estimates for riesz kernels of eigenfunction expansions of elliptic differential operators on compact manifolds let 2 and be the eigenvalues and an orthonormal system of eigenvectors of a second order elliptic differential operator on a compact n dimensional manifold m the riesz means of order of an integrable function on m are defined by r x m 1 2 2 x y y d y in this paper we study the kernels and the operator norms of he operators r on lp m 1 p 2 n 1 n 3 we also prove that if 1 p 2 n 1 n 3 and np n 1 2 then the operators r are of weak type p p uniformly with respect to

resonances for periodic jacobi operators with finitely supported perturbations we describe the resonances and the eigenvalues of a periodic jacobi operator with finitely supported perturbations in the case of small diagonal perturbations we determine their asymptotics

harmonic analysis as spectral theory of laplacians by writing the fourier inversion formula on euclidean space in polar coordinates we obtain f x 0 fgl x dx where fgl x 2fgl x which is the spectral theory of the laplacian how do properties of f relate to properties of the family of eigenfunctions fgl answers are provided for the following spaces l2 s s d in odd dimensions analogous results are obtained for harmonic analysis on hyperbolic space constant curvature semi riemannian spaces the heisenberg group and for differential forms on hyperbolic space for f l2 there is a plancherel formula f 22 limt o 1t bt z f x 2dx d where bt z denotes the ball of radius t about an arbitrary point z which is independent of the dimension and the identical formula holds in hyperbolic space if the parametrization of the eigenvalues is shifted for certain semi riemannian symmetric spaces we obtain a paley wiener theorem that explains the role of discrete series in producing functions of compact support and also involve certain non unitary representations that contain the discrete series representations for the heisenberg group the plancherel formula is of a different nature requiring that the eigenfunctions be almost periodic in one variable

asymptotic expansions and boundary values of eigenfunctions on riemannian symmetric spaces soit x g k un espace de riemann symetrique de type non compacte a bord b k m et soit d x l algebre de tous les operateurs differentiels g invariants sur x on obtient des developpements asymptotiques pour une grande classe de fonctions jointives f sur x

dispersive estimates for the dirac equation in an aharonov bohm field abstract we prove local smoothing and weighted strichartz estimates for the dirac equation with a aharonov bohm potential the proof inspired by 12 relies on an explicit representation of the solution built in terms of spectral projections

lp contractive projections and the heat semigroup for differential forms a projection operator that is contractive on lp for two distinct values of p is shown to be contractive for all values of p and the range must be of a special form this result is used to show that the heat semigroup for k forms on many manifolds with nontrivial cohomology in dimension k cannot be contractive on any lp for p 2

fourier asymptotics of fractal measures abstract a measure on r n will be called locally uniformly dimensional if br x cr for all r 1 and all x where br x denotes the ball of radius r about x for l 2 d the measure d is in i so d is well defined we show it is locally in l2 and sup r 1 r n b r y f d 2 d c f 2 under additional hypotheses we show that lim r r n b r y f d 2 2 d is comparable in size to 2 2 a number of other related results are established the special case when is an integer and is the surface measure on a c1 manifold was treated by s agmon and l hormander j analyse math 30 1976 1 38

explicit solutions of maxwell s equations on a space of constant curvature abstract an explicit solution is given to the cauchy problem for the source free maxwell s equations in a vacuum on a space time of the form r 1 x m3 where m3 is a 3 manifold of constant curvature this solution satisfies huyghens principle that all electromagnetic radiation propagates at exactly the speed of light the solution is obtained by harmonic analysis on m3 and in the process a generating class of plane wave solutions is found these solutions approximate the flat space plane wave solutions in a neighbourhood of a point but their global properties are somewhat different the solutions obtained are easily transplanted to the robertson walker models of general relativity by re scaling the time variable

carleman inequalities and the heat operator we prove a strong unique continuation theorem for functions u vanishing to infinite order in the space time variables at 0 0 and verifying the inequality 4u tu v x t u on rn 0 t when t1 n 2r 1 s v x t lt locl r x n 2r 1 s 1 n 2 r

the campbell baker hausdorff dynkin formula and solutions of differential equations the campbell baker hausdorff dynkin formula is a special case of a simpler and more general formula for the solution of nonautonomous systems of first order ordinary differential equations in terms of autonomous systems specifically suppose u t takes values in a c manifold and satisfies the initial value problem u t a t u t u 0 a where a t is a vector field on the manifold depending continuously on t then u t exp z t a here exp z a means the solution at s 1 to v s z v s v 0 a for z t r 1 pr e r2r 1e tr t x a s 1 a s 2 a s r ds as t 0 where tr t s rr 0 s1 s2 sr t pr is the set of permutations on 1 r e is the number of errors in ordering consecutive terms in 1 r and is the usual commutator of vector fields under appropriate analyticity assumptions the series for z t is convergent for small t this formula gives an explicit formulation of results of k t chen published in 1957 applications are given to problems in sub riemannian geometry and to improving convergence estimates for the campbell baker hausdorff dynkin formula in the context of banach algebras our formula can be thought of as a noncommutative generalization of the familiar formula u t a exp 0t a s ds in the scalar linear case in the same way that the campbell baker hausdorff dynkin formula is a noncommutative generalization of the familiar formula exey ex y

