existence and multiplicity of nontrivial solutions for a class of modified nonlinear fourth order elliptic equations on r n in this paper we study the following fourth order elliptic equation of the form 2 u u v x u 2 u 2 u h x u x r n where 2 is the biharmonic operator 0 n 6 v c r n r and h c r n r r under appropriate assumptions on v x and h x u some existence and multiplicity results for nontrivial solutions are obtained via variational methods

local polynomial variance function estimation the conditional variance function in a heteroscedastic nonparametric regression model is estimated by linear smoothing of squared residuals attention is focused on local polynomial smoothers both the mean and variance functions are assumed to be smooth but neither is assumed to be in a parametric family the biasing effect of preliminary estimation of the mean is studied and a degrees of freedom correction of bias is proposed the corrected method is shown to be adaptive in the sense that the variance function can be estimated with the same asymptotic mean and variance as if the mean function were known a proposal is made for using standard bandwidth selectors for estimating both the mean and variance functions the proposal is illustrated with data from the lidar method of measuring atmospheric pollutants and from turbulence model computations

local superlinearity and sublinearity for the p laplacian we study the existence nonexistence and multiplicity of positive solutions for a family of problems updelta p u f lambda x u u in w 0 1 p upomega where is a bounded domain in mathbb r n n p and 0 is a parameter the family we consider includes the well known nonlinearities of ambrosetti brezis cerami type in a more general form namely lambda a x u q b x u r where 0 leqslant q p 1 r leqslant p 1 here the coefficient a x is assumed to be nonnegative but b x is allowed to change sign even in the critical case preliminary results of independent interest include the extension to the p laplacian context of the brezis nirenberg result on local minimization in w 0 1 p text and c 0 1 a c 1 alpha estimate for equations of the form updelta p u h x u with h of critical growth a strong comparison result for the p laplacian and a variational approach to the method of upper lower solutions for the p laplacian

existence results and iterative method for solving a nonlinear biharmonic equation of kirchhoff type abstract in this paper for investigating a boundary value problem for a nonlinear biharmonic equation of kirchhoff type we propose an efficient approach by the reduction of the problem to an operator equation for the nonlinear part of the equation the result is that we have established the existence and uniqueness of a solution under some easily verified conditions also we propose an iterative method for finding the solution some examples demonstrate the applicability of the theoretical results and the efficiency of the iterative method

a reexamination of diffusion estimators with applications to financial model validation time homogeneous diffusion models have been widely used for describing the stochastic dynamics of the underlying economic variables recently stanton proposed drift and diffusion estimators based on a higher order approximation scheme and kernel regression method he claimed that higher order approximations must outperform lower order approximations and concluded nonlinearity in the instantaneous return function of short term interest rates to examine the impact of higher order approximations we develop general and explicit formulas for the asymptotic behavior of both drift and diffusion estimators we show that these estimators will reduce the numerical approximation errors in asymptotic biases but their asymptotic variances escalate nearly exponentially with the order of approximation simulation studies also confirm our asymptotic results this variance inflation problem arises not only from nonparametric fitting but also from parametric fitting stanton s work also postulates the interesting qu

asymptotic nonequivalence of garch models and diffusions this paper investigates the statistical relationship of the garch model and its diffusion limit regarding the two types of models as two statistical experiments formed by discrete observations from the models we study their asymptotic equivalence in terms of le cam s deficiency distance to our surprise we are able to show that the garch model and its diffusion limit are asymptotically equivalent only under deterministic volatility with stochastic volatility due to the difference between the structure with respect to noise propagation in their conditional variances their likelihood processes asymptotically behave quite differently and thus they are not asymptotically equivalent this stochastic nonequivalence discredits a general belief that the two types of models are asymptotically equivalent in all respects and warns against the common financial practice that applies statistical inferences derived under the garch model to its diffusion limit

high energy solutions and nontrivial solutions for fourth order elliptic equations abstract in this paper we study the existence of nontrivial solutions and infinitely many high energy solutions for a class of nonlinear fourth order elliptic equations in r n via variational methods three main theorems are obtained

