on the spectra of a class of self affine measures on r2 abstract for i 1 2 let 0 i 1 for an expanding real matrix m d i a g 1 1 2 1 and a three element digit set d z 2 with cardinality d let m d be the self affine measure defined by m d 1 d d d m d m d let f 3 2 1 3 l 1 l 2 t l 1 l 2 n 0 l 1 l 2 2 and z m d x r 2 d d e 2 i d x 0 if a f 3 2 z m d a mod z 2 and x 0 1 2 d d e 2 i d x d 0 0 t then l 2 m d has an exponential orthonormal basis if and only if i 1 3 z 0 for i 1 2

analytic smoothing effect for nonlinear schrodinger equation with quintic nonlinearity abstract we prove the global existence of analytic solutions to nonlinear schrodinger equation with quintic nonlinearity in n space dimensions for sufficiently small cauchy data with exponential decay the smallness assumption on the data is imposed in terms of the critical sobolev space h n 2 1 2 moreover a characterization of some class of analytic functions is given

asymptotic behavior of solutions to time dependent hartree equations on considere le comportement asymptotique pour t grand des solutions du probleme de cauchy pour l equation iu t u f u 2 u t x r r u o x x x r 3 ou f u 2 r 1 u 2 r 3 u t y 2 x y dy

a numerical study of the self similar solutions of the schrodinger map we present a numerical study of the self similar solutions of the localized induction approximation of a vortex filament these self similar solutions which are explicitly known constitute a one parameter family and develop a singularity in finite time we study a number of boundary conditions that allow us to reproduce the mechanism of singularity formation some related questions are also considered

on solutions of the initial value problem for the nonlinear schrodinger equations on considere le probleme aux valeurs initiales pour l equation de schrodinger non lineaire de la forme iu t u f u 2 u t x r r n u 0 x x x r n on etudie la regularite des solutions quand les fonctions initiales ne sont pas regulieres

dilation method and smoothing effects of solutions to the benjamin ono equation in this paper we study smoothing effects of solutions to the benjamin ono equation where h is the hilbcrt transform defined by we prove that if h 2 and x x 4 then the solution u of bo belongs to where

domain and range of the modified wave operator for schrodinger equations with a critical nonlinearity we study the final problem for the nonlinear schrodinger equation i partial t u frac 1 2 delta u lambda u frac 2 n u quad t x in mathbf r times mathbf r n where lambda in bf r n 1 2 3 if the final data u in bf h 0 alpha left phi in bf l 2 left 1 left vert x right vert right alpha phi in bf l 2 right with frac n 2 alpha min left n 2 1 frac 2 n right and the norm vert widehat u vert bf l infty is sufficiently small then we prove the existence of the wave operator in l2 we also construct the modified scattering operator from h0 to h0 with frac n 2 delta alpha

on the derivative nonlinear schro dinger equation abstract in this paper we discuss the cauchy problem for the derivative nonlinear schrodinger equation i t 2i x 2 0 0 x x where 0 under an explicit smallness condition of the initial data we prove the unique global existence of solutions to this problem in the usual sobolev spaces in the weighted sobolev spaces and in the schwartz class we describe the smoothing effect in detail furthermore for the data decaying exponentially at infinity we prove that the above equation has unique local solutions which are analytic in the space direction

spectral property of self affine measures on mathbb r n abstract we study spectral properties of the self affine measure m d generated by an expanding integer matrix m m n z and a consecutive collinear digit set d 0 1 q 1 v where v z n 0 and q 2 is an integer some sufficient conditions for m d to be a spectral measure or to have infinitely many orthogonal exponentials are given moreover for some special cases we can obtain a necessary and sufficient condition on the spectrality of m d our study generalizes the one dimensional results proved by dai et al 4 5

on the spectra of sierpinski type self affine measures abstract for a class of sierpinski type self affine measures on r 2 a characterization for all maximal orthogonal sets of exponentials is given by this characterization a sufficient condition for a maximal orthogonal set to be a basis of l 2 is given

