fourier transform inequalities with measure weights abstract fourier transform norm inequalities f q p are proved for measure weights on moment subspaces of lp r n density theorems are established to extend the inequalities to all of lp r n in both cases the conditions for validity are computable for n 2 and are radial and the results are applied to prove spherical restriction theorems which include power weights t t n p 1 p n p 1

wigner weyl correspondence in quantum mechanics for continuous and discrete systems a dirac inspired view drawing inspiration from dirac s work on functions of non commuting observables we develop an approach to phase space descriptions of operators and the wigner weyl correspondence in quantum mechanics complementary to standard formulations this involves a two step process introducing phase space descriptions based on placing position dependences to the left of momentum dependences or the other way around then carrying out a natural transformation to eliminate a kernel which appears in the expression for the trace of the product of two operators the method works uniformly for both continuous cartesian degrees of freedom and for systems with finite dimensional state spaces it is interesting that the kernel encountered is naturally expressible in terms of geometric phases and its removal involves extracting its square root in a suitable manner

wigner function for discrete phase space exorcising ghost images abstract we construct using simple geometrical arguments a wigner function defined on a discrete phase space of arbitrary integer hilbert space dimension that is free of redundancies ghost images plaguing other wigner functions for discrete phase spaces are thus revealed as artifacts it allows to devise a corresponding phase space propagator in an unambiguous manner we apply our definitions to eigenstates and propagator of the quantum baker map scars on unstable periodic points of the corresponding classical map become visible with unprecedented resolution

weighted norm inequalities for oscillatory integrals with finite type phases on the line abstract we obtain two weighted l 2 norm inequalities for oscillatory integral operators of convolution type on the line whose phases are of finite type the conditions imposed on the weights involve geometrically defined maximal functions and the inequalities are best possible in the sense that they imply the full l p r l q r mapping properties of the oscillatory integrals our results build on work of carbery soria vargas and the first author

equipartition of the eigenfunctions of quantized ergodic maps on the torus we give a simple proof of the equipartition of the eigenfunctions of a class of quantized ergodic area preserving maps on the torus examples are the irrational translations the skew translations the hyperbolic automorphisms and some of their perturbations

on the regularity of solutions to a class of nonlinear dispersive equations we shall study special regularity properties of solutions to some nonlinear dispersive models the goal is to show how regularity on the initial data is transferred to the solutions this will depend on the spaces where regularity is measured

weighted norm inequalities for potentials with applications to schrodinger operators fourier transforms and carleson measures on donne une nouvelle caracterisation de l inegalite de trace pour des operateurs potentiel et on l utilise pour affiner des resultats sur la distribution des valeurs propres des operateurs de schrodinger

on the propagation of regularities in solutions of the fifth order kadomtsev petviashvili ii equation abstract in this paper we study special properties of solutions of the initial value problem ivp associated to the fifth order kadomtsev petviashvili ii equation mainly for initial data h 1 r 2 whose restriction belongs to h m x 0 r for some m z m 2 and x 0 r we prove that the restriction of the corresponding solution u t belongs to h m b r for any b r and any t 0 in other words the extra regularity on the data propagates in the solutions in the direction of the dispersion

wigner functions and separability for finite systems a discussion of discrete wigner functions in phase space related to mutually unbiased bases is presented this approach requires mathematical assumptions which limits it to systems with density matrices defined on complex hilbert spaces of dimension pn where p is a prime number with this limitation it is possible to define a phase space and wigner functions in close analogy to the continuous case that is we use a phase space that is a direct sum of n two dimensional vector spaces each containing p2 points this is in contrast to the more usual choice of a two dimensional phase space containing p2n points a useful aspect of this approach is that we can relate complete separability of density matrices and their wigner functions in a natural way we discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd special attention is required for two qubits p 2 and our technique fails to establish the separability property for more than two qubits finally we give a brief discussion of hamiltonian dynamics in the language developed in the paper

discrete wigner function for finite dimensional systems a phase space approach to finite dimensional systems is developed from basic principles for a system describable by a hilbert space of dimension d we define a one to one correspondence between operators and functions on a discrete and finite phase space with points valid for any dimension d the properties fulfilled by this correspondence and its uniqueness are examined this formalism is applied to the number difference and phase difference of a two mode field this case is compared with the marginal distribution for these variables arising from a two mode wigner function for number and phase