multiple and sign changing solutions for a class of semilinear biharmonic equation abstract in this paper under an improved hardy rellich s inequality we study the existence of multiple and sign changing solutions for a biharmonic equation in unbounded domain by the minimax method and linking theorem

the ivp for a nonlocal perturbation of the benjamin ono equation in classical and weighted sobolev spaces abstract we prove that the initial value problem associated to a nonlocal perturbation of the benjamin ono equation is locally and globally well posed in sobolev spaces h s r for any s 3 2 and we establish that our result is sharp in the sense that the flow map of this equation fails to be c 2 in h s r for s 3 2 finally we study persistence properties of the solution flow in the weighted sobolev spaces z s r h s r l 2 x 2 r d x for s r 0 we also prove some unique continuation properties of the solution flow in these spaces

estimators of diffusions with randomly spaced discrete observations a general theory we provide a general method to analyze the asymptotic properties of a variety of estimators of continuous time diffusion processes when the data are not only discretely sampled in time but the time separating successive observations may possibly be random we introduce a new operator the generalized infinitesimal generator to obtain taylor expansions of the asymptotic moments of the estimators as a special case our results apply to the situation where the data are discretely sampled at a fixed nonrandom time interval we include as specific examples estimators based on maximum likelihood and discrete approximations such as the euler scheme

the cauchy problem associated to the benjamin equation in weighted sobolev spaces abstract we study the initial value problem associated to the benjamin equation our purpose here is to establish persistence properties and some unique continuation properties of the solutions of this equation in weighted sobolev spaces

estimating the jump activity index under noisy observations using high frequency data it is widely accepted that the high frequency data are contaminated by microstructure noise whose effect on the statistical inference has been of increasing interest in the literature much of it however has focused on the integrated volatility in this article we investigate another important characteristic namely the jump activity index jai of a discretely sampled semi martingale corrupted by microstructure noise we point out that ignoring the microstructure noise can have a disastrous effect on the estimation of the jai consequently we propose a two stage procedure to estimate the jai it first reduces the effect of noise by local smoothing and then estimates the index from the smoothed data the asymptotic properties such as consistency and asymptotic normality are given simulations are conducted to evaluate the performance of the procedure finally we implement our estimators to some real datasets

sign changing solutions for some fourth order nonlinear elliptic problems in this paper we consider the existence and multiplicity of sign changing solutions for some fourth order nonlinear elliptic problems and some existence and multiple are obtained the weak solutions are sought by means of sign changing critical theorems

existence of nontrivial solutions of an asymptotically linear fourth order elliptic equation abstract in this paper using the mountain pass theorem we give the existence result for nontrivial solutions for a class of asymptotically linear fourth order elliptic equations

infinitely many homoclinic orbits for hamiltonian systems with indefinite sign subquadratic potentials abstract in this paper we deal with the existence and multiplicity of homoclinic solutions of the second order hamiltonian system u t l t u t w t u t 0 where l t and w t x are neither autonomous nor periodic in t under the assumption that w t x is indefinite sign and subquadratic as x and l t is a n n real symmetric positive definite matrices for all t r we establish some existence criteria to guarantee that the above system has at least one or infinitely many homoclinic solutions by using the genus properties in critical theory

efficient estimation of conditional variance functions in stochastic regression author s fan jianqing yao qiwei abstract conditional heteroscedasticity has been often used in modelling and understanding the variability of statistical data under a general setup which includes the nonlinear time series model as a special case we propose an e cient and adaptive method for estimating the conditional variance the basic idea is to apply a local linear regression to the squared residuals we demonstrate that without knowing the regression function we can estimate the conditional variance asymptotically as well as if the regression were given this asymptotic result established under the assumption that the observations are made from a strictly stationary and absolutely regular process is also veri ed via simulation further the asymptotic result paves the way for adapting an automatic bandwidth selection scheme an application with nancial data illustrates the usefulness of the proposed techniques

multiplicity of solutions for a class of fourth elliptic equations we show that there exist at least three nontrivial solutions for a class of fourth elliptic equations under navier boundary conditions by linking approaches