local smoothing properties of dispersive equations is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data for the linear wave equation for instance the answer is no however in 10 t kato found a local smoothing property of the korteweg de vries equation the solution of the initial value problem is locally one derivative smoother than the initial datum kato s proof uses in a curcial way the algebraic properties of the symbol for the korteweg de vries equation and the fact that the underlying spatial dimension is one actually judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect one would be inclined to believe this was a special property of the k dv equation this is not however the case in this paper we attempt to describe a general local smoothing effect for dispersive equations and systems the smoothing effect is due to the dispersive nature of the linear part of the equation all the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property to mention only a few the k dv benjamin ono intermediate long wave various boussinesq and schrodinger equations are included we study thus equations and systems of the form

regularity of solutions to nonlinear dispersive equations on considere le probleme de cauchy pour les equations dispersives non lineaires de la forme t u ip d u f u x r n t u x 0 u 0 x

analysis of orthogonality and of orbits in affine iterated function systems we introduce a duality for affine iterated function systems aifs which is naturally motivated by group duality in the context of traditional harmonic analysis our affine systems yield fractals defined by iteration of contractive affine mappings we build a duality for such systems by scaling in two directions fractals in the small by contractive iterations and fractals in the large by recursion involving iteration of an expansive matrix by a fractal in the small we mean a compact attractor x supporting hutchinson s canonical measure and we ask when is a spectral measure i e when the hilbert space l 2 mu has an orthonormal basis onb of exponentials e lambda lambda in lambda we further introduce a fourier duality using a matched pair of such affine systems using next certain extreme cycles and positive powers of the expansive matrix we build fractals in the large which are modeled on lacunary fourier series and which serve as spectra for x our two main results offer simple geometric conditions allowing us to decide when the fractal in the large is a spectrum for x our results in turn are illustrated with concrete sierpinski like fractals in dimensions 2 and 3

nonexistence of asymptotically free solutions for a nonlinear schrodinger equation let u be a nontrivial smooth solution to iut u u p 1u if n 1 and 2 p 3 then there does not exist any finite energy free solution v such that u t v t 2 0 as t this extends a theorem of strauss in which the same result was proved for 1 p 2

self similar solutions for the 1 d schrodinger map on the hyperbolic plane in this paper we will study self similar solutions for x t x x ss the equivalent in the minkowski 3 space mathbb r 2 1 to the localized induction approximation flow trying to adapt some results given by gutierrez rivas and vega we will show the existence of a one parameter family of smooth solutions developing a corner in finite time the main difference with respect to the euclidean case studied by those authors will be the proof of the boundedness of t e 1 and e 2 the equivalents of t b and n in mathbb r 2 1

solvability of the localized induction equation for vortex motion the initial and the initial boundary value problems for the localized induction equation which describes the motion of a vortex filament are considered we prove the existence of solutions of both problems globally in time in the sense of distribution by the method of regularization

smooth norms in dense subspaces of banach spaces abstract in the first part of our paper we show that l has a dense linear subspace which admits an equivalent real analytic norm as a corollary every separable banach space as well as l 1 c also has a dense linear subspace which admits an analytic renorming by contrast no dense subspace of c 0 1 admits an analytic norm in the second part we prove solving in particular an open problem of guirao montesinos and zizler in 7 that every banach space with a long unconditional schauder basis contains a dense subspace that admits a c smooth norm finally we prove that there is a proper dense subspace of l c 1 that admits no g teaux smooth norm here l c 1 denotes the banach space of real valued bounded and countably supported functions on 1

remark on the scattering operator for the cubic nonlinear dirac equation in three space dimensions abstract this paper is concerned with the scattering operator s for the three dimensional dirac equation with a cubic nonlinearity it follows from known results that s is well defined on a neighborhood of 0 in the sobolev space h r 3 c 4 for any 1 in the present paper we prove that for any m n and s max m there exists some neighborhood u of 0 in the weighted sobolev space h s m r 3 c 4 such that s u h s m r 3 c 4