characterization of informational completeness for covariant phase space observables in the nonrelativistic setting with finitely many canonical degrees of freedom a shift covariant phase space observable is uniquely characterized by a positive operator of trace one and in turn by the fourier weyl transform of this operator we study three properties of such observables and characterize them in terms of the zero set of this transform the first is informational completeness for which it is necessary and sufficient that the zero set has dense complement the second is a version of informational completeness for the hilbert schmidt class equivalent to the zero set being of measure zero and the third known as regularity is equivalent to the zero set being empty we give examples demonstrating that all three conditions are distinct the three conditions are the special cases for p 1 2 of a more general notion of p regularity defined as the norm density of the span of translates of the operator in the schatten p class we show that the relation between zero sets and p regularity can be mapped completely to the corresponding relation for functions in classical harmonic analysis

estimates for operators in mixed weighted spaces a weighted marcinkiewicz interpolation theorem is proved if t is simultaneously of weak type p q i 0 1 1 1 the result is applied to obtain weighted estimates for the laplace and fourier transform as well as the riesz potential 1 in 8 we proved that if t is simultaneously of weak type pi qi i 0 1 1 po 1 as always new interpolation results imply new results for a number of classical operators here we apply the results to the laplace transform complementing those obtained in 1 theorem 2 4 the fourier transform extending results in 8 10 12 and apply it to obtain a new inequality with applications in signal analysis 6 further we obtain a weighted extension of sobolev s theorem in 9 theorem 1 1 it was shown that the only bounded translation invariant operators from lp to lq q 1 in the next section we introduce notation give a preliminary result and prove the main result theorem 2 2 3 contains applications throughout p denotes the conjugate index of p and is related top by p p pp with p 00 if p 1 similarly for other letters further constants are denoted by c and may be different at different appearances but are always independent of the function in question z denotes the set of integers 2 let u and v be nonnegative weight functions defined on s2 c r and t a sublinear operator defined on lebesgue measurable functions on u2 we denote by lp l p 9 0 1 or 0 o it is well known 2 4 that p e lip lq 1 o 0 where here and in the sequel xe x 1 if x e xe j 0 i f x f is the characteristic function the corresponding result for the dual operator also holds only now the characteristic functions in the above integrals are interchanged in case 0 0 if p e lp lq fix a positive increasing sequence xk k z and a given sequence of nonnegative numbers ak this content downloaded from 207 46 13 124 on wed 22 jun 2016 05 32 16 utc all use subject to http about jstor org terms operators in mixed weighted lp spaces 485 and define f x fakxk x v x k

on the generalized benjamin ono equation we study well posedness of the initial value problem for the generalized benjamin ono equation tu ukdxu 0 k e in sobolev spaces hs r for small data and higher nonlinearities k 2 new local and global including scattering results are established our method of proof is quite general it combines several estimates concerning the associated linear problem with the contraction principle hence it applies to other dispersive models in particular it allows us to extend the results for the generalized benjamin ono to nonlinear schrodinger equations or systems of the form atu i2u p u oxu ui axi 0

weighted decay estimates for the wave equation abstract in this work we study weighted sobolev spaces in r n generated by the lie algebra of vector fields 1 x 2 1 2 x j j 1 n interpolation properties and sobolev embeddings are obtained on the basis of a suitable localization in r n as an application we derive weighted l q estimates for the solution of the homogeneous wave equation for the inhomogeneous wave equation we generalize the weighted strichartz estimate established by v georgiev 1997 amer j math 119 1291 1319 and establish global existence results for the supercritical semilinear wave equation with non compact small initial data in these weighted sobolev spaces

weighted norm estimates for the fourier transform with a pair of weights we prove weighted norm inequalities of the form iifiilq ciifiihp 0 p q 0 for the fourier transform on rn for some weight functions v the hardy space hp on the right can be replaced by l the proof depends on making an atomic decomposition of f and using cancellation properties of the atoms

