exponents of this system are non universal under a conformal mapping onto a strip of width l the al 1 inhomogeneity transforms into a l pi sin pi l l 1 for the square lattice the spectrum of the transfer matrix in the strip geometry is calculated exactly in the extreme anisotropic limit the analytical results and numerical results for the triangular lattice are compared with predictions of conformal invariance

local critical behaviour at aperiodic surface extended perturbation in the ising quantum chain the surface critical behaviour of the semi infinite one dimensional quantum ising model in a transverse field is studied in the presence of an aperiodic surface extended modulation the perturbed couplings are distributed according to a generalized fredholm sequence leading to a marginal perturbation and varying surface exponents the surface magnetic exponents are calculated exactly whereas the expression of the surface energy density exponent is conjectured from a finite size scaling study the system displays surface order at the bulk critical point above a critical value of the modulation amplitude it may be considered as a discrete realization of the hilhorst van leeuwen model

polynomial growth estimates for multilinear singular integral operators on demontre un theoreme pour des mesures de carleson et on l applique a l operateur de kato en dimension 1 on demontre un critere de limitation general on donne une application aux commutateurs de coolderon on etudie les d commutateurs

critical behaviour in aperiodic systems the author considers layered systems with aperiodically modulated couplings and study the effect of inhomogeneity on the critical behaviour using scaling arguments a relevance irrelevance type criterion is formulated and for relevant inhomogeneities the scaling form of singular quantities is determined the author has performed exact calculations for the surface magnetization of aperiodic quantum ising chains and for a 1 1 dimensional directed polymer in a longitudinal aperiodic environment the results for both systems are in accord with scaling considerations for a marginal aperiodicity the critical exponents are found to be non universal

coupled contact systems and rigidity of maximal dimensional variations of hodge structure in this article we prove that locally griffiths horizontal distribution on the period domain is given by a generalized version of the familiar contact differential system as a consequence of this description we obtain strong local rigidity properties of maximal dimensional variations of hlodge structure for example we prove that if the weight is odd greater than one then there is a unique germ of maximal dimensional variation of hodge structure through every point of the period domain similar results hold if the weight is even with the exception of one case

schrodinger invariance and strongly anisotropic critical systems the extension of strongly anisotropic or dynamical scaling to local scale invariance is investigated for the special case of an anisotropy or dynamical exponent z 2 the group of local scale transformation considered is the schrodinger group which can be obtained as the nonrelativistic limit of the conformal group the requirement of schrodinger invariance determines the two point function in the bulk and reduces the three point function to a scaling form of a single variable scaling forms are also derived for the two point function close to a free surface which can be either spacelike or timelike these results are reproduced in several exactly solvable statistical systems namely the kinetic ising model with glauber dynamics lattice diffusion lifshitz points in the spherical model and critical dynamics of the spherical model with a nonconserved order parameter for generic values of evidence from higher order lifshitz points in the spherical model and from directed percolation suggests a simple scaling form of the two point function

finite lattice extrapolation algorithms two algorithms for sequence extrapolation due to van den broeck and schwartz 1979 and bulirsch and stoer 1964 are reviewed and critically compared applications to three state and six state quantum chains and to the 2 1 d ising model show that the algorithm of bulirsch and stoer is superior in particular if only very few finite lattice data are available

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

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

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

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

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

radial fredholm perturbation in the two dimensional ising model and gap exponent relation we consider concentric circular defects in the two dimensional ising model which are distributed according to a generalized fredholm sequence i e at exponentially increasing radii this type of aperiodicity does not change the bulk critical behaviour but introduces a marginal extended perturbation the critical exponent of the local magnetization is obtained through finite size scaling using a corner transfer matrix approach in the extreme anisotropic limit it varies continuously with the amplitude of the modulation and is closely related to the magnetic exponent of the radial hilhorst van leeuwen model through a conformal mapping of the system onto a strip the gap exponent relation is shown to remain valid for such an aperiodic defect

hypersurface variations are maximal ii the variation of hodge structure defined by the natural family of hypersurfaces of degreed and dimensionn is maximal if the cohomology has hodge level 1 there is a small list of hypersurfaces of level one which give non maximal variations plane curves of degreed 5 cubics of dimension 3 and 5 and quartic threefolds

almost global wellposedness of the 2 d full water wave problem we consider the problem of global in time existence and uniqueness of solutions of the 2 d infinite depth full water wave equation it is known that this equation has a solution for a time period 0 t e for initial data of the form e where t depends only on in this paper we show that for such data there exists a unique solution for a time period 0 e t e this is achieved by better understandings of the nature of the nonlinearity of the full water wave equation

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

generic integral manifolds for weight two period domains we define the notion of a generic integral element for the griffiths distribution on a weight two period domain draw the analogy with the classical contact distribution and then show how to explicitly construct an infinite dimensional family of integral manifolds tangent to a given element

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 variations of hodge structure of maximal dimension with associated hodge numbers ℎ 2 0 2 and ℎ 1 1 2 1 do not arise from geometry the specified variations are proved to be covered by a bounded contractible domain q2 after classifying the analytic boundary components of q with respect to a fixed realization the group of the biholomorphic automorphisms aut q2 and the aut q orbit structure of q2 are found explicitly then n is shown to admit no quasiprojective arithmetic quotients whereas the lack of geometrically arising variations covered by q2 the weight 1 period domain sp n r u arises from geometry in as much as it classifies the first cohomologies of the n dimensional abelian varieties with fixed polarization the geometric locus on the weight 2 period domain is shown to satisfy the griffiths horizontality cf 9

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

bounds on the dimension of variations of hodge structure we derive upper bounds on the dimension of a variation of hodge structure of weight two and show that these bounds are sharp using them we exhibit maximal geometric variations of hodge structure analogous results for higher weight are obtained in the presence of a nondegeneracy hypothesis and variations coming from hypersurfaces are shown to be nondegenerate maximal geometric variations of higher weight are also constructed