fourth order navier boundary value problem with combined nonlinearities abstract existence and multiplicity of solutions are proved for the fourth order navier boundary value problem with indefinite nonlinearity by using the least action principle the ekeland variational principle and the mountain pass theorem

existence and multiplicity of solutions for fourth order elliptic equations in rn abstract in this paper we study the fourth order elliptic equations 2 u u v x u f x u in r n u h 2 r n where 1 and the nonlinearity f is either superlinear or sublinear at infinity we establish the existence and multiplicity results without the compactness of embedding of the working space which unify and sharply improve the recent results of liu chen and wu j liu s chen x wu existence and multiplicity of solutions for a class of fourth order elliptic equations in r n j math anal appl 395 2012 608 615

multiplicity results for some fourth order elliptic equations abstract this paper deals with some fourth order elliptic equations with navier boundary condition by using the variational method some existence and multiplicity results are established

existence results for some fourth order nonlinear elliptic problems of local superlinearity and sublinearity in this paper we study the existence of positive solutions for the problem 0 1 2u c u f x u in u 0 u 0in u u 0on where c 1 and f x u satisfies the local superlinearity and sublinearity condition

infinitely many nontrivial solutions for a class of biharmonic equations via variant fountain theorems abstract by a variant version of fountain theorem due to zou w zou variant fountain theorems and their applications manuscripta math 104 2001 343 358 the existence of infinitely many solutions is obtained for a class of biharmonic equations where the nonlinearity involves a combination of superlinear and asymptotically linear terms

the cauchy problem for dissipative benjamin ono equation in weighted sobolev spaces abstract we study the well posedness in weighted sobolev spaces for the initial value problem ivp associated with the dissipative benjamin ono dbo equation we establish persistence properties of the solution flow in the weighted sobolev spaces z s r h s r l 2 x 2 r d x s r 0 we also prove some unique continuation properties in these spaces in particular such results of unique continuation show that our results of well posedness are sharp

estimating stochastic volatility diffusion using conditional moments of integrated volatility we exploit the distributional information contained in high frequency intraday data in constructing a simple conditional moment estimator for stochastic volatility diffusions the estimator is based on the analytical solutions of the first two conditional moments for the latent integrated volatility the realization of which is effectively approximated by the sum of the squared high frequency increments of the process our simulation evidence indicates that the resulting gmm estimator is highly reliable and accurate our empirical implementation based on high frequency five minute foreign exchange returns suggests the presence of multiple latent stochastic volatility factors and possible jumps

calibrating the degrees of freedom for automatic data smoothing and effective curve checking curve fitting and curve checking based on the local polynomial regression technique are commonly used data analytic methods in statistics this article examines in nonparametric settings both the asymptotic expressions and empirical formulas for degrees of freedom df a notion introduced by hastie and tibshirani of linear smoothers the asymptotic results give useful insights into the nonparametric modeling complexity meanwhile by substituting the exact dfs by the empirical formula an empirical version of the generalized cross validation egcv is obtained an automatic bandwidth selection method based on minimizing egcv is proposed for conducting local smoothing this procedure preserves full benefits of the ordinary and generalized cross validation but offers a substantial reduction in computational burden furthermore the egcv minimizing bandwidth can be extended in a very simple manner to fit multivariate models such as the varying coefficient models applications of calibrating dfs to impo

infinitely many solutions for fourth order elliptic equations abstract in this paper we study a class of fourth order elliptic equations setting on r n based on the minimax methods in critical point theory we obtain the existence of infinitely many large energy and small energy solutions which unifies and improves the recent results of yin and wu y yin x wu high energy solutions and nontrivial solutions for fourth order elliptic equations j math anal appl 375 2011 699 705

mathematical analysis of dynamic models of suspension bridges in this paper we present some mathematical analysis of dynamic pde models of suspension bridges as proposed by lazer and mckenna our results are illustrated by numerical simulation with physical interpretation