nonlocal models for nonlinear dispersive waves abstract considered herein are model equations for the unidirectional propagation of small amplitude nonlinear dispersive long waves such as those governed by the classical korteweg de vries equation of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial so that the operator modelling dispersive effect is nonlocal particular cases in view here are the benjamin ono equation and the intermediate long wave equation which arise in internal wave theory and smith s equation which governs certain types of continental shelf waves the initial value problem for these equations is shown to be globally well posed in the classical sense including continuous dependence upon the initial data and in certain cases upon the modelling of nonlinear and dispersive effects whilst the results are stated for the specific equations listed above the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear dispersive evolution equations particularly worthy of note is our theorem implying that solutions of the intermediate long wave equation converge strongly to solutions of the korteweg de vries equation or to solutions of the benjamin ono equation in appropriate asymptotic limits

on the propagation of regularity for solutions of the fractional korteweg de vries equation abstract we consider the initial value problem ivp for the fractional korteweg de vries equation fkdv 0 1 t u d x x u u x u 0 x t r 0 1 u x 0 u 0 x it has been shown that the solutions to certain dispersive equations satisfy the propagation of regularity phenomena more precisely it deals in determine whether regularity of the initial data on the right hand side of the real line is propagated to the left hand side by the flow solution this property was found originally in solutions of korteweg de vries kdv equation and it has been also verified in other dispersive equations as the benjamin ono bo equation recently it has been shown that the solutions of the dispersive generalized benjamin ono dgbo equation this is 2 3 in 0 1 also satisfy the propagation of regularity phenomena this is achieved by introducing a commutator decomposition to handle the dispersive part in the equation following the approach used in the dgbo we prove that the solutions of the fkdv also satisfies the propagation of regularity phenomena consequently this type of regularity travels with infinite speed to its left as time evolves

on the propagation of regularities in solutions of the benjamin ono equation abstract we shall deduce some special regularity properties of solutions to the ivp associated to the benjamin ono equation mainly for datum u 0 h 3 2 r whose restriction belongs to h m b for some m z m 2 and b r we shall prove that the restriction of the corresponding solution u t belongs to h m for any r and any t 0 therefore this type of regularity of the datum travels with infinite speed to its left as time evolves

a note on rearrangements spectral concentration and the zero order prolate spheroidal wavefunction if the measure of the support of a function f is small its symmetric decreasing rearrangement f is more nearly bandlimited to low frequencies than f while their norms are equal an immediate corollary is that the time limited zero order prolate spheroidal wavefunction is the extremal function for a new optimization problem involving time and bandlimiting the result has an application in exploration seismology

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

sequential measurement of conjugate variables as an alternative quantum state tomography it is shown how it is possible to reconstruct the initial state of a one dimensional system by sequentially measuring two conjugate variables the procedure relies on the quasicharacteristic function the fourier transform of the wigner quasiprobability the proper characteristic function obtained by fourier transforming the experimentally accessible joint probability of observing position then momentum or vice versa can be expressed as a product of the quasicharacteristic function of the two detectors and that unknown of the quantum system this allows state reconstruction through the sequence 1 data collection 2 fourier transform 3 algebraic operation and 4 inverse fourier transform the strength of the measurement should be intermediate for the procedure to work

weighted norm inequalities for the hankel and transformations we give conditions on pairs of non negative weight functions u and v which are sufficient that for 1 where t is the hankel or the k transformation the proofs are based on a weighted marcinkiewicz interpolation theorem for linear operators in the case that t is the hankel transformation and 1 the result is similar to a weighted estimate of heywood and rooney 9 but with different weight conditions

wigner weyl isomorphism for quantum mechanics on lie groups the wigner weyl isomorphism for quantum mechanics on a compact simple lie group g is developed in detail several features are shown to arise which have no counterparts in the familiar cartesian case notable among these is the notion of a semiquantized phase space a structure on which the weyl symbols of operators turn out to be naturally defined and figuratively speaking located midway between the classical phase space t g and the hilbert space of square integrable functions on g general expressions for the star product for weyl symbols are presented and explicitly worked out for the angle angular momentum case

on the propagation of regularity and decay of solutions to the benjamin equation in this paper we investigate some special regularities and decay properties of solutions to the initial value problem of the benjamin equation the main result shows that for initial datum u0 hs r with s 3 4 if the restriction of u0 belongs to hl x0 for some l z and x0 r then the restriction of the corresponding solution u t belongs to hl for any r and any t 0 t consequently this type of regularity travels with infinite speed to its left as time evolves

on well posedness for the benjamin ono equation we prove existence and uniqueness of solutions for the benjamin ono equation with data in h s mathbb r s 1 4 moreover the flow is holder continuous in weaker topologies