sobolev estimates for operators given by averages over cones we prove a result related to work by a greenleaf and g uhlmann concerning sobolev estimates for operators given by averages over cones this is done using the almost orthogonality lemma of cotlar and stein and the van der corput lemma on oscillatory integrals 0 introduction this paper is inspired by gu1 theorem 3 23 to state the result we need to introduce some notions following gu1 let m g be an n dimensional riemannian manifold we consider a c map ty m x sm 1 x ilt s tm with s tm the unitary tangent bundle on m such that for any x e iltn the map sm 1 3 s ey x s 0 e s txm is an embedding with image eyx c s tzm we define a m x sm 1 x 2 m by x r expx r x r let r x x r as the variables x r vary in the support of the distribution k defined below consider now k x r e zd m x sm 1 x r a distributiorl with very small support whith the following properties first of all we fix a number u with 0 u m then k x x r is a smooth function for r 7 0 satisfying the inequalities 0 1 a l0 x 0l k x a r i cog i r 8 l moreover if u m for any pair of small positive nurnbers o e ry 0 1 b a az g x x u r drl ca

intrinsic ultracontractivity and eigenfunction estimates for schrodinger operators let d be a domain in rn n 2 and let h h0 v where h0 is a divergence form uniformly elliptic operator with dirichlet boundary conditions and v is in the kato class intrinsic ultracontractivity is proved for the semigroup of h when d is holder domain of order 0 or a uniformly holder domain of order for 0 2 for every 2 there exists a uniformly holder domain of order for which the dirichlet laplacian is not intrinsically ultracontractive for a large class of domains it is shown that the heat kernels for h0 and h decay at the same rate at the boundary applications are given to the lifetime of conditioned brownian motion some of our results seem to be new even for smooth domains

stein compacts in levi flat hypersurfaces we explore connections between geometric properties of the levi foliation of a levi flat hypersurface m and holomorphic convexity of compact sets in m or bounded in part by m applications include extendability of cauchy riemann functions solvability of the b equation approximation of cauchy riemann and holomorphic functions and global regularity of the neumann operator

rank and regularity for averages over submanifolds abstract this paper establishes endpoint l p l q and sobolev mapping properties of radon like operators which satisfy a homogeneity condition similar to semiquasihomogeneity and a condition on the rank of a matrix related to rotational curvature for highly degenerate operators the rank condition is generically satisfied for algebraic reasons similar to an observation of greenleaf pramanik and tang a greenleaf m pramanik w tang oscillatory integral operators with homogeneous polynomial phases in several variables j funct anal 244 2 2007 444 487 concerning oscillatory integral operators

geometric incidence theorems via fourier analysis we show that every non trivial sobolev bound for generalized radon transforms which average functions over families of curves and surfaces yields an incidence theorem for suitably regular discrete sets of points and curves or surfaces in euclidean space this mechanism allows us to deduce geometric results not readily accessible by combinatorial methods

a stationary planar random graph with singular stationary dual dyadic lattice graphs dyadic lattice graphs and their duals are commonly used as discrete approximations to the hyperbolic plane we use them to give examples of random rooted graphs that are stationary for simple random walk but whose duals have only a singular stationary measure this answers a question of curien and shows behaviour different from the unimodular case the consequence is that planar duality does not combine well with stationary random graphs we also study harmonic measure on dyadic lattice graphs and show its singularity

on the stability of limit cycles for planar differential systems abstract we consider a planar differential system x p x y y q x y where p and q are c 1 functions in some open set u r 2 and d dt let be a periodic orbit of the system in u let f x y u r 2 r be a c 1 function such that p x y f x x y q x y f y x y k x y f x y where k x y is a c 1 function in u and x y f x y 0 we assume that if p u is such that f p 0 and f p 0 then p is a singular point we prove that 0 t p x q y t dt 0 t k t dt where t 0 is the period of as an application we take profit from this equality to show the hyperbolicity of the known algebraic limit cycles of quadratic systems