smooth partitions of unity on banach spaces abstract we show a new characterisation of the existence of smooth partitions of unity on a banach space this leads to a slight generalisation of some as well as a very easy recovery of most of the known results using a unified treatment

l c ℝn decay of classical solutions for nonlinear schrodinger equations on etudie le probleme aux valeurs initiales pour l equation de schrodinger non lineaire iu t u f u 2 u u o x x x r n t r 1 n 11 sous de bonnes hypotheses de regularite sur f et et des conditions de croissance et de signe sur f on montre que les normes maximum des solutions decroissent quand t au meme taux que celles des solutions de l equation de schrodinger libre

non spectral problem for the planar self affine measures abstract in this paper we consider the non spectral problem for the planar self affine measures m d generated by an expanding integer matrix m m 2 z and a finite digit set d z 2 let p 2 be a positive integer e p 2 1 p i j t 0 i j p 1 and z d 2 x 0 1 2 d d e 2 i d x 0 we show that if z d 2 e p 2 0 and gcd det m p 1 then there exist at most p 2 mutually orthogonal exponential functions in l 2 m d in particular if p is a prime then the number p 2 is the best

geometric optics and long range scattering for one dimensional nonlinear schrodinger equations with the methods of geometric optics used in 2 we provide a new proof of some results of 10 to construct modified wave operators for the one dimensional cubic schrodinger equation we improve the rate of convergence of the nonlinear solution towards the simplified evolution and get better control of the loss of regularity in sobolev spaces in particular using the results of 9 we deduce the existence of a modified scattering operator with small data in some sobolev spaces we show that in terms of geometric optics this gives rise to a random phase shift at a caustic

smoothing effects and dispersion of singularities for the schrodinger evolution group we describe smoothing effects and dispersion of singularities for the schrodinger evolution group in the weighted sobolev spaces under a fairly general assumption on the potential it is shown that all singularities in the wavefunction vanish instantly whenever the initial state has sufficient decay we measure the regularity gained by the wavefunction by the decay property of the initial state no assumptions on the regularity of the initial state are imposed throughout the paper

on densely isomorphic normed spaces abstract in the first part of our note we prove that every weakly lindelof determined wld in particular every reflexive non separable banach x space contains two dense linear subspaces y and z that are not densely isomorphic this means that there are no further dense linear subspaces y 0 and z 0 of y and z which are linearly isomorphic our main result theorem b concerns the existence of biorthogonal systems in normed spaces in particular we prove under the continuum hypothesis ch that there exists a dense linear subspace of l 2 1 or more generally every wld space of density 1 which contains no uncountable biorthogonal system this result lies between two fundamental results concerning biorthogonal systems namely the construction of kunen under ch of a non separable banach space which contains no uncountable biorthogonal system and the construction of todorcevic under martin maximum of an uncountable biorthogonal system in every non separable banach space

space time behavior of propagators for schrodinger evolution equations with stark effect abstract we give a detailed description of the space time behavior of the unitary propagator u t s t s r determined by the schrodinger evolution equation i t u 1 2 u e xu v t x u t r x r n where e is a homogeneous electric field and v is a time dependent potential

stability of the self similar dynamics of a vortex filament in this paper we continue our investigation of self similar solutions of the vortex filament equation also known as the binormal flow or the localized induction equation our main result is the stability of the self similar dynamics of small perturbations of a given self similar solution the proof relies on finding precise asymptotics in space and time for the tangent and the normal vectors of the perturbations a main ingredient in the proof is the control of the evolution of weighted norms for a cubic one dimensional schrodinger equation connected to the binormal flow by hasimoto s transform