meromorphic continuations of finite gap herglotz functions and periodic jacobi matrices we find a necessary and sufficient condition for a herglotz function m to be the borel transform of the spectral measure of an exponential decaying perturbation of a periodic jacobi matrix the condition is in terms of meromorphic continuation of m to a natural riemann surface and the structure of its zeros and poles

image of the schwartz space under spectral projection let x g k be a riemannian symmetric space of non compact type and rank one the spectral projection p f of a function f on x can be written p f f where is the elementary spherical function corresponding to the complex parameter we characterize the image of the schwartz space mathcal s p x under the spectral projection for 0 p 2

regularity of the neumann problem at point of infinite type abstract we introduce general estimates for gain of regularity of solutions of the neumann problem and relate it to the existence of weights with large levi form at the boundary this enables us to discuss in a unified framework the classical results on fractional ellipticity subellipticity superlogarithmic ellipticity and compactness for each case we exhibit a corresponding class of domains

l p eigenfunction bounds for the hermite operator author s koch herbert tataru daniel abstract we obtain l p eigenfunction bounds for the harmonic oscillator in r n and for other related operators improving earlier results of thangavelu and karadzhov we also construct suitable counterexamples which show that our estimates are sharp

periodic jacobi operator with finitely supported perturbations the inverse resonance problem abstract we consider a periodic jacobi operator h with finitely supported perturbations on z we solve the inverse resonance problem we prove that the mapping from finitely supported perturbations to the scattering data the inverse of the transmission coefficient and the jost function on the right half axis is one to one and onto we consider the problem of reconstruction of the scattering data from all eigenvalues resonances and the set of zeros of r 1 where r is the reflection coefficient

bochner riesz profile of anharmonic oscillator l d2dx2 x we investigate spectral multipliers bochner riesz means and the convergence of eigenfunction expansion corresponding to the schrodinger operator with anharmonic potential l d2dx2 x we show that the bochner riesz profile of the operator l completely coincides with such profile of the harmonic oscillator h d2dx2 x2 it is especially surprising because the bochner riesz profile of the one dimensional standard laplace operator is known to be essentially different and the case of operators h and l resembles more the profile of multidimensional laplace operators another surprising element of the main obtained result is the fact that the proof is not based on restriction type estimates and instead an entirely new perspective has to be developed to obtain the critical exponent for bochner riesz means convergence

random weighted sobolev inequalities and application to quantum ergodicity this paper is a continuation of poiret et al ann henri poincare 16 651 689 2015 where we studied a randomisation method based on the laplacian with harmonic potential here we extend our previous results to the case of any polynomial and confining potential v on mathbb r d we construct measures under concentration type assumptions on the support of which we prove optimal weighted sobolev estimates on mathbb r d this construction relies on accurate estimates on the spectral function in a non compact configuration space then we prove random quantum ergodicity results without specific assumption on the classical dynamics finally we prove that almost all bases of hermite functions are quantum uniquely ergodic

perturbations of orthogonal polynomials with periodic recursion coefficients the results of denisov rakhmanov szeg shohat nevai and killip simon are extended from asymptotically constant orthogonal polynomials on the real line oprl and unit circle opuc to asymptotically periodic oprl and opuc the key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations

the weak type l1 convergence of eigenfunction expansions for pseudodifferential operators soit m une variete compacte connexe lisse sans bord de dimension n 2 soit p d un operateur pseudodifferentiel elliptique d ordre 1 sur m qui est auto adjoint par rapport a une densite positive lisse dx on demontre la convergence faible de type l 1 pour les developpements en fonctions propres

local harmonic analysis on spheres abstract a continuous expansion x x h d is established for functions or distributions supported on a sufficiently small set k in the sphere sn valid in a neighborhood of k where x are eigenfunctions of the laplacian on with eigenvalue n 1 2 2 2 the function h and the operators p that give from are given by explicit formulas when n is odd the function x can be uniquely characterized by analyticity and growth conditions in of paley wiener type the expansion is applied to study functions of the laplacian and operators that commute with the laplacian