understanding exponential smoothing via kernel regression exponential smoothing is the most common model free means of forecasting a future realization of a time series it requires the specification of a smoothing factor which is usually chosen from the data to minimize the average squared residual of previous one step ahead forecasts in this paper we show that exponential smoothing can be put into a nonparametric regression framework and gain some interesting insights into its performance through this interpretation we also use theoretical developments from the kernel regression field to derive for the first time asymptotic properties of exponential smoothing forecasters

critical points at infinity in a fourth order elliptic problem with limiting exponent abstract this paper is concerned with the following nonlinear elliptic problem p 2 u u q u 0 in u u 0 on where is a bounded smooth domain in r n n 6 and q 1 2 n n 4 is the critical sobolev exponent we provide a version of morse lemma at infinity for this problem as application we characterize the critical points at infinity of the associated variational problem we also compute the difference of the topology that these critical points at infinity induce between the level sets of the functional corresponding to the problem

on the well posedness of higher order viscous burgers equations abstract we consider higher order viscous burgers equations with generalized nonlinearity and study the associated initial value problems for given data in the l 2 based sobolev spaces we introduce appropriate time weighted spaces to derive multilinear estimates and use them in the contraction mapping principle argument to prove local well posedness for data with sobolev regularity below l 2 we also prove ill posedness for this type of models and show that the local well posedness results are sharp in some particular cases viz when the orders of dissipation p and nonlinearity k 1 satisfy a relation p 2 k 1

the zakharov kuznetsov equation in weighted sobolev spaces abstract in this work we consider the initial value problem ivp associated to the two dimensional zakharov kuznetsov equation u t x 3 u x y 2 u u x u 0 x y r 2 t r u x y 0 u 0 x y we study the well posedness of the ivp in the weighted sobolev spaces h s r 2 l 2 1 x 2 y 2 r d x d y with s r r

the ivp for the benjamin ono zakharov kuznetsov equation in weighted sobolev spaces abstract in this paper we study the initial value problem associated with the benjamin ono zakharov kuznetsov equation we prove that the ivp for such equation is locally well posed in the usual sobolev spaces h s r 2 s 2 and in the anisotropic spaces h s 1 s 2 r 2 s 2 2 s 1 s 2 we also study the persistence properties of the solution and local well posedness in the weighted sobolev class z s r h s r 2 l 2 1 x 2 y 2 r d x d y where s 2 r 0 and s 2 r unique continuation properties of the solution are also established these continuation principles show that our persistence properties are sharp

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

critical semilinear biharmonic equations in rn an existence theorem is obtained for a fourth order semilinear elliptic problem in r n involving the critical sobolev exponent n 4 n 4 n 4 a preliminary result is that the best constant in the sobolev embedding l 2n n 4 r n is attained by all translations and dilations of 1 x 2 4 n 2 the best constant is found to be

high energy solutions of modified quasilinear fourth order elliptic equations with sign changing potential abstract in the present paper we consider the following modified quasilinear fourth order elliptic equation 1 1 2 u u v x u 1 2 u 2 u f x u for x r n u x h 2 r n where 2 is the biharmonic operator v c r n r and f c r n r r n 5 are allowed to be sign changing two main theorems on the existence of nontrivial solutions and infinitely many high energy solutions for eq 1 1 are obtained via variational methods

multiple nontrivial solutions for some fourth order semilinear elliptic problems abstract in this paper we consider the existence of multiple nontrivial solutions for some fourth order semilinear elliptic boundary value problems the weak solutions are sought by means of morse theory and local linking

multiple solutions for some fourth order nonlinear elliptic variational inequalities abstract we consider the solvability for fourth order nonlinear elliptic variational inequalities some existence results are given by the variational methods

on solutions of second and fourth order elliptic equations with power type nonlinearities abstract we study second and fourth order semilinear elliptic equations with a power type nonlinearity depending on a power p and a parameter 0 for both equations we consider dirichlet boundary conditions in the unit ball b r n regularity of solutions strictly depends on the power p and the parameter we are particularly interested in the radial solutions of these two problems and many of our proofs are based on an ordinary differential equation approach

existence and multiplicity of solutions for a class of fourth order elliptic equations in r n abstract in this paper we study the existence and multiplicity of nontrivial solutions for the following fourth order elliptic equations 2 u u v x u f x u in r n u h 2 r n via variational methods two main theorems are obtained