extensions of vector valued functions with preservation of derivatives abstract let x and y be banach or normed linear spaces and f x a closed set we apply our recent extension theorem for vector valued baire one functions to obtain an extension theorem for vector valued functions f f y with pre assigned derivatives with preservation of differentiability at every point where the pre assigned derivative is actually a derivative preservation of continuity preservation of point wise lipschitz property and for finite dimensional domain x preservation of strict differentiability and global eventually local lipschitz continuity this work depends on the paper extensions of vector valued baire one functions with preservation of points of continuity koc and kola 2016 20

smoothing of bump functions abstract let x be a separable banach space with a schauder basis admitting a continuous bump which depends locally on finitely many coordinates then x admits also a c smooth bump which depends locally on finitely many coordinates

μm d orthogonality and compatible pair abstract let m d be a self affine measure associated with an expanding integer matrix m m n z and a finite subset d z n in the present paper we study the m d orthogonality and compatible pair conditions as well as relations between them the research here is based on the structure of vanishing sums of roots of unity and is closely related to the problem of spectral self affine measure

non spectral self affine measure problem on the plane domain the self affine measure m d corresponding to an expanding integer matrix m abcd andd 00 10 01 is supported on the attractor or invariant set of the iterated function system d x m 1 x d d d in the present paper we show that if a d 2 4 ad bc and ad bc is not a multiple of 3 then there exist at most 3 mutually orthogonal exponential functions in l2 m d and the number 3 is the best this extends several known results on the non spectral self affine measure problem the proof of such result depends on the characterization of the zero set of the fourier transform m d and provides a way of dealing with the non spectral problem

long range scattering for nonlinear schrodinger equations in one space dimension we consider the scattering problem for the nonlinear schrodinger equation in 1 1 dimensions open image in new window where x r 0 r p 3 we show that modified wave operators for exist on a dense set of a neighborhood of zero in the lebesgue spacel2 r or in the sobolev spaceh1 r the modified wave operators are introduced in order to control the long range nonlinearity u 2u

weighted sobolev spaces and rapidly decreasing solutions of some nonlinear dispersive wave equations where d is a positive integer a is a d dimensional laplacian and the subscripts x t denote partial differentiations the initial value problems for 1 i 1 3 require the asymptotic conditions as x tends to infinity and one must specify spaces of functions satisfying the desired asymptotic conditions the usual choices of spaces are the sobolev spaces z p well posedness in hs of the initial value problems for 1 1 1 3 was established by many authors 3 5 8 12 181 it is of interest to know that eqs l l 1 3 have solutions decreasing faster than hs convergence as x tends to infinity in particular solutions in the schwartz spaces s for each t provided that their initial data are in s this is motivated by tanaka s result for the korteweg de vries equation 131 which establishes the existence of s solutions by the inverse scattering techniques and by the fact that eqs l l 1 3 have travelling solutions in s solitary waves which may play an important role of asymptotic behavior of solutions as t tends to infinity

the cardinality of certain μm d orthogonal exponentials abstract the self affine measures m d corresponding to the case i m p i 3 d 0 e 1 e 2 e 3 in the space r 3 and the case ii m p i 2 d 0 e 1 e 2 e 1 e 2 in the plane r 2 are non spectral where p 1 is odd i n is the n n identity matrix and e 1 e n are the standard basis of unit column vectors in r n one of the non spectral problem on m d is to estimate the number of orthogonal exponentials in l 2 m d and to find them in the present paper we show that in both cases i and ii there are at most 4 mutually orthogonal exponentials in l 2 m d each and the number 4 is the best

smoothing properties for the higher order nonlinear schrodinger equation with constant coefficients abstract we study local and global existence and smoothing properties for the initial value problem associated to a higher order nonlinear schrodinger equation with constant coefficients which appears as a model for propagation of pulse in an optical fiber