mean value properties of the laplacian via spectral theory soit z 2 une fonction entiere paire de type exponentiel temperee l une realisation auto adjointe de c x ou est l operateur de laplace beltrami sur une variete de riemann et l l operateur donne par la theorie spectrale on demontre un theoreme de paley wiener sur le support de l et on l utilise pour montrer que lu u sur un bon domaine implique l u u

the spherical transform of a schwartz function on the heisenberg group suppose thatk u n is a compact lie group acting on the 2n 1 dimensional heisenberg grouphn we say that k hn is a gelfand pair if the convolution algebral1k hn of integrablek invariant functions onhnis commutative in this case the gelfand space k hn is equipped with the godement plancherel measure and the spherical transform l2k hn l2 k hn is an isometry the main result in this paper provides a complete characterization of the set sk hn f f sk hn of spherical transforms ofk invariant schwartz functions onhn we show that a functionfon k hn belongs to sk hn if and only if the functions obtained fromfvia application of certain derivatives and difference operators satisfy decay conditions we also consider spherical series expansions fork invariant schwartz functions onhnmodulo its center

finite range perturbations of finite gap jacobi and cmv operators abstract necessary and sufficient conditions are presented for a measure to be the spectral measure of a finite range perturbation of a jacobi or cmv operator from a finite gap isospectral torus the special case of eventually periodic operators solves an open problem of simon 28 d 2 7 we also solve the inverse resonance problem it is shown that an operator is completely determined by the set of its eigenvalues and resonances and we provide necessary and sufficient conditions on their configuration for such an operator to exist

fourier restriction theorem and characterization of weak l2 eigenfunctions of the laplace beltrami operator abstract in this paper we prove that for r 0 the poisson transform p f of f l 2 k m belongs to l 2 g k this complements an earlier result of lohoue and rychener using this we then prove an analogue of tomas stein restriction theorem for riemannian symmetric spaces of noncompact type with real rank one by using this estimate of the poisson transform we also characterize all weak l 2 eigenfunction of the laplace beltrami operator of riemannian symmetric spaces of noncompact type with real rank one and eigenvalue 2 2 for r 0

necessary conditions for subellipticity of the overline partial neumann problem let a be a a closed form of type p q with l2 coefficients on a smoothly bounded domain q in c the problem of determining both the existence and regularity properties of the solution u of au a where u is orthogonal to the null space of a on p q l forms is known as the a neumann problem one of the principal methods used in the investigation of this problem is the proof of certain a priori subelliptic estimates let u be a neighborhood of a point z0 in the boundary of q a subelliptic estimate is said to hold in u if the estimate

scattering theory for the laplacian on symmetric spaces of noncompact type and its application to a conjecture of strichartz abstract in this paper we develop the scattering theory for the laplacian on symmetric spaces of noncompact type we study the asymptotic properties of the resolvent in the framework of the agmon hormander space our approach is based on a detailed analysis of the helgason fourier transform and generalized spherical functions on symmetric spaces of noncompact type as an application of our scattering theory we prove a conjecture by strichartz concerning a characterization of a family of generalized eigenfunctions of the laplacian

riesz lp summability of spectral expansions related to the schrodinger operator with constant magnetic field abstract for the spectral expansions related to the schrodinger operator with constant magnetic field we establish riesz bochner summability in lp

subellipticity of the neumann problem on a weakly q pseudoconvex concave domain abstract for a domain d of c n which is weakly q pseudoconvex or q pseudoconcave we give a sufficient condition for subelliptic estimates for the neumann problem this extends to domains which are not necessarily pseudoconvex the results and the techniques of catlin 1987 3

a characterization of the l2 range of the poisson transform related to strichartz conjecture on symmetric spaces of noncompact type abstract the main purpose of this paper is to solve strichartz conjecture concerning an image characterization of the poisson transform in the l 2 theory on symmetric spaces of noncompact type we prove that the poisson transform provides an isomorphism between the l 2 space on the boundary and a certain weighted l 2 space consisting of joint eigenfunctions on symmetric spaces of noncompact type our approach is based on the scattering theory for the schrodinger operator moreover we give a fourier restriction estimate and an asymptotic formula for the poisson transform

eigenspaces of the laplacian on hyperbolic spaces composition series and integral transforms abstract let x be a projective real complex or quaternion hyperbolic space realized as the pseudo riemannian symmetric space x g h with g o p q u p q or sp p q these are the classical isotropic symmetric spaces let be the g invariant laplace beltrami operator on x a complete description by k types for each c of all closed g invariant subspaces of the eigenspace f c x f f is given the eigenspace representations are compared with principal series representations using poisson transformations similar results are obtained also for the exceptional isotropic symmetric space the langlands parameters of the spherical discrete series representations are determined