weighted estimates for the helmholtz equation and some applications we characterize the radial functionsvin rnfor which the a priori inequality u1 2 2 k uv1 2 2 c v 1 2 k2 u 2holds with constant independent ofk the condition is forvto have the x rays transform everywhere bounded we apply these estimates to the well posedness of evolution schrodinger equations with time dependent drift terms and to the restriction of the fourier transform to euclidean spheres

quantitative uniqueness of elliptic equations based on a variant of frequency function we improve the vanishing order of solutions for schr o dinger equations which describes quantitative behavior of strong uniqueness continuation property for the first time we investigate the quantitative uniqueness of higher order elliptic equations and show the vanishing order of solutions furthermore strong unique continuation is established for higher order elliptic equations using this variant of frequency function

carleman inequalities for the dirac and laplace operators and unique continuation on demontre les meilleures inegalites de type carleman possibles dans le cas de l operateur de dirac on obtient la propriete de prolongement unique pour d v ou v l loc r n 3n 2 2 n 3

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

sharp vanishing order of solutions to stationary schrodinger equations on carnot groups of arbitrary step abstract based on a variant of the frequency function approach of almgren 1 under appropriate assumptions we establish an optimal upper bound on the vanishing order of solutions to stationary schrodinger equations associated to sub laplacian on carnot groups of arbitrary step such a bound provides a quantitative form of strong unique continuation and can be thought of as a subelliptic analogue of the recent results obtained by bakri 3 and zhu 27 for the standard laplacian

on l p resolvent estimates and the density of eigenvalues for compact riemannian manifolds we address an interesting question raised by dos santos ferreira kenig and salo forum math 2014 about regions mathcal r g subset mathbb c for which there can be uniform l frac 2n n 2 to l frac 2n n 2 resolvent estimates for delta g zeta zeta in mathcal r g where delta g is the laplace beltrami operator with metric g on a given compact boundaryless riemannian manifold of dimension n geq 3 this is related to earlier work of kenig ruiz and the third author duke math j 55 329 347 1987 for the euclidean laplacian in which case the region is the entire complex plane minus any disc centered at the origin presently we show that for the round metric on the sphere s n the resolvent estimates in dos santos ferreira et al in forum math 2014 involving a much smaller region are essentially optimal we do this by establishing sharp bounds based on the distance from zeta to the spectrum of delta s n in the other direction we also show that the bounds in dos santos ferreira et al in forum math 2014 can be sharpened logarithmically for manifolds with nonpositive curvature and by powers in the case of the torus mathbb t n mathbb r n mathbb z n with the flat metric the latter improves earlier bounds of shen int math res not 1 1 31 2001 the work of dos santos ferreira et al in forum math 2014 and shen in int math res not 1 1 31 2001 was based on hadamard parametrices for delta g zeta 1 ours is based on the related hadamard parametrices for cos t sqrt delta g and it follows ideas in sogge in ann math 126 439 447 1987 of proving l p multiplier estimates using small time wave equation parametrices and the spectral projection estimates from sogge in j funct anal 77 123 138 1988 this approach allows us to adapt arguments in berard math z 155 249 276 1977 and hlawka monatsh math 54 1 36 1950 to obtain the aforementioned improvements over dos santos ferreira et al in forum math 2014 and shen in int math res not 1 1 31 2001 further improvements for the torus are obtained using recent techniques of the first author bourgain in israel j math 193 1 441 458 2013 and his work with guth bourgain and guth in geom funct anal 21 1239 1295 2011 based on the multilinear estimates of bennett carbery and tao math z 2 261 302 2006 our approach also allows us to give a natural necessary condition for favorable resolvent estimates that is based on a measurement of the density of the spectrum of sqrt delta g and moreover a necessary and sufficient condition based on natural improved spectral projection estimates for shrinking intervals as opposed to those in sogge in j funct anal 77 123 138 1988 for unit length intervals we show that the resolvent estimates are sensitive to clustering within the spectrum which is not surprising given sommerfeld s original conjecture sommerfeld in physikal zeitschr 11 1057 1066 1910 about these operators

quantitative uniqueness of some higher order elliptic equations abstract we study the quantitative unique continuation property of some higher order elliptic operators we prove a lower bound for nontrivial solutions of the equation m u v x u 0 m n v x is bounded the bound shows that nontrivial solutions can not decay faster than e x 4 3 ln x at infinity moreover we obtain an improved lower bound of nontrivial solutions for a special forth order elliptic operators in dimension 2 the bound is shown to be essentially sharp by constructing a meshkov type example

eigenvalue bounds for dirac and fractional schrodinger operators with complex potentials abstract we prove lieb thirring type bounds for fractional schrodinger operators and dirac operators with complex valued potentials the main new ingredient is a resolvent bound in schatten spaces for the unperturbed operator in the spirit of frank and sabin 11

a partial answer to a conjecture of b simon concerning unique continuation in his survey paper on schrijdinger semigroups s b simon formulated the following conjecture see s p 5193 let ve kfp the local kato class of potentials then the time independent schrddinger operator h d v has the unique continuation property ucp by this it is meant that given any connected open subset 52 c r the only solution u of hu 0 in 52 that can vanish in an open subset 52 c sz is u 0 a measurable function v on r is said to belong to the kato class k if

positive solutions for a weakly coupled nonlinear schrodinger system abstract existence of a nontrivial solution is established via variational methods for a system of weakly coupled nonlinear schrodinger equations the main goal is to obtain a positive solution of minimal action if possible with all vector components not identically zero generalizations for nonautonomous systems are considered

inverse scattering problem for a two dimensional random potential we study an inverse problem for the two dimensional random schrodinger equation q k 2 u 0 the potential q x is assumed to be a gaussian random function whose covariance operator is a classical pseudodifferential operator we show that the backscattered field obtained from a single realization of the random potential q determines uniquely the principal symbol of the covariance operator of q the analysis is carried out by combining harmonic and microlocal analysis with stochastic methods

on the fractional landis conjecture abstract in this paper we study a landis type conjecture for fractional schrodinger equations of fractional power s 0 1 with potentials we discuss both the cases of differentiable and non differentiable potentials on the one hand it turns out for differentiable potentials with some a priori bounds if a solution decays at a rate e x 1 then this solution is trivial on the other hand for s 1 4 1 and merely bounded non differentiable potentials if a solution decays at a rate e x with 4 s 4 s 1 then this solution must again be trivial remark that when s 1 4 s 4 s 1 4 3 which is the optimal exponent for the standard laplacian for the case of non differentiable potentials and s 1 4 1 we also derive a quantitative estimate mimicking the classical result by bourgain and kenig

least energy solitary waves for a system of nonlinear schrodinger equations in mathbb r n in this paper we consider systems of coupled schrodinger equations which appear in nonlinear optics the problem has been considered mostly in the one dimensional case here we make a rigorous study of the existence of least energy standing waves solitons in higher dimensions we give conditions on the parameters of the system under which it possesses a solution with least energy among all multi component solutions conditions under which the system does not have positive solutions and the associated energy functional cannot be minimized on the natural set where the solutions lie

l p l q estimates for electromagnetic helmholtz equation abstract in space dimension n 3 we consider the electromagnetic schrodinger hamiltonian h i a x 2 v and the corresponding helmholtz equation i a x 2 u u v x u f in r n we extend the well known l p l q estimates for the solution of the free helmholtz equation to the case when the electromagnetic hamiltonian h is considered

two coupled nonlinear schrodinger equations involving a non constant coupling coefficient abstract in this paper we study the existence and multiplicity of vector positive solutions for two coupled nonlinear schrodinger equations involving a non constant coupling coefficient

decay estimates and strichartz estimates of fourth order schrodinger operator abstract we study time decay estimates of the fourth order schrodinger operator h 2 v x in r d for d 3 and d 5 we analyze the low energy and high energy behaviour of resolvent r h z and then derive the jensen kato dispersion decay estimate and local decay estimate for e i t h p a c under suitable spectrum assumptions of h based on jensen kato type decay estimate and local decay estimate we obtain the l 1 l estimate of e i t h p a c in 3 dimension by ginibre argument and also establish the endpoint global strichartz estimates of e i t h p a c for d 5 furthermore using the local decay estimate and the georgescu larenas soffer conjugate operator method we prove the jensen kato type decay estimates for some functions of h

uniform resolvent estimates for schrodinger operator with an inverse square potential abstract we study the uniform resolvent estimates for schrodinger operator with a hardy type singular potential let l v v x where is the usual laplacian on r n and v x v 0 r 2 where r x x x and v 0 c 1 s n 1 is a real function such that the operator v 0 n 2 2 4 is a strictly positive operator on l 2 s n 1 we prove some new uniform weighted resolvent estimates and also obtain some uniform sobolev estimates associated with the operator l v

unique continuation and absence of positive eigenvalues for schrodinger operators on considere l operateur de laplace sur r n et une fonction v x sur un sous ensemble connexe ouvert de r n on montre que si n 3 une propriete de prolongement unique est vraie pour v l loc n 2 r n dans l espace de sobolev h loc 2 q r n avec q 2n n 2

unique continuation for elliptic operators a geometric variational approach dans un sous ensemble connexe ouvert de r n n 3 on considere l equation d ordre 2 lu div a x u b x u v x u 0 ou a x a id j x est une matrice symetrique reelle on etudie les proprietes d unicite de l operateur

dual variational methods and nonvanishing for the nonlinear helmholtz equation we set up a dual variational framework to detect real standing wave solutions of the nonlinear helmholtz equation u k2u q x u p 2u u w2 p rn with n 3 2 n 1 n 1 p 2nn 2 and nonnegative q l rn we prove the existence of nontrivial solutions for periodic q as well as in the case where q x 0 as x in the periodic case a key ingredient of the approach is a new nonvanishing theorem related to an associated integral equation the solutions we study are superpositions of outgoing and incoming waves and are characterized by a nonlinear far field relation

three circle theorems for eigenfunctions on complete shrinking gradient ricci solitons with constant scalar curvature abstract the classical hadamard s three circle theorem is generalized to complete shrinking gradient ricci solitons with constant scalar curvature for eigenfunctions of the laplace operator based on a variant of frequency function

duality of positive currents and plurisubharmonic functions in calibrated geometry recently the authors showed that there is a robust potential theory attached to any calibrated manifold x phi in particular on x there exist phi plurisubharmonic functions phi convex domains phi convex boundaries etc all inter related and having a number of good properties in this paper we show that in a strong sense the plurisubharmonic functions are the polar duals of the phi submanifolds or more generally the phi currents studied in the original paper on calibrations in particular we establish an analogue of duval sibony duality which characterizes points in the phi convex hull of a compact set k subset x in terms of phi positive green s currents on x and jensen measures on k we also characterize boundaries of phi currents entirely in terms of phi plurisubharmonic functions specific calibrations are used as examples throughout analogues of the hodge conjecture in calibrated geometry are considered

the singular set of 1 1 integral currents we prove that 2 dimensional integer multiplicity 2 dimensional rectifiable currents which are almost complex cycles in an almost complex manifold admitting locally a compatible positive symplectic form are smooth surfaces aside from isolated points and therefore are j holomorphic curves

uniqueness of tangent cones to positive p p integral cycles we prove that every positive p p integral cycle in an arbitrary almost complex manifold possesses at every point a unique tangent cone the argument relies on an algebraic blow up perturbed in order to face the analysis issues of this problem in the almost complex setting primary subjects 53c38 secondary subjects 49q05 49q20 35j99

quantitative uniqueness of solutions to parabolic equations abstract we investigate the quantitative uniqueness of solutions to parabolic equations with lower order terms on compact smooth riemannian manifolds quantitative uniqueness is a quantitative form of strong unique continuation property we characterize quantitative uniqueness by the rate of vanishing we can obtain the sharp vanishing order for solutions in term of the c 1 1 norm of the potential functions as well as the l norm of the coefficient functions some new quantitative carleman estimates and three cylinder inequalities are established