uniqueness properties of solutions to schrodinger equations nx j 1 2 u x 2 x 0 harmonic functions in the unit ball x r n x 2 since harmonic functions are still real analytic in x r n x 1 in fact it is well known that if p x d is a linear elliptic differential operator with real analytic coefficients andp x d u 0 in a open set r n then u is real analytic in hence the s u c p also holds for such solutions through the work of hadamard 28 on the uniqueness of the cauchy problem which is closely related to the strong unique continuation property discussed earlier it became clear for applications in nonlinear problems that it would be desirable to establish the strong unique continuation property for operators whose coefficients are not necessarily real analytic oreven c the first results in this direction were found in the pioneering work of carleman 9 when

smoothness in weakly compactly generated banach spaces received august 25 1982 revised april 15 1983 if x is a weakly compactly generated wcg banach space then x admits an equivalent c smooth norm if a wcg banach space x admits a ck smooth function with bounded support then x admits ck smooth partitions of unity

on solutions of the initial value problem for the nonlinear schrodinger equations in one space dimension t 0 xe has a regularizing property that the solutions become smooth instantly for t 0 even if their initial data are not smooth for example in l n our aim of this paper is to show that the nls equation 1 1 1 2 has a similar regularizing effect more precisely all solutions of finite energy to 1 1 1 2 are smooth for tq 0 provided the initial functions in hi n or in

spectrality of the planar sierpinski family abstract let be a borel probability measure with compact support in r 2 is called a spectral measure if there exists a countable set r 2 such that e e 2 i x is an orthonormal basis for l 2 in this note we prove that the integral sierpinski measure a d is a spectral measure if and only if a d is admissible this completely settles the spectrality of integral sierpinski measures in r 2

spectral property of the bernoulli convolutions for 0 1 let be the bernoulli convolution associated with jorgensen and pedersen p jorgensen s pedersen dense analytic subspaces in fractal l2 spaces j anal math 75 1998 185 228 proved that if 1 q where q is an even integer then l2 has an exponential orthonormal basis we show that for any 0 1 l2 contains an infinite orthonormal set of exponential functions if and only if is the nth root of a fraction p q where p is an odd integer and q is an even integer

smoothing effect for nonlinear schrodinger equations in exterior domains abstract we consider the following nonlinear schrodinger equations in exterior domains i t u 1 2 u u 2 u t x r x d u 0 x x x d u t x 0 or u t6v 0 t x r x d where d x rn x r d r 0 and v denotes the outward normal unit vector at x d in this paper we prove the radially symmetric solutions of have a smoothing property

microlocal dispersive smoothing for the schrodinger equation this paper establishes a connection between the microlocal smoothness of solutions of the initial value problem for schrodinger s equation and the global behavior of bicharacteristics of the principal symbol in particular localized initial data gives rise to solutions which are microlocally smooth at all points which are not trapped backwards by the bicharacteristic flow the origin of the phenomenon is in the dispersive nature of the equation the results imply microlocal regularity properties of the fundamental solution 1995 john wiley sons inc

non spectrality of planar self affine measures with three elements digit set abstract the self affine measure m d associated with an affine iterated function system d x m 1 x d d d is uniquely determined the problems of determining the spectrality or non spectrality of a measure m d have been received much attention in recent years one of the non spectral problem on m d is to estimate the number of orthogonal exponentials in l 2 m d and to find them in the present paper we show that for an expanding integer matrix m m 2 z and the three elements digit set d given by m a b d c and d 0 0 1 0 0 1 if a c b d 3 z then there exist at most 3 mutually orthogonal exponentials in l 2 m d and the number 3 is the best this confirms the three elements digit set conjecture on the non spectrality of self affine measures in the plane