asymptotic stability at infinity for differentiable vector fields of the plane let x r2 d r2 be a differentiable but not necessarily c1 vector field where 0 and d z r2 z denote by r z the real part of z c if for some 0 and for all p r2 d no eigenvalue of dpx belongs to 0 z c r z 0 then a for all p r2 d there is a unique positive semi trajectory of x starting at p b it is associated to x a well defined number i x of the extended real line called the index of x at infinity such that for some constant vector v r2 the following is satisfied if i x is less than zero respectively greater or equal to zero then the point at infinity of the riemann sphere r2 is a repellor respectively an attractor of the vector field x v

strongly pseudoconvex cr structures over small balls part i an a priori estimate the purpose of the present series of papers is to develop the theory of harmonic integrals on strongly pseudoconvex cr structures over small balls along the line developed by d c spencer c b morrey j j kohn and l nirenberg 5 6 7 8 for strongly pseudoconvex complex manifolds we consider a strongly pseudoconvex cr structure on a manifold of real dimension 2n 1 in part i we establish the a priori estimate for the neumann boundary value problem on the complex associated with the structure in the case the structure is induced by an embedding in c and restricted to a small ball of special type provided i c q c n 3 where q is the degree of differential forms we consider in part ii we derive the regularity theorem of solutions of the neumann boundary value problem based on the a priori estimate of part i as an application we prove in part il that when n 5 the structure is realized on a neighborhood of a reference point by an embedding in cn we note here that l nirenberg gave an example of a strongly pseudoconvex cr structure which cannot be induced by such an embedding when n 2 9 l boutet de monvel proved that in the case of such structures over a compact manifold we can find such an embedding when n 3 1 there are two major differences between the cases of strongly pseudoconvex integrable almost complex structures and cr structures firstly the principal parts of the associated laplacians in cr structures are no longer elliptic secondly we have characteristic points on the boundary as a result when we follow over the predecessor s calculation in our case we end up with terms which may not be absorbed in the main non negative terms fortunately these leftover terms depend mostly on the levi form of the cr structure and the hessian form with respect to the cr structure of the equation of the boundary of our

a note on relaxation oscillators with exact limit cycles abstract in this note we give a family of planar polynomial differential systems with a prescribed hyperbolic limit cycle this family constitutes a corrected and wider version of an example given in the work m a abdelkader relaxation oscillators with exact limit cycles j math anal appl 218 1998 308 312 the result given in this note may be used to construct models of lienard differential equations exhibiting a desired limit cycle

fredholm property of a family of operators on the five dimensional heisenberg group abstract an expansion in euclidean spherical harmonics on the ball in the heisenberg group of dimension five decomposes the dirichlet problem for the laplacian in an infinite number of two dimensional problems fundamental solutions are obtained for each of the partial differential operators in these problems thus reducing them further via layer potentials to one dimensional integral equations the main result in this article states that the corresponding integral operators are fredholm in appropriate weighted l 2 spaces

from harmonic analysis to arithmetic combinatorics arithmetic combinatorics or additive combinatorics is a fast developing area of research combining elements of number theory combinatorics harmonic analysis and ergodic theory its arguably best known result and the one that brought it to global prominence is the proof by ben green and terence tao of the long standing conjecture that primes contain arbitrarily long arithmetic progressions there are many accounts and expositions of the green tao theorem including the articles by kra 119 and tao 182 in the bulletin the purpose of the present article is to survey a broader highly interconnected network of questions and results built over the decades and spanning several areas of mathematics of which the green tao theorem is a famous descendant an old geometric problem lies at the heart of key conjectures in harmonic analysis a major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles an unexpected argument points harmonic analysts towards additive number theory with consequences that could have hardly been anticipated we will not try to give a comprehensive survey of harmonic analysis combinatorics or additive number theory we will not even be able to do full justice to our specific areas of focus instead referring the reader to the more complete expositions and surveys listed in section 7 our goal here is to emphasize the connections between these areas we will thus concentrate on relatively few problems chosen as much for their importance to their fields as for their links to each other the article is written from the point of view of an analyst who in the course of her work was gradually introduced to the questions discussed here and found them fascinating we hope that the reader will enjoy a taste of this experience

invariant algebraic curves and rational first integrals for planar polynomial vector fields we present three main results the first two provide sufficient conditions in order that a planar polynomial vector field in c2 has a rational first integral and the third one studies the number of multiple points that an invariant algebraic curve of degree n of a planar polynomial vector field of degree m can have in function of m and n