arch models as diffusion approximations abstract this paper investigates the convergence of stochastic difference equations e g arch to stochastic differential equations as the length of the discrete time intervals between observations goes to zero these results are applied to the garch 1 1 model of bollerslev 1986 and to the ar 1 exponential arch model of nelson 1989 in their continuous time limits the conditional variance processes in these models have stationary distributions that are inverted gamma and lognormal respectively in addition a class of diffusion approximations based on the exponential arch model is developed

on some fourth order semilinear elliptic problems in r n the existence of two solutions for a fourth order semilinear elliptic problem involving critical growth from the viewpoint of sobolev embedding was established the basic tools used in the analysis were the mountain pass theorem constrained minimization and concentration compactness principle the existence of positive principle eigenvalues for the corresponding linear elliptic problem was also investigated

existence of nontrivial solutions for fourth order asymptotically linear elliptic equations abstract in this paper we discuss the following fourth order semilinear elliptic problem 2 u c u f x u x u u 0 x where f x t is asymptotically linear with respect to t at infinity is a smooth bounded domain in r n and n 4 in this case f x t does not satisfy the following ambrosetti rabinowitz type condition see ambrosetti and rabinowitz 1973 9 for short which is called the ar condition that is for some 0 and m 0 0 f x t 0 t f x s d s 1 2 f x t t uniformly a e x and t m which is important in applying the mountain pass theorem by a variant version of the mountain pass theorem we obtain the existence of nontrivial solutions to the above problem under suitable assumptions of f x t which generalizes and improves the results in liu and wang 2007 12 and an and liu 2008 13

on a fourth order elliptic equation with critical nonlinearity in dimension six abstract this paper is devoted to the study of the following nonlinear elliptic problem 2 u ku 5 u 0 in u u 0 on where k is a positive function and is a bounded and smooth domain in r 6 we prove a version of morse lemma at infinity for this problem using this morse lemma at infinity we characterize the critical points at infinity of the associated functional and we give an existence result

a test for model specification of diffusion processes we propose a test for model specification of a parametric diffusion process based on a kernel estimation of the transitional density of the process the empirical likelihood is used to formulate a statistic for each kernel smoothing bandwidth which is effectively a studentized l2 distance between the kernel transitional density estimator and the parametric transitional density implied by the parametric process to reduce the sensitivity of the test on smoothing bandwidth choice the final test statistic is constructed by combining the empirical likelihood statistics over a set of smoothing bandwidths to better capture the finite sample distribution of the test statistic and data dependence the critical value of the test is obtained by a parametric bootstrap procedure properties of the test are evaluated asymptotically and numerically by simulation and by a real data example

non gaussian ornstein uhlenbeck based models and some of their uses in financial economics non gaussian processes of ornstein uhlenbeck ou type offer the possibility of capturing important distributional deviations from gaussianity and for flexible modelling of dependence structures this paper develops this potential drawing on and extending powerful results from probability theory for applications in statistical analysis their power is illustrated by a sustained application of ou processes within the context of finance and econometrics we construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive ou processes and we study these models in relation to financial data and theory

econometric analysis of realized volatility and its use in estimating stochastic volatility models summary the availability of intraday data on the prices of speculative assets means that we can use quadratic variation like measures of activity in financial markets called realized volatility to study the stochastic properties of returns here under the assumption of a rather general stochastic volatility model we derive the moments and the asymptotic distribution of the realized volatility error the difference between realized volatility and the discretized integrated volatility which we call actual volatility these properties can be used to allow us to estimate the parameters of stochastic volatility models without recourse to the use of simulation intensive methods

local superlinearity and sublinearity for indefinite semilinear elliptic problems in this paper the usual notions of superlinearity and sublinearity for semilinear problems like du f x u are given a local form and extended to indefinite nonlinearities here f x s is allowed to change sign or to vanish for s near zero as well as for s near infinity some of the well known results of ambrosetti bre zis cerami are partially extended to this context