local smoothing effects for a nonlinear timoshenko type equation where x e ir t r 0 here a denotes the usual laplace operator vu is the spatial gradient of u and m s is a nonlinear smooth function the term m jr vu12 du au is usually known as timoshenko s type nonlinearity and appears in a variety of models mostly associated with nonlinear vibrations of beams l results on the existence of global solutions of 1 1 are available in the literature 2 we are concerned here with further properties of the solutions more specifically we show that there is a class of solutions of 1 1 which are locally more regular than their initial data we prove the result assuming a suitable decay of the data at infinity our main results theorems 3 1 and 3 2 in this paper will only be given in the one dimensional case however as will become clear during our proofs several ingredients are valid in any dimension there is a close relationship between solutions of 1 l and solutions of a nonlinear schrodinger equation at the end of this section we will give more details on this fact thanks to this observation our strategy was to combine classical energy estimates with the fact that the operator l a2 at2 avax4 6 almost commutes with the operators pm xu 2td u and

scalar spectral measures associated with an operator fractal we study a spectral theoretic model on a hilbert space l2 where is a fixed cantor measure in addition to we also consider an independent scaling operator u acting in l2 to make our model concrete we focus on explicit formulas we take to be the bernoulli infinite convolution measure corresponding to scale number 14 we then define the unitary operator u in l2 from a scale by 5 operation the spectral theoretic and geometric properties we have previously established for u are as follows i u acts as an ergodic operator ii the action of u is not spatial and finally iii u is fractal in the sense that it is unitarily equivalent to a countable infinite direct sum of twisted copies of itself in this paper we prove new results about the projection valued measures and scalar spectral measures associated to u and its constituent parts our techniques make use of the representations of the cuntz algebra o2 on l2

non spectral problem for a class of planar self affine measures abstract the self affine measure m d corresponding to an expanding matrix m m n r and a finite subset d r n is supported on the attractor or invariant set of the iterated function system d x m 1 x d d d the spectral and non spectral problems on m d including the spectrum tiling problem implied in them have received much attention in recent years one of the non spectral problem on m d is to estimate the number of orthogonal exponentials in l 2 m d and to find them in the present paper we show that if a b c z a 1 c 1 and a c z 3 z m a b 0 c and d 0 0 1 0 0 1 then there exist at most 3 mutually orthogonal exponentials in l 2 m d and the number 3 is the best this extends several known conclusions the proof of such result depends on the characterization of the zero set of the fourier transform m d and provides a way of dealing with the non spectral problem

on the μm d orthogonal exponentials abstract let m d be a self affine measure corresponding to a given affine iterated function system d x m 1 x d d d in the present paper we will study the problem of how to determine the l 2 m d space has finite or infinite orthogonal exponentials such research is motivated by a conjecture on the non spectrality of m d we first obtain a non spectral criterion which extends the result of dutkay and jorgensen in opposition to the condition of this criterion we then obtain some conditions which imply the infinite orthogonal systems in l 2 m d these are necessary for further investigation on the spectrality of m d as an application we completely settle the corresponding problem for the generalized planar sierpinski family

the asymptotic behavior of nonlinear schrodinger equations on considere l equation de schrodinger non lineaire ns i u t u 2 u p 1 u dans r n n 2 avec 0 soit h o 2 et u l 2 r n u 2 u 2 xu 2 ou u q est la norme l 9 de u soit 1 2 n p 1 4 n 2 alors pour tout u 0 il existe un unique u l 2 r n tel que la solution u t de ns avec u 0 u 0 ait l asymptote libre u quand t lim t u t exp ith 0 u 2 0

sierpinski type spectral self similar measures abstract we consider the sierpinski type self similar measure on r 2 with contraction ratio 0 1 we show that is a spectral measure if and only if 1 3 p for some integer p 0 a similar characterization for bernoulli convolution is due to dai 2 over which 1 2 p

gain of regularity for the kp i equation abstract in this paper we study the smoothness properties of solutions to the kp i equation we show that the equation s dispersive nature leads to a gain in regularity for the solution in particular if the initial data possesses certain regularity and sufficient decay as x then the solution u t will be smoother than for 0 t t where t is the existence time of the solution