on the stability of equivariant embedding of compact cr manifolds with circle action we prove the stability of the equivariant embedding of compact strictly pseudoconvex cr manifolds with transversal cr circle action under circle invariant deformations of the cr structures

boundary holder and estimates for local solutions of the tangential cauchy riemann equation nous etudions la resolubilite locale de l operateur de cauchy riemann tangentiel sur un voisinage w d un point z 0 d une sous variete cr generique q concave m de codimension quelconque de c n nous construisons une formule d homotopie pour le b sur lorsque est l intersection de m et d un domaine strictement pseudoconvexe nous obtenons ainsi un operateur de resolution pour toute forme b fermee sur nous en deduisons des estimations l p et des estimations holderiennes jusqu au bord pour la solution de l equation de cauchy riemann tangentielle sur

heat kernel eigenfunctions and conditioned brownian motion in planar domains abstract some recent results concerning uniform convergence of the shape of the heat kernel to that of the first eigenfunction and the lifetime of h processes in bounded lipschitz domains in r n are extended to all planar domains of finite area

a conformal inequality related to the conditional gauge theorem we prove the inequality h x 1 g x y h y cg x y c where g is the green function of a plane domain d h is positive and harmonic on d and c is a constant whose value depends on the topological nature of the domain in particular for the class of proper simply connected domains c may be taken to be an absolute constant as an application we prove the conditional gauge theorem for plane domains of finite area for which the constant c in the above inequality is finite

potential theory for the schrodinger equation soit a a ij x x i x j un operateur uniformement elliptique dont les coefficients sont des fonctions bornees mesurables sur un domaine de lipschitz borne d r d on definit la classe de kato k d on demontre des resultats sur l operateur l a v en utilisant des resultats connus pour a et en etudiant la jauge conditionnelle

lifetimes of conditioned diffusions we investigate when an upper bound on expected lifetimes of conditioned diffusions associated with elliptic operators in divergence and non divergence form can be found the critical value of the parameter is found for each of the following classes of domains l p domains p n 1 uniformly regular twistedl p domains p n 1 and twisted holder domains 1 3 a related parabolic boundary harnack principle is proved

divergence and poincare liapunov constants for analytic differential systems abstract we consider a planar autonomous real analytic differential system with a monodromic singular point p we deal with the center problem for the singular point p our aim is to highlight some relations between the divergence of the system and the poincare liapunov constants of p when these are defined

conditioned brownian motion in planar domains on etudie un mouvement brownien complexe standard qui part d un point x dans un domaine greenien connexe du plan complexe c et soit conditionne pour sortir en un point donne y de sa frontiere de martin soit conditionne pour frapper un point y dans avant de laisser

the solution of the neumann problem in a strictly pseudoconvex siegel domain in this paper we solve the o neumann problem on 0 q forms q 0 in the strictly pseudoconvex siegel domain the metric we use is invariant under the action of the heisenberg group on the domain we give a formula for a kernel of the neumann operator the fundamental solution of the problem in 1 6 and theorem 4 2 and we prove uniqueness of the neumann operator as far as we know this is the first example of an explicit solution of the neumann problem greiner and stein gs constructed a parametrix for the problem on 0 1 forms in bounded strictly pseudoconvex domains phong p has announced an explicit construction of a parametric on 03 forms in the siegel domain in c 1 n 1 harvey and polking hp have since constructed a neumann kernel for the neumann problem on p q forms on the ball we obtain the kernel of the neumann operator by integrating the fundamental solution of the corresponding heat equation s1 2 with respect to time to carry out the integration we require estimates on the fundamental solution for large time we did not need such estimates in 1 and 2 also to prove that the resulting kernel is locally integrable we require more precise estimates for small time in sect 1 after some preliminaries we state our main theorem we then summarize the results of s1 and 2 we solve the corresponding elliptic problem on c 1 in sect 2 in sect 3 we study the operator which corrects the boundary values of the solution we obtained in sect 2 we solve the o neumann problem in sect 4 most of the work for this paper was done while i was a member of the institute for advanced study i wish to thank the institute for its hospitality

the range of the tangential cauchy riemann operator over a small ball in this paper we study the closed range property of tangential cauchy riemann equations of an abstract cr structure over a small ball let d be the set x e r 1 1x1 4 kuranishi 141 and akahori 11 have studied the l theory

