the limits of stochastic integrals of differential forms this paper is concerned with the integration of 1 forms against the markov stochastic process associated with a second order elliptic differential operator in divergence form it focuses on the limiting behavior of the integral as the process leaves a fixed point or goes to infinity this extends previous work in the area where advantage was usually taken of the fact that the operator was self adjoint and started with the associated invariant measure applications are given for example it is a trivial consequence that the diffusion associated to a uniformly elliptic operator on a negatively curved cartan hadamard manifold has an asymptotic direction recovering and strengthening the previous arguments of pratt sullivan and others the approach can also be used to construct a levy area for such processes to study the thinness of sets for the elliptic operator and probably has wider applications

the number of limit cycles from a cubic center by the melnikov function of any order abstract in this paper we consider the system x y 1 x 2 p x y y x 1 x 2 q x y where p x y and q x y are arbitrary quadratic polynomials we study the maximum number of limit cycles bifurcating from the periodic orbits by using the melnikov function of any order we prove that the upper bound for the number of limit cycles is 3 and reached

generalized liouville property for schrodinger operator on riemannian manifolds in this paper we prove that the dimension of the space of positive bounded respectively mathcal l harmonic functions on a complete riemannian manifold with mathcal l regular ends is equal to the number of ends mathcal l nonparabolic ends respectively this result is a solution of an open problem of grigor yan related to the liouville property for the schrodinger operator mathcal l we also prove that if a given complete riemannian manifold is roughly isometric to a complete riemannian manifold satisfying the volume doubling condition the poincare inequality and the finite covering condition on each end then the dimension of the space of positive bounded respectively solutions for the schrodinger operator with a potential satisfying a certain decay rate on the manifold is equal to the number of ends mathcal l nonparabolic ends respectively this is a partial answer of the question suggested by li related to the regularity of ends of a complete riemannian manifold especially our results directly generalize various earlier results of yau of li and tam of grigor yan and of present authors but with different techniques that the peculiarity of the schrodinger operator demands

harnack inequality for quasilinear elliptic equations on riemannian manifolds abstract we study viscosity solutions to degenerate and singular elliptic equations div f u u u h of p laplacian type on riemannian manifolds where an even function f c 1 r c 2 0 is supposed to be strictly convex on 0 under the assumption that either f c 2 r or its convex conjugate f c 2 r with some structural condition we establish a locally uniform abp type estimate and the krylov safonov type harnack inequality on riemannian manifolds with the use of an intrinsic geometric quantity to the operator here the c 2 regularities of f and f account for degenerate and singular operators respectively

some new results on eigenvectors via dimension diameter and ricci curvature abstract we generalise for a general symmetric elliptic operator the different notions of dimension diameter and ricci curvature which coincide with the usual notions in the case of the laplace beltrami operators on riemannian manifolds if 1 denotes the spectral gap that is the first nonzero eigenvalue we investigate in this paper the best lower bound on 1 one can obtain under an upper bound on the dimension an upper bound on the diameter and a lower bound of the ricci curvature two cases are known namely if the ricci curvature is bounded below by a constant r 0 then 1 nr n 1 and this estimate is sharp for the n dimensional spheres lichnerowicz s bound if the ricci curvature is bounded below by zero then zhong yang s estimate asserts that 1 2 d 2 where d is an upper bound on the diameter this estimate is sharp for the 1 dimensional torus in the general case many interesting estimates have been obtained this paper provides a general optimal comparison result for 1 which unifies and sharpens lichnerowicz and zhong yang s estimates together with other comparison results concerning the range of the associated eigenfunctions and their derivatives

heat flow on finsler manifolds this paper studies the heat flow on finsler manifolds a finsler manifold is a smooth manifold m equipped with a minkowski norm f x txm on each tangent space mostly we will require that this norm be strongly convex and smooth and that it depend smoothly on the base point x the particular case of a hilbert norm on each tangent space leads to the important subclasses of riemannian manifolds where the heat flow is widely studied and well understood we present two approaches to the heat flow on a finsler manifold as gradient flow on l2 m m for the energy as gradient flow on the reverse l2 wasserstein space 2 m of probability measures on m for the relative entropy both approaches depend on the choice of a measure m on m and then lead to the same nonlinear evolution semigroup we prove 1 regularity for solutions to the nonlinear heat equation on the finsler space m f m typically solutions to the heat equation will not be 2 moreover we derive pointwise comparison results a la cheeger yau and integrated upper gaussian estimates a la davies 2008 wiley periodicals inc

