maximal theorems for the directional hilbert transform on the plane for a schwartz function f on the plane and a non zero v r 2 define the hilbert transform of f in the direction v to be h v f x p v r f x vy dy y let be a schwartz function with frequency support in the annulus 1 2 the l 2 estimate is sharp the method of proof is based upon techniques related to the pointwise convergence of fourier series indeed our main theorem implies this result on fourier series

on the schrodinger equation for the gaussian potential a exp λr2 the energy eigenvalues of the attractive radial gaussian potential for various eigenstates are very accurately determined within the framework of the hypervirial pade scheme

algebraic recurrence formulae for matrix elements between the solutions of the transformed jacobi eigenequation the algebraic recursive determination of matrix elements of some selected families of functions q1 x between the solutions psi jm x of factorisable equations is reinvestigated the possibilities of the procedure outlined in a previous paper are enlarged by using the connection between factorisation types i e the different possible factorisations of the same differential equation the computation of matrix elements between the jacobi eigenfunctions psi jm x approximately sin ax 2 alpha 1 2 cos ax 2 beta 12 pv alpha beta cos ax where a is a real or pure imaginary constant is studied in detail algebraic recurrence formulae satisfied by matrix elements of qt x cos ax 2 p sin ax 2 q tan ax 2 t qt x sin ax p tan ax q cos ax t qt x psi jt x and qt x psi tm x are given and for the particular cases qt x tan ax 2 t and qt x cos ax t closed form expressions are obtained as an illustrative application it is briefly shown how the expressions can serve to derive analytical approximations of the bound state energies for the potential v x a exp x2 l l 1 x2 some further applications are pointed out

compactness for nonlinear continuity equations abstract we prove compactness and hence existence for solutions to a class of nonlinear transport equations the corresponding models combine the features of linear transport equations and scalar conservation laws we introduce a new method which gives quantitative compactness estimates compatible with both frameworks

on interpolation of bilinear operators abstract in this paper we study interpolation of bilinear operators between products of banach spaces generated by abstract methods of interpolation in the sense of aronszajn and gagliardo a variant of bilinear interpolation theorem is proved for bilinear operators from corresponding weighted c 0 spaces into banach spaces of non trivial the periodic fourier cotype this result is then extended to the spaces generated by the well known minimal and maximal methods of interpolation determined by quasi concave functions in the case when a maximal construction is generated by hilbert spaces we obtain a general variant of bilinear interpolation theorem combining this result with the abstract grothendieck theorem of pisier yields further results the results are applied in deriving a bilinear interpolation theorem for calderon lozanovsky for orlicz spaces and an embedding interpolation formula for e p summing operators

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

reconstruction of singularities from full scattering data by new estimates of bilinear fourier multipliers we prove that the singularities of a complex valued potential q in the schrodinger hamiltonian q can be reconstructed from the linear born approximation for full scattering data by averaging in the extra variables we prove that with this procedure the accuracy in the reconstruction improves the previously known accuracy obtained from fixed angle or backscattering data in particular for q in w alpha 2 for 0 in 2d we recover the main singularity of q with an accuracy of one derivative in 3d the accuracy is epsilon 1 2 increasing with this gives a mathematical basis for diffraction tomography the proof is based on some new estimates for multidimensional bilinear fourier multipliers of independent interest

stabilizing controllers for bilinear systems the problem of finding stabilizing feedback controllers for continuous bilinear systems where the controls act additively and multiplicatively simultaneously is treated the applicability of the quadratic control law of jacobson 5 and others is extended to the case when the a matrix has arbitary eigenvalues under certain conditions a class of bilinear systems common among biochemical flow systems is defined the dyadic bilinear systems a control scheme the so called division controller for dyadic bilinear systems is suggested the practicality of the control schemes is demonstrated on the problem of controlling the neutron level in a fission reactor

on multilinear oscillatory integrals nonsingular and singular basic questions concerning nonsingular multilinear operators with oscillatory factors are posed and partially answered l p norm inequalities are established for multilinear integral operators of calderon zygmund type which incorporate oscillatory factors e ip where p is a real valued polynomial

stability of discrete time bilinear systems with constant inputs in this paper discrete time bilinear systems are considered necessary and sufficient conditions are developed which ensure that for any constant input in an unbounded control domain the trajectories of the system tend to the origin as time increases

a note on the schrodinger equation for the potential a exp x2 l l 1 x2 the bound states energies and eigenfunctions of the schrodinger equation with a radial gaussian potential are obtained using a perturbational and also a variational treatment on a conveniently chosen basis of transformed jacobi functions the accuracy of the results is fairly good

on calderon s conjecture with constants cfi p1 p2 depending only on fi p1 p2 and p p1p2 p1 p2 hold the flrst result of this type is proved in 4 and the purpose of the current paper is to extend the range of exponents p1 and p2 for which 2 is known in particular the case p1 2 p2 1 is solved to the a rmative this was