the ivp for the benjamin ono equation in weighted sobolev spaces abstract in this work we continue our study initiated in fonseca and ponce 2011 11 on the uniqueness properties of real solutions to the ivp associated to the benjamin ono bo equation in particular we shall show that the uniqueness results established in fonseca and ponce 2011 11 do not extend to any pair of non vanishing solutions of the bo equation also we shall prove that the uniqueness result established in fonseca and ponce 2011 11 under a hypothesis involving information of the solution at three different times cannot be relaxed to two different times

infinitely many solutions for schrodinger maxwell equations with indefinite sign subquadratic potentials in this paper we deal with multiplicity of solutions of a class of sublinear schrodinger maxwell equations du v x u fu f x u inr 3 d f u 2 lim x f x 0 inr 3 under more relaxed assumptions on v and f we establish some existence criteria to guarantee that the above problem has at least one or infinitely many nontrivial solutions by using the genus properties in critical theory

existence of solutions for some fourth order nonlinear elliptic problems in this paper we consider the existence of positive negative and sign changing solutions for some fourth order semilinear elliptic boundary value problems we present new results on invariant sets of the gradient flows of the corresponding variational functionals the structure of the invariant sets will be built into minimax procedures to construct the sign changing solutions

on a nonlinear fourth order elliptic equation involving the critical sobolev exponent in this paper we use an algebraic topological argument due to bahri and coron to show how the topology of the domain influences the existence of positive solutions of the following problem involving the bilaplacian operator with the critical sobolev exponent 2u un 4 n 4 in u 0 in u u 0 on where is a bounded domain of rn n 5 with a smooth boundary

on sign changing solution for a fourth order asymptotically linear elliptic problem in this paper we deal with a fourth order elliptic problem whose nonlinear term is asymptotically linear at both zero and infinity by using the variational method we obtain an existence result of sign changing solutions as well as positive and negative solutions

well posedness for the two dimensional generalized zakharov kuznetsov equation in anisotropic weighted sobolev spaces abstract we consider the well posedness of the initial value problem associated to the k generalized zakharov kuznetsov equation in fractional weighted sobolev spaces h s r 2 l 2 x 2 r 1 y 2 r 2 d x d y s r 1 r 2 r our method of proof is based on the contraction mapping principle and it mainly relies on the well posedness results recently obtained for this equation in the sobolev spaces h s r 2 and a new pointwise commutator type formula involving the group induced by the linear part of the equation and the fractional anisotropic weights to be considered

solutions for fourth order elliptic equations on rn involving uδ u2 and sign changing potentials abstract we obtain existence and multiplicity results for fourth order elliptic equations on r n involving u u 2 and sign changing potentials our results generalize some recent results on this kind of problems to study this kind of problems we first consider the case that the potential v is coercive so that the working space can be compactly embedded into lebesgue spaces then we studied the case that the potential v is bounded so that the working space is exactly h 2 r n which can not be compactly embedded into lebesgue spaces anymore to deal with this more difficult case we study the weak continuity of the term in the energy functional corresponding to the term u u 2 in the equation

dynamic integration of time and state domain methods for volatility estimation time and state domain methods are two common approaches to nonparametric prediction whereas the former uses data predominantly from recent history the latter relies mainly on historical information combining these two pieces of valuable information is an interesting challenge in statistics we surmount this problem by dynamically integrating information from both the time and state domains the estimators from these two domains are optimally combined based on a data driven weighting strategy which provides a more efficient estimator of volatility asymptotic normality is separately established for the time domain the state domain and the integrated estimators by comparing the efficiency of the estimators we demonstrate that the proposed integrated estimator uniformly dominates the other two estimators the proposed dynamic integration approach is also applicable to other estimation problems in time series extensive simulations are conducted to demonstrate that the newly proposed procedure outper

infinitely many solutions for fourth order elliptic equations with general potentials abstract this paper is concerned with the following fourth order elliptic equations 2 u u v x u f x u in r n u h 2 r n where u r n r under general assumption on potential v x and f x u is indefinite sign and sublinear as u we establish the existence of infinitely many solutions by using the genus properties in critical point theory recent results from the literature are extended

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation