a note on self adjoint extensions of the laplacian on weighted graphs abstract we study the uniqueness of self adjoint and markovian extensions of the laplacian on weighted graphs we first show that for locally finite graphs and a certain family of metrics completeness of the graph implies uniqueness of these extensions moreover in the case when the graph is not metrically complete and the cauchy boundary has finite capacity we characterize the uniqueness of the markovian extensions

a sampling theorem on homogeneous manifolds we consider a generalization of entire functions of spherical exponential type and lagrangian splines on manifolds an analog of the paleywiener theorem is given we also show that every spectral entire functionl on a manifold is uniquely determined by its values on some discrete sets of points the main result of the paper is a formula for reconstruction of spectral entire functions from their values on discrete sets using lagrangian splines

l p bernstein estimates and approximation by spherical basis functions the purpose of this paper is to establish l p error estimates a bernstein inequality and inverse theorems for approximation by a space comprising spherical basis functions located at scattered sites on the unit n sphere in particular the bernstein inequality estimates l p bessel potential sobolev norms of functions in this space in terms of the minimal separation and the l p norm of the function itself an important step in its proof involves measuring the l p stability of functions in the approximating space in terms of the l p norm of the coefficients involved as an application of the bernstein inequality we derive inverse theorems for sbf approximation in the l p norm finally we give a new characterization of besov spaces on the n sphere in terms of spaces of sbfs

hamilton s gradient estimates and a monotonicity formula for heat flows on metric measure spaces abstract in this paper we extend the hamilton s gradient estimates hamilton 1993 and a monotonicity formula of entropy ni 2004 for heat flows from smooth riemannian manifolds to non smooth metric measure spaces with appropriate riemannian curvature dimension condition

random dynamical systems generated by stochastic navier stokes equations on a rotating sphere abstract in this paper we first prove the existence and uniqueness of the solution to the stochastic navier stokes equations on the rotating 2 dimensional unit sphere then we show the existence of an asymptotically compact random dynamical system associated with the equations

weak type 1 1 of some operators for the laplacian with drift let v v 1 ldots v n be a vector in mathbb r n setminus 0 consider the laplacian on mathbb r n with drift delta v sum i 1 n big frac partial 2 partial x i 2 2 v i frac partial partial x i big and the measure d mu x e 2 langle v x rangle dx with respect to which delta v is self adjoint let d and nabla denote the euclidean distance and the gradient operator on mathbb r n consider the space mathbb r n d d mu which has the property of exponential volume growth we obtain weak type 1 1 for the riesz transform nabla delta v frac 1 2 and for the heat maximal operator with respect to d mu further we prove that the uncentered hardy littlewood maximal operator is bounded on l p for 1 p le infty but not of weak type 1 1 if n ge 2

bilinear embedding for real elliptic differential operators in divergence form with potentials we present a simple bellman function proof of a bilinear estimate for elliptic operators in divergence form with real coefficients and with nonnegative potentials the constants are dimension free the p range of applicability of this estimate is 1 for any real accretive nonsymmetric matrix a of coefficients

some properties of laplacians on fractals abstract kigami has defined an analog of the laplacian on a class of self similar fractals including the familiar sierpinski gasket we study properties of this operator we show that there is a maximal principle for solutions of certain nonlinear equations of the form u x f x u x we discuss the extension of the laplacian to non compact fractal blow ups and show that it is essentially self adjoint and we prove an analog of liouville s theorem in some cases we also give an explicit algorithm for solving the dirichlet problem for certain domains in the sierpinski gasket and give a characterization of all harmonic functions on those domains

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

riesz transforms for bounded laplacians on graphs we study several problems related to the ell p boundedness of riesz transforms for graphs endowed with so called bounded laplacians introducing a proper notion of the gradient of a function we prove for p in 1 2 an ell p estimate for the gradient of the continuous time heat semigroup an ell p interpolation inequality as well as the ell p boundedness of the modified littlewood paley stein function for a graph with bounded laplacian this yields an analogue to dungey s results in 21 while removing some additional assumptions coming back to the classical notion of the gradient we give a counterexample to the interpolation inequality and hence to the boundedness of riesz transforms for bounded laplacians for 1 p 2 finally we prove the boundedness of the riesz transform for 1 p infty under the assumption of positive spectral gap

harmonic analysis on sl 2 r and smoothness of the density of states in the one dimensional anderson model we consider infinite jacobi matrices with ones off diagonal and independent identically distributed random variables with distributionf v dv on diagonal iff has compact support and lies in some sobolev spacel 1 then we prove that the integrated density of states k e isc ine

differential operators and finite dimensional algebras abstract let r be a dedekind domain that is finitely generated over k an algebraically closed field of characteristic zero let m be a torsionfree module of rank one over a subalgebra of r with integral closure r this paper investigates the structure of d m the ring of differential operators on m it is shown that d m has a unique minimal non zero ideal j m and that the factor d m j m is a finite dimensional k algebra this factor is realised as the algebra of all endomorphisms of an associated vector space that preserve certain subspaces the main result is that given any finite dimensional k algebra a there exists such an m with a d m j m

hydrodynamic limit of the symmetric exclusion process on a compact riemannian manifold we consider the symmetric exclusion process on suitable random grids that approximate a compact riemannian manifold we prove that a class of random walks on these random grids converge to brownian motion on the manifold we then consider the empirical density field of the symmetric exclusion process and prove that it converges to the solution of the heat equation on the manifold

functional inequalities on arbitrary riemannian manifolds abstract for any connected not necessarily complete riemannian manifold we construct a probability measure of type e v x d x where d x is the riemannian volume measure and v is a function c smooth outside a closed set of zero volume satisfying poincare sobolev type functional inequalities in particular v is c smooth on the whole manifold when the poincare and the super poincare inequalities are considered the sobolev inequality for infinite measures are also studied

riesz transforms and spectral multipliers of the hodge laguerre operator abstract on r d endowed with the laguerre probability measure we define a hodge laguerre operator l acting on differential forms here is the laguerre exterior differentiation operator defined as the classical exterior differential except that the partial derivatives x i are replaced by the laguerre derivatives x i x i and is the adjoint of with respect to inner product on forms defined by the euclidean structure and the laguerre measure we prove dimension free bounds on l p 1 p for the riesz transforms l 1 2 and l 1 2 as applications we prove the strong hodge de rahm kodaira decomposition for forms in l p and deduce existence and regularity results for the solutions of the hodge and de rham equations in l p we also prove that for suitable functions m the operator m l is bounded on l p 1 p

a gaussian estimate for the heat kernel on differential forms and application to the riesz transform let m g be a complete riemannian manifold which satisfies a sobolev inequality of dimension n and on which the volume growth is comparable to the one of mathbb r n for big balls if there is no non zero l 2 harmonic 1 form and the ricci tensor is in l frac n 2 varepsilon cap l infty for an varepsilon 0 then we prove a gaussian estimate on the heat kernel of the hodge laplacian acting on 1 forms this allows us to prove that under the same hypotheses the riesz transform d vardelta 1 2 is bounded on l p for all 1 3

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

erratum to on the weak l p hodge decomposition and beurling ahlfors transforms on complete riemannian manifolds based on a new martingale representation formula we prove some quantitative upper bound estimates of the l p norm of some singular integral operators on complete riemannian manifolds this leads us to establish the weak l p hodge decomposition theorem and to prove the l p boundedness of the beurling ahlfors transforms on complete non compact riemannian manifolds with non negative weitzenbock curvature operator

stochastic volterra equations in banach spaces and stochastic partial differential equation abstract in this paper we study the existence uniqueness and large deviation estimate for stochastic volterra integral equations with singular kernels in 2 smooth banach spaces then we apply them to a large class of semilinear stochastic partial differential equations spde and obtain the existence of unique maximal strong solutions in the sense of sde and pde under local lipschitz conditions moreover stochastic navier stokes equations are also investigated

bessel capacities on compact manifolds and their relation to poisson capacities a motivation for this paper comes from the role of choquet capacities in the study of semilinear elliptic partial differential equations in particular the recent progress in the classification of all positive solutions of lu u in a bounded smooth domain e rd was achieved by using as a tool capacities on a smooth manifold e either the poisson capacities associated with the poisson kernel in e or the bessel capacities related to the bessel kernel have been used in this and many other applications there is no advantage in choosing any special member in a class of equivalent capacities two capacities are called equivalent if their ratio is bounded away from 0 and in the literature bessel capacities are considered mostly in the space rd we introduce two versions of bessel capacities on a compact n dimensional manifold a class capl p of equivalent capacities is defined for lp n on every compact lipschitz manifold another class cbl p is defined for all l 0 p 1 in terms of a diffusion process on a c2 manifold these classes coincide when both are defined if the manifold is the boundary of a bounded c2 domain e rd then both versions of the bessel capacities are equivalent to the poisson capacities

martingale solutions for the stochastic nonlinear schrodinger equation in the energy space we consider a stochastic nonlinear schrodinger equation with multiplicative noise in an abstract framework that covers subcritical focusing and defocusing stochastic nlse in h 1 on compact manifolds and bounded domains we construct a martingale solution using a modified faedo galerkin method based on the littlewood paley decomposition for the 2d manifolds with bounded geometry we use the strichartz estimates to show the pathwise uniqueness of solutions

a kernel based embedding method and convergence analysis for surfaces pdes we analyze a least squares strong form kernel collocation formulation for solving second order elliptic pdes on smooth connected and compact surfaces with bounded geometry the methods do not require any partial derivatives of surface normal vectors or metric based on some standard smoothness assumptions for high order convergence we provide the sufficient denseness conditions on the collocation points to ensure the methods are convergent in addition to some convergence verifications we also simulate some reaction diffusion equations to exhibit the pattern formations

stochastic completeness for graphs with curvature dimension conditions abstract we prove pointwise gradient bounds for heat semigroups associated to general possibly unbounded laplacians on infinite graphs satisfying the curvature dimension condition cd k using gradient bounds we show stochastic completeness for graphs satisfying the curvature dimension condition

gundy varopoulos martingale transforms and their projection operators on manifolds and vector bundles this paper proves the l p boundedness of generalized first order riesz transforms obtained as conditional expectations of martingale transforms a la gundy varopoulos for quite general diffusions on manifolds and vector bundles several specific examples and applications are presented lie groups of compact type the heisenberg group mathbb su 2 and riesz transforms on forms and spinors

boundary behavior of harmonic functions on manifolds in this paper we mainly set up a kind of representation theorem of harmonic functions on manifolds with ricci curvature bounded below and study non tangential limits of harmonic functions

lp spectral theory of kleinian groups let be a geometrically finite group acting on the hyperbolic space hn 1 and let h denote the laplace beltrami operator on m hn 1 which we assume is not compact if m is either of finite volume or cusp free we determine completely the lp spectrum of h for 1 p finding that it depends upon p in a nontrivial manner we also obtain a number of pointwise and lp decay properties of the l2 eigenfunctions associated with eigenvalues e in the range 0 e n24

regularities for semilinear stochastic partial differential equations abstract in this paper we study the regularities of solutions to semilinear stochastic partial differential equations in general settings and prove that the solution can be smooth arbitrarily when the data is sufficiently regular as applications we also study several classes of semilinear stochastic partial differential equations on abstract wiener space complete riemannian manifold as well as bounded domain in euclidean space

lp theory of semi linear spdes on general measure spaces and applications abstract in general settings applying evolutional semigroup arguments we prove the existence and uniqueness of l p solutions to semi linear spdes of the type d u t x l u t x f t x u t d t k g k t x u t d w t k u 0 x u 0 x x e where l is an unbounded linear negative operator on l p e b w t k t 0 k 1 2 is a sequence of independent brownian motions and e b is a general measure space we also discuss the regularities of solutions in sobolev spaces moreover a time discretized approximation for above equation is proved to convergence in holder spaces as applications we study several classes of solutions for different types spdes on abstract wiener space and riemannian manifold

cases of equality in certain multilinear inequalities of hardy riesz rogers brascamp lieb luttinger type abstract cases of equality in certain hardy riesz rogers brascamp lieb luttinger rearrangement inequalities are characterized

quantitative c1 estimates by bismut formulae abstract for a c 2 function u and an elliptic operator l we prove a quantitative estimate for the derivative du in terms of local bounds on u and lu an integral version of this estimate is then used to derive a condition for the zero mean value property of u an extension to differential forms is also given our approach is probabilistic and could easily be adapted to other settings

fundamental solution of a higher step grushin type operator abstract we examine a class of grushin type operators p k where k n 0 defined in 1 1 the operators p k are non elliptic and degenerate on a sub manifold of r n l geometrically they arise via a submersion from a sub laplace operator on a nilpotent lie group of step k 1 we explain the geometric framework and prove some analytic properties such as essential self adjointness the main purpose of the paper is to give an explicit expression of the fundamental solution of p k our methods rely on an appropriate change of coordinates and involve the theory of bessel and modified bessel functions together with weber s second exponential integral

some function spaces on symmetric spaces related to convolution operators soit g un groupe de lie semi simple connexe non compact de rang 1 une mesure positive k droite invariante a support compact soit 1 p 2 alors definit un operateur de convolution borne sur l p g si et seulement si il y a une fonction h dans l p 1 g telle que pour toute fonction c sur g k u avec u 4p 2 pp u g u g o d g g u g o h g dg la norme convolution de est equivalente a la norme l p 1 de h

logarithmic sobolev inequalities for pinned loop groups abstract letgbe a connected compact type lie group equipped with anadg invariant inner product on the lie algebra g ofg given this data there is a well known left invariant h1 riemannian structure on l l g the infinite dimensional group of continuous based loops ing using this riemannian structure we define and construct a heat kernel t g0 associated to the laplace beltrami operator on l g heret 0 g0 l g and t g0 is a certain probability measure on l g for fixedg0 l g andt 0 we use the measure t g0 and the riemannian structure on l g to construct a classical pre dirichlet form the main theorem of this paper asserts that this pre dirichlet form admits a logarithmic sobolev inequality

on lp contractivity of semigroups generated by linear partial differential operators abstract this paper is devoted to the study of contraction semigroups generated by linear partial differential operators it is shown that linear partial differential operators of order higher than two cannot generate contraction semigroups on l p n for p 1 unless p 2 if p 1 and the l p dissipativity criterion is restricted to the cone of nonnegative functions for differential operators with real valued coefficients it is proven that the criterion still fails for operators of order higher than two except for some fourth order operators if 3 2 p 3 a class of such fourth order operators is also presented

the mourre theory for analytically fibered operators abstract we develop in this paper the mourre theory for an abstract class of fibered self adjoint operators which we call analytically fibered operators we construct a conjugate operator for which we prove that a mourre estimate holds examples of analytically fibered operators are given and finally perturbations of such operators are considered

harmonic analysis on semigroups wherever t 0 and 1 p so e m stein st 3 developed a littlewoodpaley theory for such semigroups with some additional hypotheses r r coifman r rochberg and g weiss crw pointed out that some of stein s estimates could be obtained more simply by the use of transference techniques which were developed by coifman and weiss cw our object is to push these transference methods to present an alternative and simpler approach to some of the principal results of stein s book we shall also use these results to prove estimates for the wave equation

n 2 supersymmetric quantum mechanics on riemann surfaces with meromorphic superpotentials we constructn 2 supersymmetric quantum hamiltonians with meromorphic superpotentials on compact riemann surfaces and investigate the topological properties of these hamiltonians l2 cohomology groups for supercharge a deformed bar partial operator are considered and the witten index for the supersymmetric hamiltonian with meromorphic superpotential is calculated in terms of euler characteristic of the riemann surface and the degree of a divisor of poles for the differential of the superpotential

orlicz spaces and the large scale geometry of heintze groups we consider an orlicz space based cohomology for metric measured spaces with bounded geometry we prove the quasi isometry invariance for a general young function in the hyperbolic case we prove that the degree one cohomology can be identified with an orlicz besov function space on the boundary at infinity we give some applications to the large scale geometry of homogeneous spaces with negative curvature heintze groups as our main result we prove that if the heintze group is not of carnot type any self quasi isometry fixes a distinguished point on the boundary and preserves a certain foliation on the complement of that point

non self adjoint jacobi matrices with a rank one imaginary part we develop direct and inverse spectral analysis for finite and semi infinite non self adjoint jacobi matrices with a rank one imaginary part it is shown that given a set of n not necessarily distinct nonreal numbers in the open upper lower half plane uniquely determines an n n jacobi matrix with a rank one imaginary part having those numbers as its eigenvalues counting algebraic multiplicity algorithms of reconstruction for such finite jacobi matrices are presented a new model complementing the well known livsic triangular model for bounded linear operators with a rank one imaginary part is obtained it turns out that the model operator is a non self adjoint jacobi matrix we show that any bounded prime non self adjoint linear operator with a rank one imaginary part acting on some finite dimensional respectively separable infinite dimensional hilbert space is unitarily equivalent to a finite respectively semi infinite non self adjoint jacobi matrix this obtained theorem strengthens a classical result of stone established for self adjoint operators with simple spectrum we establish the non self adjoint analogs of the hochstadt and gesztesy simon uniqueness theorems for finite jacobi matrices with nonreal eigenvalues as well as an extension and refinement of these theorems for finite non self adjoint tri diagonal matrices to the case of mixed eigenvalues real and nonreal a unique jacobi matrix unitarily equivalent to the operator of integration ff x 2i xlf t dt in the hilbert space l2 0 l is found as well as spectral properties of its perturbations and connections with the well known bernoulli numbers we also give the analytic characterization of the weyl functions of dissipative jacobi matrices with a rank one imaginary part

asymptotic finite propagation speed for heat diffusion on certain riemannian manifolds using pointwise upper bounds recently obtained by the first author we first show that the heat kernel ht x y on a riemannian symmetric space of the noncompact type gk is asymptotically concentrated in an annulus centered at y and moving to infinity with finite speed 2 being as usual the half sum of all positive roots of gk in the higher rank case we prove moreover that heat not only concentrates in an annulus but also along the k orbit of the axis by applying wave equation techniques developed by m e taylor we partially extend the above results to riemannian manifolds with exponential volume growth and l2 spectrum of the laplacian bounded away from 0 some extensions to the vector bundle case are also considered

surface measures and tightness of r p capacities on poisson space abstract we prove tightness of r p sobolev capacities on configuration spaces equipped with poisson measure by using this result we construct surface measures on configuration spaces in the spirit of the malliavin calculus a related gauss ostrogradskii formula is obtained

trudinger moser inequalities on harmonic an groups under lorentz norms abstract the main purpose of this paper is to establish sharp trudinger moser type inequalities on harmonic a n groups s for functions whose gradient is in the lorentz space l n q our results show that even if s is not with strictly negative sectional curvature one can also replace the norm s s u n d v s u n d v by s s u n d v in the trudinger moser type inequalities

lp estimates and existence theorems for the operator on complete kahler manifolds abstract in this paper we establish some l p estimates and existence theorems for the operator on a holomorphic hermitian vector bundle over a complete kahler manifold as applications we prove some vanishing theorems of the l p dolbeault cohomology on complete kahler manifolds with suitable geometric conditions

martingale transforms and lp norm estimates of riesz transforms on complete riemannian manifolds under the condition that the bakry emery ricci curvature is bounded from below we prove a probabilistic representation formula of the riesz transforms associated with a symmetric diffusion operator on a complete riemannian manifold using the burkholder sharp l p inequality for martingale transforms we obtain an explicit and dimension free upper bound of the l p norm of the riesz transforms on such complete riemannian manifolds for all 1 p in the euclidean and the gaussian cases our upper bound is asymptotically sharp when p 1 and when p

boundary value problems in spaces of distributions on smooth and polygonal domains we study boundary value problems of the form email protected f on w and bu g on the boundary email protected with either dirichlet or neumann boundary conditions where w is a smooth bounded domain in r n and the data f g are distributions this problem has to be first properly reformulated and for practical applications it is of crucial importance to obtain the continuity of the solution u in terms of f and g for f 0 taking advantage of the fact that u is harmonic on w we provide four formulations of this boundary value problem one using nontangential limits of harmonic functions one using green functions one using the dirichlet to neumann map and a variational one we show that these four formulations are equivalent we provide a similar analysis for f 0 and discuss the roles of f and g which turn to be somewhat interchangeable in the low regularity case the weak formulation is more convenient for numerical approximation whereas the nontangential limits definition is closer to the intuition and easier to check in concrete situations we extend the weak formulation to polygonal domains using weighted sobolev spaces we also point out some new phenomena for the concentrated loads at the vertices in the polygonal case

sum of squares manifolds the expressibility of the laplace beltrami operator on pseudo riemannian manifolds as a sum of squares of vector fields in this paper we investigate under what circumstances the laplace beltrami operator on a pseudo riemannian manifold can be written as a sum of squares of vector fields as is naturally the case in euclidean space we show that such an expression exists globally on one dimensional manifolds and can be found at least locally on any analytic pseudo riemannian manifold of dimension greater than two for two dimensional manifolds this is possible if and only if the manifold is flat these results are achieved by formulating the problem as an exterior differential system and applying the cartan kahler theorem to it

the poisson equation on complete manifolds with positive spectrum and applications abstract in this paper we investigate the existence of a solution to the poisson equation on complete manifolds with positive spectrum and ricci curvature bounded from below we show that if a function f has decay f o r 1 e for some e 0 where r is the distance function to a fixed point then the poisson equation u f has a solution u with at most exponential growth we apply this result on the poisson equation to study the existence of harmonic maps between complete manifolds and also existence of hermitian einstein metrics on holomorphic vector bundles over complete manifolds thus extending some results of li tam and ni assuming moreover that the manifold is simply connected and of ricci curvature between two negative constants we can prove that in fact the poisson equation has a bounded solution and we apply this result to the ricci flow on complete surfaces

heat equation derivative formulas for vector bundles we use martingale methods to give bismut type derivative formulas for differentials and co differentials of heat semigroups on forms and more generally for sections of vector bundles the formulas are mainly in terms of weitzenbock curvature terms in most cases derivatives of the curvature are not involved in particular our results improve b k driver s formula in 1997 j math pures appl 9 76 703 737 for logarithmic derivatives of the heat kernel measure on a riemannian manifold our formulas also include the formulas of k d elworthy and x m li 1998 c r acad sci paris ser i math 327 87 92

covering of spheres by spherical caps and worst case error for equal weight cubature in sobolev spaces we prove that the covering radius of an n point subset xnxn of the unit sphere sd rd 1sd rd 1 is bounded above by a power of the worst case error for equal weight cubature 1n x xnf x sdfd d for functions in the sobolev space wps sd where d d denotes normalized area measure on sdsd these bounds are close to optimal when s is close to d pd p our study of the worst case error along with results of brandolini et al motivate the definition of quasi monte carlo qmc design sequences for wps sd which have previously been introduced only in the hilbert space setting p 2p 2 we say that a sequence xn xn of n point configurations is a qmc design sequence for wps sd with s d ps d p provided the worst case equal weight cubature error for xnxn has order n s dn s d as n n a property that holds in particular for a sequence of spherical t designs in which each design has order tdtd points for the case p 1p 1 we deduce that any qmc design sequence xn xn for w1s sd with s ds d has the optimal covering property i e the covering radius of xnxn has order n 1 dn 1 d as n n a significant portion of our effort is devoted to the formulation of the worst case error in terms of a bessel kernel and showing that this kernel satisfies a bernstein type inequality involving the mesh ratio of xnxn as a consequence we prove that any qmc design sequence for wps sd is also a qmc design sequence for wp s sd for all 1 p s d ps s d p

rings of operators on modules over commutative rings and their right ideals suppose that there is an inclusion ofk algebrasr e endkmwithrcommutative andenon commutative we introduce and impose conditions under which the finitely generated essential right ideals ofemay be classified in terms ofk submodules ofm this yields a classification of the domains morita equivalent toewheneis a noetherian domain for example a special case of our results is theorem let r be a commutative noetherian k algebra which is domain let e be a simple ore extension of r of the form r x x 1 or r x in the latter case we must also assume r 3 then for a certain sublatticelof the lattice of k submodules of r a every non zero right ideal of e is isomorphic to one of the form formula for some v l b every domain morita equivalent to e is isomorphic to formula for some v l conversely if r is dedekind then e v is morita equivalent to e for v l