ground state of n coupled nonlinear schrodinger equations in mathbb r n n leq 3 we establish some general theorems for the existence and nonexistence of ground state solutions of steady state n coupled nonlinear schrodinger equations the sign of coupling constants ij s is crucial for the existence of ground state solutions when all ij s are positive and the matrix is positively definite there exists a ground state solution which is radially symmetric however if all ij s are negative or one of ij s is negative and the matrix is positively definite there is no ground state solution furthermore we find a bound state solution which is non radially symmetric when n 3

a variational characterization of complex submanifolds in this note we generalize our results in arezzo and sun reine angew math doi 10 1515 crelle 2013 0097 2012 to integer p currents of any degree we prove that if the mass of a current as a functional of the ambient metric has a critical or stable point in some special directions then the current is complex this holds for any dimension and codimension we also study a natural functional on the space of currents representing a fixed homology class closely related to the first derivative of the mass in our new approach detecting the deviation of a surface from being holomorphic

unique continuation from infinity for linear waves abstract we prove various uniqueness results from null infinity for linear waves on asymptotically flat space times assuming vanishing of the solution to infinite order on suitable parts of future and past null infinities we derive that the solution must vanish in an open set in the interior we find that the parts of infinity where we must impose a vanishing condition depend strongly on the background geometry in particular for backgrounds with positive mass such as schwarzschild or kerr the required assumptions are much weaker than the ones in the minkowski space time the results are nearly optimal in many respects they can be considered analogues of uniqueness from infinity results for second order elliptic operators this work is partly motivated by questions in general relativity

improved eigenvalue bounds for schrodinger operators with slowly decaying potentials we extend a result of davies and nath j comput appl math 148 1 1 28 2002 on the location of eigenvalues of schrodinger operators with slowly decaying complex valued potentials to higher dimensions in this context we also discuss various examples related to the laptev and safronov conjecture laptev and safronov in commun math phys 292 1 29 54 2009

singular solutions of elliptic equations and the determination of conductivity by boundary measurements abstract we improve some results on uniqueness and stability in the determination of the coefficient a in the equation i div a grad u 0 in when all possible pairs of dirichlet and neumann data on are known we also treat cases of anisotropic equations our method relies on the construction of solutions to i having an isolated singularity with prescribed asymptotic behaviour

non trivial l q solutions to the ginzburg landau equation it is shown that the contour problem for the stationary ginzburg landau equation where x r with r x is well posed in l 4 n for a class of small data f l 2 s n 1

agmon kato kuroda theorems for a large class of perturbations we prove asymptotic completeness for operators of the form h l on l2 rd d 2 where l is an admissible perturbation our class of admissible perturbations contains multiplication operators defined by real valued potentials v lq rd q d 2 d 1 2 if d 2 then we require q 1 3 2 as well as real valued potentials v satisfying a global kato condition the class of admissible perturbations also contains first order differential operators of the form a a for suitable vector potentials a our main technical statement is a new limiting absorption principle which we prove using techniques from harmonic analysis related to the stein tomas restriction theorem

nodal sets of eigenfunctions on riemannian manifolds soit m n une variete connexe compacte a metrique riemannienne c soit le laplacien de m soit f une fonction propre reelle de de valeur propre on demontre que la fonction propre f s annule au plus a l ordre c 1 2 pour tout point dans m

sharp resolvent estimates outside of the uniform boundedness range in this paper we are concerned with resolvent estimates for the laplacian delta in euclidean spaces uniform resolvent estimates for delta were shown by kenig et al duke math j 55 2 329 347 1987 who established rather a complete description of the lebesgue spaces allowing such estimates however the problem of obtaining sharp l p l q bounds depending on z has not been considered in a general framework which admits all possible p q in this paper we present a complete picture of sharp l p l q resolvent estimates which may depend on z we also obtain the sharp resolvent estimates for the fractional laplacians and a new result for the bochner riesz operators of negative index