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

operator valued typically real functions induced by a contraction let h be a complex hilbert space and let l h denote the algebra of all bounded linear operators on h letf z n 0 znanbe an operator valued analytic function whose coefficients are bounded operators in l h forzin the open unit disc d and the series is convergent in the strong operator topology in this paper we discuss operator valued typically real functionsf z n 0 znanwhich generalize complex valued typically real functions we characterize operator valued typically real functions and study such functionsf z induced by a contraction on h in addition we considerm n tuple operator valued typically real functions and positively real functions finally we charaterize operator valued typically real functions in the finite dimensional cases

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

large time behavior of schrodinger heat kernels and applications we obtain global in time and qualitatively sharp bounds for the heat kernel g of the schrodinger operator v the potential v satisfies v x c d x b near infinity with b 0 2 when v 0 our result can be described as follows g is bounded from above and below by the multiples of standard gaussians with a weight function if b 2 then the weight is bounded between two positive it constants if b 2 the weight is bounded between two positive functions of t d x and d y which have polynomial decay if b 2 or b 0 an application to a semilinear elliptic problem is also given

heat kernel on smooth metric measure spaces with nonnegative curvature we derive a local gaussian upper bound for the f heat kernel on complete smooth metric measure space m g e f dv with nonnegative bakry emery ricci curvature as applications we obtain a sharp l f 1 liouville theorem for f subharmonic functions and an l f 1 uniqueness property for nonnegative solutions of the f heat equation assuming f is of at most quadratic growth in particular any l f 1 integrable f subharmonic function on gradient shrinking and steady ricci solitons must be constant we also provide explicit f heat kernel for gaussian solitons

the boundary behavior of heat kernels of dirichlet laplacians we prove the following complete and qualitatively sharp description of heat kernels g of dirichlet laplacians on bounded c1 1 domains d there exist positive constants c1 c2 and t 0 depending on d such that for x dist x d x y t 1c1tn 2e c2 x y 2 t g x t y 0 x y t 11c1tn 2e x y 2 c2t for all x y d and 0 t are known bounds when d is unbounded are also given

on poincare type inequalities using estimates of the heat kernel we prove a poincare inequality for star shape domains on a complete manifold the method also gives a lower bound for the gap of the first two neumann eigenvalues of a schrodinger operator

heat extensions optimal atomic decompositions and sobolev embeddings in presence of symmetries on manifolds our purpose is twofold we give the heat semi group characterization of spaces of besov and triebel lizorkin types defined on riemannian manifolds of bounded geometry moreover we study boundedness and compactness of sobolev embeddings of the spaces in presence of symmetries to make it possible we first construct an optimal atomic decomposition for the above function spaces

localized upper bounds of heat kernels for diffusions via a multiple dynkin hunt formula we prove that for a general diffusion process certain assumptions on its behavior only within a fixed open subset of the state space imply the existence and sub gaussian type off diagonal upper bounds of the global heat kernel on the fixed open set the proof is mostly probabilistic and is based on a seemingly new formula which we call a multiple dynkin hunt formula expressing the transition function of a hunt process in terms of that of the part process on a given open subset this result has an application to heat kernel analysis for the liouville brownian motion the canonical diffusion in a certain random geometry of the plane induced by a massive gaussian free field

the heat kernel of a schrodinger operator with inverse square potential we consider the schrodinger operator h v x with radial potential v which may have singularity at 0 and a quadratic decay at infinity first we study the structure of positive harmonic functions of h and give their precise behavior second under quite general conditions we prove an upper bound for the correspond heat kernel p x y t of the type 0 p x y t ct n 2u min x t u min y t u t 2exp x y 2ct for all x y rn and t 0 where u is a positive harmonic function of h third if u2 is an a2 weight on rn then we prove a lower bound of a similar type

positive solutions to schrodinger equations and geometric applications a variant of li tam theory which associates to each end of a complete riemannian manifold a positive solution of a given schrodinger equation on the manifold is developed it is demonstrated that such positive solutions must be of polynomial growth of fixed order under a suitable scaling invariant sobolev inequality consequently a finiteness result for the number of ends follows in the case when the sobolev inequality is of particular type the finiteness result is proven directly as an application an estimate on the number of ends for shrinking gradient ricci solitons and submanifolds of euclidean space is obtained

sobolev spaces laplacian and heat kernel on alexandrov spaces we prove the compactness of the imbedding of the sobolev space w 1 2 0 omega into l 2 omega for any relatively compact open subset omega of an alexandrov space as a corollary the generator induced from the dirichlet energy form has discrete spectrum the generator can be approximated by the laplacian induced from the dc structure on the alexandrov space we also prove the existence of the locally holder continuous heat kernel

using melnikov functions of any order for studying limit cycles abstract in this work we study a class of planar hamiltonian systems perturbed with general quadratic polynomials in order to identify the number of limit cycles employing melnikov functions of any order we show that the class of systems can have at most three limit cycles and the limit is reached

the global behavior of heat kernels in exterior domains abstract we establish the global boundary behavior of dirichlet heat kernels on some unbounded domains these combined with the interior estimate in the recent paper by grigoryan and saloff coste comm pure appl math 55 1 2002 93 provide a complete and qualitatively sharp description of heat kernels g of dirichlet laplacians on some unbounded c1 1 domains d m here m is certain complete noncompact manifold with nonnegative ricci curvature for example there exist positive constants c1 c2 depending on the unbounded domain d m such that for x dist x d x t 1 1 y t 1 1 c 1 e c 2 d x y 2 t b x t g x t y 0 x t 1 1 y t 1 1 e d x y 2 c 2 t c 1 b x t for all x y d and t 0 existence of these global bounds had been an unresolved problem even in the euclidean case

surgery of the faber krahn inequality and applications to heat kernel bounds 2 cutting faber krahn inequalities 3 2 1 fk functions 3 2 2 cutting a manifold 6 2 3 non parabolic case 7 2 4 fk functions in balls 9 2 5 relative fk function 16

harnack inequality for degenerate and singular operators of p laplacian type on riemannian manifolds we study viscosity solutions to degenerate and singular elliptic equations of p laplacian type on riemannian manifolds the krylov safonov type harnack inequality for the p laplacian operators with 1 p infty is established on the manifolds with ricci curvature bounded from below based on abp type estimates we also prove the harnack inequality for nonlinear p laplacian type operators assuming that a nonlinear perturbation of ricci curvature is bounded below

the third order melnikov function of a quadratic center under quadratic perturbations abstract we study quadratic perturbations of the integrable system 1 x d h where h x 2 y 2 2 we prove that the first three melnikov functions associated to the perturbed system give rise at most to three limit cycles

extremal of log sobolev inequality and w entropy on noncompact manifolds abstract let m be a complete connected noncompact manifold with bounded geometry under a condition near infinity we prove that the log sobolev functional 1 1 has an extremal function decaying exponentially near infinity we also prove that an extremal function may not exist if the condition is violated this result has the following consequences 1 it seems to give the first example of connected complete manifolds with bounded geometry where a standard log sobolev inequality does not have an extremal 2 it gives a negative answer to the open question on the existence of extremal of perelman s w entropy in the noncompact case which was stipulated by perelman 2002 22 p 9 3 2 remark 3 it helps to prove in some cases that noncompact shrinking breathers of ricci flow are gradient shrinking solitons

space of ricci flows i based on the compactness of the moduli of non collapsed calabi yau spaces with mild singularities we set up a structure theory for polarized kahler ricci flows with proper geo metric bounds our theory is a generalization of the structure theory of non collapsed kahler einstein manifolds as applications we prove the hamilton tian conjecture and the partial c 0 conjecture of tian

global bounds of schrodinger heat kernels with negative potentials abstract we obtain global in time bounds for the heat kernel g of the schrodinger operator l v the potential v satisfies v x c d x b near infinity with b 0 the result can be described as follows suppose l is positive and b 2 then g g x t y 0 has a global upper bound which is a standard gaussian times a polynomial function of x y t however when l is not nonnegative and negative eigenvalues exist g g x t 0 0 is bounded below by a standard gaussian times ectwithc 0 in other words the growth rate of g x t 0 0 is comparable with the heat kernel of c regardless how fast v decays near infinity

on the lp independence of the spectrum of the hodge laplacian on non compact manifolds abstract the central aim of this paper is the study of the spectrum of the hodge laplacian on differential forms of any order k in l p the underlying space is a c smooth open manifold m n with ricci curvature bounded below and uniformly subexponential volume growth it will be demonstrated that on such manifolds the l p spectrum of the hodge laplacian on differential k forms is independent of p for 1 p whenever the weitzenbock tensor on k forms is also bounded below it follows as a corollary that the isolated eigenvalues of finite multiplicity are l p independent the proof relies on the existence of a gaussian upper bound for the heat kernel of the hodge laplacian by considering the l p spectra on the hyperbolic space h n 1 we conclude that the subexponential volume growth condition is necessary in the case of one forms as an application we will show that the spectrum of the laplacian on one forms has no gaps on certain manifolds with a pole or that are in a warped product form this will be done under less strict curvature restrictions than what has been known so far and it was achieved by finding the l 1 spectrum of the laplacian

geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature we apply alexandrov geometry methods to study geometric analysis aspects of infinite semiplanar graphs with nonnegative combinatorial curvature we obtain the metric classification of these graphs and construct the graphs embedded in the projective plane minus one point moreover we show the volume doubling property and the poincare inequality on such graphs the quadratic volume growth of these graphs implies the parabolicity finally we prove the polynomial growth harmonic function theorem analogous to the case of riemannian manifolds

fredholm theory for elliptic operators on quasi asymptotically conical spaces we consider the mapping properties of generalized laplace type operators l r on the class of quasi asymptotically conical qac spaces which provide a riemannian generalization of the qale manifolds considered by joyce joy00 our main result gives conditions under which such operators are fredholm when between

configurations of limit cycles and planar polynomial vector fields abstract we show that every finite configuration of disjoint simple closed curves of the plane is topologically realizable as the set of limit cycles of a polynomial vector field moreover the realization can be made by algebraic limit cycles and we provide an explicit polynomial vector field exhibiting any given finite configuration of limit cycles

the diffusion phenomenon for dissipative wave equations in metric measure spaces abstract we study the long time behavior of the solution to a type of dissipative wave equation where the operator in the equation is time dependent and the solution is defined on a metric measure space x m satisfying appropriate conditions the operator is assumed to be self adjoint and is related to a time dependent dirichlet form we link hyperbolic pdes with the firmly established theories for parabolic pdes in metric measure spaces and dirichlet forms subsequently deriving the asymptotic behavior of the solution to the dissipative wave equation we present several nontrivial examples of dissipative wave equations in metric measure spaces where our theory works

a note on a result of g s petrov about the weakened 16th hilbert problem abstract g s petrov functional anal appl 22 1988 72 73 and p mardesic ergodic theory dynamical systems10 1990 523 529 have proved that for system y p x y 1 3x2 q x y where p x y and q x y are polynomials of x y with degree n if the first order melnikov function m1 h 0 then the lowest upper bound b n of the number of limit cycles of the above system is n 1 we prove that if m1 h 0 and the second order melnikov function m2 h 0 then formula

a class of reversible quadratic systems with piecewise polynomial perturbations abstract this paper investigates a class of reversible quadratic systems perturbed inside piecewise polynomial differential systems of arbitrary degree n all possible phase portraits of the reversible quadratic systems on the plane are obtained by analytic techniques then we present the algebraic structure of its corresponding first order melnikov function which can be used to study the limit cycle bifurcation problem in this direction we use it to investigate the hopf bifurcation of the perturbed systems and obtain an upper bound for the hopf cyclicity meanwhile we employ it to discuss the maximal number of limit cycles emerging from the periodic orbits around the origin for a special case and also provide its upper bound

rough metrics on manifolds and quadratic estimates we study the persistence of quadratic estimates related to the kato square root problem across a change of metric on smooth manifolds by defining a class of rough riemannian like metrics that are permitted to be of low regularity and degenerate on sets of measure zero we also demonstrate how to transmit quadratic estimates between manifolds which are homeomorphic and locally bi lipschitz as a consequence we demonstrate the invariance of the kato square root problem under lipschitz transformations and obtain solutions to this problem on functions and forms on compact manifolds with a rough metric furthermore we show that a lower bound on the injectivity radius is not a necessary condition to solve the kato square root problem

holder estimates and regularity for holomorphic and harmonic functions in this paper we proved that if a singular manifold satisfies a weak mean value property for positive subharmonic functions then one can derive an oscillation bound for bounded holomorphic functions moreover if we further assume that the volume decays at most polynomially at a singular point

on the geometry of semiclassical limits on dirichlet spaces abstract this paper is a contribution to semiclassical analysis for abstract schrodinger type operators on locally compact spaces let x be a metrizable separable locally compact space let be a radon measure on x with a full support let t x y p t x y be a strictly positive pointwise consistent heat kernel and assume that the generator h p 0 of the corresponding self adjoint contraction semigroup in l 2 x induces a regular dirichlet form then given a function 0 1 0 such that the limit lim t 0 p t x x t exists for all x x we prove that for every potential w x r one has lim t 0 t tr e t h p w e w x lim t 0 p t x x t d x for the schrodinger type operator h p w provided w satisfies very mild conditions at that are essentially only made to guarantee that the sum of quadratic forms h p w t is self adjoint and bounded from below for small t and to guarantee that e w x lim t 0 p t x x t d x the proof is probabilistic and relies on a principle of not feeling the boundary for p t x x in particular this result implies a new semiclassical limit result for partition functions valid on arbitrary connected geodesically complete riemannian manifolds and one also recovers a previously established semiclassical limit result for possibly locally infinite connected weighted graphs