uniform estimates for bilinear hilbert transforms and bilinear maximal functions associated to polynomials we study the bilinear hilbert transforms and bilinear maximal functions associated to non flat polynomial curves and obtain uniform l r estimates for r d 1 over d where d is the degree of the polynomial in addition the lower bound of r is sharp up to the end point

l p bounds for a maximal dyadic sum operator the authors prove lp bounds in the range 1 p for a maximal dyadic sum operator on rn this maximal operator provides a discrete multidimensional model of carleson s operator its boundedness is obtained by a simple twist of the proof of carleson s theorem given by lacey and thiele 7 adapted in higher dimensions 9 in dimension one the lp boundedness of this maximal dyadic sum implies in particular an alternative proof of hunt s extension 4 of carleson s theorem on almost everywhere convergence of fourier integrals

the jacobi eigenfunctions and the quantum mechanical hypervirial theorems method for bound state problems the jacobi eigenfunctions are suitably adjusted by means of the hypervirial theorems hvt scheme in order to obtain in many cases a very satisfactory analytic approximation of the s state eigenfunctions of the radial gaussian potential the results obtained show that the hvt scheme is a powerful method for obtaining analytically not only approximate eigenvalues but also the corresponding eigenfunctions

the energy levels of the schrodinger equation for various types of potentials using a renormalized method a perturbative method is used to compute energy levels for many different forms of potentials results of high accuracy are produced for various eigenstates

on the schrodinger equation with a gaussian potential eigenvalues and approximate eigenfunctions of the schrodinger equation with an attractive radial gaussian potential are obtained from a first order perturbation treatment based on a scale harmonic oscillator model the bound state energies are of comparable accuracy to those obtained using high order perturbation theory by numerical integration

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

stability of discrete deterministic and stochastic nonlinear systems abstract in this paper we derive some sufficient conditions for local asymptotic stability and instability for discrete nonlinear deterministic and stochastic systems these theorems are easy to apply to specific discrete nonlinear systems some applications of our theorems to bilinear systems illustrate the above

stability analysis of hybrid composite dynamical systems descriptions involving operators and differential equations we address the stability analysis of composite hybrid dynamical feedback systems of the type depicted in fig 1 consisting of a block usually the plant which is described by an operator l and of a finite dimensional block described by a system of ordinary differential equations usually the controller we establish results for the well posedness attractivity asymptotic stability uniform boundedness asymptotic stability in the large and exponential stability in the large for such systems the hypotheses of these results are phrased in terms of the i o properties of l and in terms of the lyapunov stability properties of the subsystem described by the indicated ordinary differential equations the applicability of our results is demonstrated by means of general specific examples involving c 0 semigroups partial differential equations or integral equations which determine l

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

almost orthogonal operators on the bitorus the operators sp f x y for the sum of which we prove an l2 estimate act as a kind of fourier coefficients on one variable and a kind of truncated hilbert transforms with a phase n x y on the other variable this result is an extension to two dimensions of an argument of almost orthogonality in fefferman s proof of a e convergence of fourier series under the basic assumption n x y mainly a function of y and the additional assumption n x y non decreasing in x for every y fixed

stabilization analysis of discrete nonlinear systems in this paper we set a new framework for the analysis of nonlinear discrete systems to carry out the stabilization theorem and to design stable controllers of the systems within that framework this framework includes many practically important systems described for example by lipschitz functions functions with arbitrary order bounded derivatives bilinear functions and all polynomial functions an important existence theorem for stabilizing has been proven for this framework the theorem basically states that under definite rather broad conditions the zero solution of the plants can be made stable by a nonlinear output feedback in addition this nonlinear feedback is from a larger class of functions than these classes given by current literature following these stabilization theorems a new design procedure for the output feedback control law is presented all results presented are easy to use and convenient to apply the application is illustrated by several examples including the design of stabilizing control for a bilinear control system

feedback stabilization of discrete time homogeneous semi linear systems for discrete time semi linear systems satisfying an accessibility condition asymptotic null controllability is equivalent to exponential feedback stabilizability using a piecewise constant feedback a constructive procedure that yields such a feedback is presented

stability analysis of bilinear systems the stability of the differential bilinear systems is studied the authors consider time varying bilinear systems with output feedback the input or control u t is not only a signal but also an input with output feedback i e u t f y t it is assumed that the feedback function f is in a function space which contains the class of linear functions functions satisfying the lipschitz condition and quadratic functions

quantitative estimates and extrapolation for multilinear weight classes in this paper we prove a quantitative multilinear limited range extrapolation theorem which allows us to extrapolate from weighted estimates that include the cases where some of the exponents are infinite this extends the recent extrapolation result of li martell and ombrosi we also obtain vector valued estimates including ell infty spaces and in particular we are able to reprove all the vector valued bounds for the bilinear hilbert transform obtained through the helicoidal method of benea and muscalu moreover our result is quantitative and in particular allows us to extend quantitative estimates obtained from sparse domination in the banach space setting to the quasi banach space setting our proof does not rely on any off diagonal extrapolation results and we develop a multilinear version of the rubio de francia algorithm adapted to the multisublinear hardy littlewood maximal operator as a corollary we obtain multilinear extrapolation results for some upper and lower endpoints estimates in weak type and mathrm bmo spaces