cohomology of q convex spaces in top degrees it is shown that every strongly q complete subvariety of a complex analytic space has a fundamental system of strongly q complete neighborhoods as a consequence we find a simple proof of ohsawa s result that every non compact irreducible n dimensional analytic space is strongly n complete finally it is shown that l cohomology theory readily implies ohsawa s hodge decomposition and lefschetz isomorphism theorems for absolutely q convex manifolds

the lifetime of conditioned brownian motion in planar domains of infinite area in this paper it is shown that all expected lifetimes ofh processes ind are finite if and only if the area ofd is finite ifd x y o x y o x x where o x o are two lipschitz functions we show that if is a bounded convex region in the plane there is anh process in with expected lifetime at leastc area we also give an example of a planar domaind of infinite area such that the expected lifetime of eachh process ind is finite

hilbert integrals singular integrals and radon transforms i on introduit deux nouvelles classes d operateurs on etudie leur relation et on montre comment on peut les appliquer a l etude des problemes aux valeurs limites

multiplicity of limit cycles and analytic m solutions for planar differential systems abstract this work deals with limit cycles of real planar analytic vector fields it is well known that given any limit cycle of an analytic vector field it always exists a real analytic function f 0 x y defined in a neighborhood of and such that is contained in its zero level set in this work we introduce the notion of f 0 x y being an m solution which is a merely analytic concept our main result is that a limit cycle is of multiplicity m if and only if f 0 x y is an m solution of the vector field we apply it to study in some examples the stability and the bifurcation of periodic orbits from some non hyperbolic limit cycles

radon transforms and finite type conditions the purpose of this paper is to study averaging operators of radon transform type we shall formulate suitable finite type conditions and prove lp sobolev and lplq estimates the results will be essentially sharp for operators associated with families of curves in the plane let xc and t be smooth manifolds dimx3 nl dim2 nr and let m be a submanifold in xc x t with conormal bundle n m we denote by t the codimension of m we shall always assume that the projections lry m x and 7rt m 42 are surjective with rank dirx nl rank dr9 nr this in particular implies that n m c t x o x t o where 0 refers to the zero sections in t x and t 2 respectively this is the usual assumption for the canonical relation associated with lagrangian distributions arising as kernels of fourier integral operators the assumptions on dirx d7rt imply that for fixed x e x y e 2 the sets amx y e t x y e mi m x ye2x x y em i

invariant algebraic curves of large degree for quadratic system in this paper we present for the first time examples of algebraic limit cycles and saddle loops of degree greater than 4 for planar quadratic systems in particular we give examples of algebraic limit cycles of degree 5 and 6 and algebraic saddle loops of degree 3 and 5 surrounding a strong focus we also give an example of an invariant algebraic curve of degree 12 for which the quadratic system has no darboux integrating factors or first integrals

strongly pseudoconvex cr structures over small balls part iii an embedding theorem the purpose of the present paper is to prove that any germ of strongly pseudoconvex cr structure on a manifold of real dimension 9 is realized by an embedding in the complex euclidean space of real codimension 1 it is well known that the above is true as formal power series hence we find an embedding such that the induced structure is very close to the given one on a small neighborhood of the reference point we correct our embedding by newton s method we then apply the moser nash procedure cf 4 to prove that the sequence of embeddings obtained by the successive correction converges on a sufficiently small neighborhood of the reference point to an embedding which realizes the given structure to explain how the correction is made let d be the differential operator in the complex associated with the given structure when f f f is an approximating embedding defined on u we set ox dfx if we can find gx such that dga on and f g is still an embedding f is completely corrected unfortunately we do not know how to find such a g however the neumann boundary value problem in parts i and ii for the differential operator df which is associated with the induced structure is a tool to handle such question by newton s method we mean that we make the best correction using df instead of d this is done as follows oa is not a differential form in the complex of df hence we replace 01 oj by its projection o on to the complex of df note that 0a may no longer be in the kernel of df and our domain may not be distinguished we first construct a distinguished distance function tf on u and consider a ball u tf r

hilbert integrals singular integrals and radon transforms ii etude des transformations de radon singulieres et des integrales de hilbert appliquees aux problemes aux valeurs limites probleme de neumann

the h path distribution of the lifetime of conditioned brownian motion for non smooth domains consider the h paths of doob and let d be the lifetime of brownian motion killed at d where d is a bounded lipschitz or nta domain in n then log p x h tau d t sim lambda d t as t to infty where d is the first positive eigenvalue of 1 2 on d

on the local solvability and the local integrability of systems of vector fields on etudie des systemes surdetermines de champs vectoriels dans un sous ensemble ouvert de r n 1 de type l j t j j t x x j 1 m ayant des coefficients analytiques on suppose que les champs de vecteurs l j satisfont la condition de frobenius

uniqueness of algebraic limit cycles for quadratic systems abstract we know five different families of algebraic limit cycles in quadratic systems one of degree 2 and four of degree 4 moreover if there are other families of algebraic limit cycles for quadratic systems then their degrees must be larger than 4 it is known that if a quadratic system has an algebraic limit cycle of degree 2 then this is the unique limit cycle of the system the first main goal of this paper is to prove that if a quadratic system has an algebraic limit cycle of degree 4 then this is the unique limit cycle of the system for the yablonskii and filiptsov algebraic limit cycles the problem of proving their uniqueness has been open since 1966 and 1973 respectively the second main goal is to provide sufficient conditions in order that all the limit cycles of a quadratic system to be algebraic

maximal operators and differentiation theorems for sparse sets we study maximal averages associated with singular measures on r our main result is a construction of singular cantor type measures supported on sets of hausdorff dimension 1 e with 0 e 1 e 1 e as a consequence we are able to answer a question of aversa and preiss on density and differentiation theorems for singular measures in one dimension our proof combines probabilistic techniques with the methods developed in multidimensional euclidean harmonic analysis in particular there are strong similarities to bourgain s proof of the circular maximal theorem in two dimensions

weighted bergman projections on the hartogs triangle abstract we prove the l p regularity of the weighted bergman projections on the hartogs triangle where the weights are powers of the distance to the singularity at the boundary the restricted range of p is proved to be sharp by using a two weight inequality on the upper half plane with muckenhoupt weights we can consider a slightly wider class of weights

nonexistence of homotopy formula for 0 1 forms on hypersurfaces in mathbb c 3 soit g une 0 1 forme definie sur une hypersurface strictement convexe dans c 3 et on suppose que b g 0 mais b g est petit on considere la resolution locale approchee de l equation b u g

local existence theorems with estimates for b on weakly pseudo convex cr manifolds in particular we prove the solvability for 0 1 on co for 1 q n 2 when m is a pseudo convex cr hypersurface of finite type in the sense defined in d angelo 11 and c co is the intersection of m with a flat hypersurface m0 see theorem 1 in sect i near a point zo of finite type we also show that there exists a neighborhood basis co of z0 such that 0 1 is solvable in co for any ct satisfying 0 2 see corol lary 1 1 we also obtain the interior regularity of the solutions for 0 1 in the sobolev spaces and show that the range of b is closed in the frechet norm l p q co loc see theorem 2 and corol lary 2 1 in sect 1 since the existence of the solution u of 0 1 was proved on co for any c satisfying 0 2 in co theorems 1 and 2 are semi global results when we assume that mo is only levi flat and with a stein ne ighborhood basis then we prove that for any relatively compact subset co c co there exists a u satisfying 0 2 in co see theorem 3 we note that the local solvability for 0 1 was not known previously even on a small subset in co on general weakly pseudo convex cr manifolds in this paper we shall prove the

transversal conics and the existence of limit cycles this paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems we provide a method to construct poincar e bendixson regions by using transversal conics we present several examples of known systems in the literature showing different features about limit cycles hyperbolicity hopf bifurcation sky blue bifurcation rotated vector fields ldots for which the obtained poincar e bendixson region allows to locate the limit cycles our method gives bounds for the bifurcation values of parametrical families of planar vector fields and intervals of existence of limit cycles

sharp martingale inequalities and applications to riesz transforms on manifolds lie groups and gauss space abstract we prove new sharp l p logarithmic and weak type inequalities for martingales under the assumption of differential subordination the l p estimates are feynman kac type versions of burkholder s celebrated martingale transform inequalities from the martingale l p inequalities we obtain that riesz transforms on manifolds of nonnegative bakry emery ricci curvature have exactly the same l p bounds as those known for riesz transforms in the flat case of r d from the martingale logarithmic and weak type inequalities we obtain similar inequalities for riesz transforms on compact lie groups and spheres combining the estimates for spheres with poincare s limiting argument we deduce the corresponding results for riesz transforms associated with the ornstein uhlenbeck semigroup thus providing some extensions of p a meyer s l p inequalities