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

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

on boundedness of discrete multilinear singular integral operators let m be a measurable locally bounded function defined in r2 let 1 p1 q1 p2 q2 such that pi 1 implies qi let also 0 p3 q3 and 1 p 1 p1 1 p2 1 p3 we prove the following transference result the operator cm f g x r rf g m e2 ix d d initially defined for integrable functions with compact fourier support extends to a bounded bilinear operator from lp1 q1 r lp2 q2 r into lp3 q3 r if and only if the family of operators dm t p a b n t1p 1212 1212p q m t t e2 in d d initially defined for finite sequences a ak1 k1 z b bk2 k2 z where p k1 zak1e 2 ik1 and q k2 zbk2e 2 ik2 extend to bounded bilinear operators from lp1 q1 z lp2 q2 z into lp3 q3 z with norm bounded by uniform constant for all t 0 we apply this result to prove boundedness of the discrete bilinear hilbert transforms and other related discrete multilinear singular integrals including the endpoints

variational estimates for the bilinear iterated fourier integral abstract we prove pointwise variational l p bounds for a bilinear fourier integral operator in a large but not necessarily sharp range of exponents this result is a joint strengthening of the corresponding bounds for the classical carleson operator the bilinear hilbert transform the variation norm carleson operator and the bi carleson operator terry lyon s rough path theory allows for extension of our result to multilinear estimates we consider our result a proof of concept for a wider array of similar estimates with possible applications to ordinary differential equations

bound state energy eigenvalues for a general class of one dimensional problems on the whole axis via the prufer transformation an extension of the prufer phase function method for the bound state energy calculation is presented it is applicable to one dimensional problems described by the schrodingere quation on the whole axis with a general class of potentials theorems are given which are a generalization of the analogous ones concerning the half axis 0 problems that have been presented in previous papers the method is suitable especially for numerical calculations of the bound state energy eignevalues

1 n expansion for gaussian potential the method of large n expansion has been applied to an attractive radial gaussian potential to obtain its bound state energy levels it has been shown that the shifted 1 n expansion technique provides a better approximation than the method of unshifted 1 n expansion and yields energy values which are in good agreement with the accurate numerical and analytic results

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

polynomial filtering of discrete time stochastic linear systems with multiplicative state noise the problem of finding an optimal polynomial state estimate for the class of stochastic linear models with a multiplicative state noise term is studied for such models a technique of state augmentation is used leading to the definition of a general polynomial filter the theory is developed for time varying systems with nonstationary and non gaussian noises moreover the steady state polynomial filter for stationary systems is also studied numerical simulations show the high performances of the proposed method with respect to the classical linear filtering techniques

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

the exponential cosine screened potential the concept of forward nested multiplication is used to develop a unified power series hill determinant method for the calculation of schrodinger equation energy levels a numerical application is made to the exponential cosine screened potential

fast eigenvalue algorithm for central potentials an algorithm is described that seeks out schrodinger eigenvalues enl for a given central potential v r the algorithm has in principle arbitrary precision a particular implementation has absolute error mod e enl mod decreasing as t 3 where t is machine computing time the method is tested on the central gaussian potential v r a exp lambda r2 of recent interest

on exact solutions of the schrodinger equation a new approach has been used to show what kind of potentials can be used to obtain a series of exact solutions in closed forms of the schrodinger equation for bound states in particular the well known coulombic and oscillator solutions and others are reproduced as special cases

stability of discrete bilinear systems with output feedback abstract time varying discrete bilinear systems with output feedback are considered sufficient conditions ensuring the stability and asymptotic stability of discrete bilinear systems with output feedback are derived these sufficient conditions depend only on the coefficient matrices of the bilinear systems thus they are convenient to check and easy to apply in engineering practice

on the schrodinger equation for the interaction x2 λx2 1 gx2 by forming the 6 6 pade approximants to the energy perturbation series for the interaction x2 lambda x2 1 gx2 as obtained from the hypervirial relations the ground and the first three excited state energy levels are calculated for g or 2 the results of the present calculation are in good agreement with those of the variational calculations the authors also present another set of exact solutions to the schrodinger equation for the interaction x2 lambda x2 1 gx2