algebraic limit cycles of degree 4 for quadratic systems abstract yablonskii differential equations 2 1996 335 and filipstov differential equations 9 1973 983 proved the existence of two different families of algebraic limit cycles of degree 4 in the class of quadratic systems it was an open problem to know if these two algebraic limit cycles where all the algebraic limit cycles of degree 4 for quadratic systems chavarriga a new example of a quartic algebraic limit cycle for quadratic sytems universitat de lleida preprint 1999 found a third family of this kind of algebraic limit cycles here we prove that quadratic systems have exactly four different families of algebraic limit cycles the proof provides new tools based on the index theory for algebraic solutions of polynomial vector fields

strictly pseudoconvex domains in c n the theory which we have developed so far will be expanded in two ways 1 in this chapter we shall use the cauchy fantappie calculus to transform the bochner martinelli kernel into a kernel adapted to strictly pseudoconvex domains in n and prove a number of finiteness and vanishing theorems for these domains 2 in the next chapter we shall pass from n to arbitrary hermitian manifolds and establish the analogue of the bmk formula in that context obtaining along the way the hodge decomposition theorem on compact manifolds both lines of thought will merge in the subsequent chapters and lead to a solution of the levi problem and finiteness theorems for strictly pseudoconvex manifolds as well as to a solution of the overline partial neumann problem to be formulated later

local extension of boundary holomorphic forms on real hypersurfaces in cn abstract let m be a real hypersurface in c n with n 3 and the levi form at p m has q 1 positive eigenvalues 1 q n 2 we solve one sided local closed extension problem near p for 0 r forms for 1 r q

levi foliations in pseudoconvex boundaries and vector fields that commute approximately with boas and straube proved a general sufficient condition for global regularity of the neumann problem in terms of families of vector fields that commute approximately with in this paper we study the existence of these vector fields on a compact subset of the boundary whose interior is foliated by complex manifolds this question turns out to be closely related to properties of interest from the point of view of foliation theory

on an estimate of cranston and mcconnell for elliptic diffusions in uniform domains we show that ifd n n 3 n 3 is a bounded uniform domain then the lifetime of the doobh paths ind for elliptic diffusions in divergence form is finite this result holds for any bounded domaind in the plane

lifetime of conditioned brownian motion in lipschitz domains if d d d 3 is bounded and has lipschitz boundary then the expected lifetime of any brownian h path process in d is finite

strongly pseudoconvex cr structures over small balls part ii a regularity theorem the purpose of the present paper is to prove the regularity theorem for the solution of the generalized neumann boundary value problem for a strongly pseudoconvex cr structure say t over a small ball this will be based on the a priori estimate established in i namely we take an admissible distance function t to a reference point cf 2 12 in i and consider a sufficiently small ball ur t r denote by d the differential operator in the complex associated with 0t and consider the metric given by a levi form set b dt which is a function of u we consider the hilbert space of l2 norm with weight factor b2a 11 11 a and a denoting its norm and inner product respectively let ad be the adjoint of d set qa u v du dv a ad u ad v a for 0t q forms u v the generalized neumann boundary value problem is to solve the following equation for a given 0t q form a and an unknown 0t q form u withuldt 0

estimates for singular convolution operators on the heisenberg group soit h n le groupe de heisenberg l operateur a co c h n donne par af f k s etend a un operateur borne de l p dans l p pour 1 p les operateurs ae co c donnes par aef f ke s etendent aux operateurs bornes sur l p pour 1 p avec des normes independantes de e

singular integrals with rough kernels along real analytic submanifolds in r n l p mapping properties will be established in this paper for singular radon transforms with rough kernels defined by translated of a real analytic submanifold in r n

singular integrals related to the radon transform and boundary value problems abstract two classes of integral operators are introduced that combine features of calderon zygmund singular integrals the hilbert integral and the radon transform examples and applications arising from integral geometry and several complex variables are discussed