endpoint estimates for one dimensional oscillatory integral operators abstract the one dimensional oscillatory integral operator associated to a real analytic phase s is given by t f x e i s x y x y f y d y in their fundamental work phong and stein established sharp l 2 estimates for t the goal of this paper is to extend their results to all endpoints in particular we obtain a complete characterization for the mapping properties for t on l p r more precisely we show that t f p f p holds for some 0 if and only if 1 p 1 p lies in the reduced newton polygon of s

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

fourier integral operators fractal sets and the regular value theorem we prove that if e r2d for d 2 is an ahlfors david regular product set of sufficiently large hausdorff dimension denoted by dimh e and is a sufficiently regular function then the upper minkowski dimension of the set w e l w tl 1 l m does not exceed dimh e m in line with the regular value theorem from the elementary differential geometry our arguments are based on the mapping properties of the underlying fourier integral operators and are intimately connected with the falconer distance conjecture in geometric measure theory we shall see that our results are in general sharp in the sense that if the hausdorff dimension is smaller than a certain threshold then the dimensional inequality fails in a quantifiable way the constructions used to demonstrate this are based on the distribution of lattice points on convex surfaces and have connections with combinatorial geometry

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

an implicit function theorem for banach spaces and some applications we prove a generalized implicit function theorem for banach spaces without the usual assumption that the subspaces involved being complemented then we apply it to the problem of parametrization of fibers of differentiable maps the lie subgroup problem for banach lie groups as well as weil s local rigidity for homomorphisms from finitely generated groups to banach lie groups

propagation of singularities and maximal functions in the plane in this work we mainly generalize bourgain s circular maximal function to include variable coefficient averages our techniques involve a combination of bourgain s basic ideas plus microlocal analysis in particular to see the role of curvature we rely heavily on methods used in studying propogation of singularities for hyperbolic differential equations we also show that forp 2 there is local smoothing inl p for solutions to the wave equation

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

sobolev estimates for fractional and singular radon transforms abstract we prove sobolev inequalities for singular and fractional radon transforms which are defined as in a paper of phong and stein under certain hypothesis on the corresponding lagrangian n c which does not necessarily have to be a canonical graph in the proof we use oscillatory integrals the cotlar stein almost orthogonality theorem a sort of littlewood paley decomposition for a certain operator some basic facts about fourier integral operators and pseudodifferential operators the main ideas come from papers by phong and stein acta math 157 1986 99 157 and sogge and stein j analyse math 54 1990 165 188

l2 estimates for singular oscillatory integral operators in this note we study singular oscillatory integrals with linear phase function over hypersurfaces which may oscillate and prove estimates of l2l2 type for the operator as well as for the corresp

on the image of a fuzzy subgroup abstract it is shown that the image of a fuzzy subgroup can be any subset of the unit interval containing its least upper bound

analysis of second order differential operators on heisenberg groups i given a matrixs sp n one finds a second order bi invariant differential operator s on the one dimensional extensiong s of the heisenberg grouph n induced bys we construct solutions to certain cauchy problems for s and fundamental solutions for these operators this analysis is connected with the discussion of the oscillator semigroup by r howe

maximal operators associated to families of flat curves in the plane bourgain showed that if c has non vanishing curvature the inequality 1 1 holds for p 2 see b in this paper we shall consider a situation when the curvature is allowed to vanish of finite order on a finite set of isolated points we shall need the following definition definition1 1 let c i r where i is a compact interval in r and c is smooth we say that c is finite type if c x c x0 does not vanish of infinite order for any x0 i and any unit vector we shall also need a more precise definition which would specify the order of vanishing at each point let a0 denote a point in the compact interval i we can always find a smooth function such that in a small neighborhood of a0 c s s s where s i definition1 2 let c be defined as before let c s s s in a small neighborhood of a0 we say that c is finite type m at a0 if a0 0 for 1 k m and a0 6 0 our main result is the following theorem 1 1 let c be a finite type curve which is finite type m at a0 let mtf x

boundedness of singular radon transforms on l p spaces under a finite type condition in this paper a new method is presented for proving boundedness of singular radon transforms on l p for 1 p two closely related results generalizing the corresponding theorem of christ nagel stein and wainger are proven 1 introduction in this paper a new method is presented for proving boundedness of singular radon transforms on l p for 1 p two closely related results generalizing the corresponding theorem of christ nagel stein and wainger in 3 are proved singular radon transforms are a type of singular integral operator a prototype for which is the hilbert transform along curves let 1 1 r n be a curve with 0 0 and let f be a schwarz function on r n 1

singular radon transforms and oscillatory integrals on considere une reduction des transformees de radon singulieres a certaines integrales oscillatoires

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

analysis of the level subgroups of a fuzzy group abstract since the level subgroups of a fuzzy subgroup of a group fail to classify certain subgroups of the group which crop up naturally an open level subgroup is defined these are further subdivided into tangible and hidden subgroups a tangible subgroup can be realized as a level subgroup defined by a positive image point in the unit interval whereas a hidden subgroup cannot be so expressed a necessary and sufficient condition to identify a hidden subgroup through a particular type of a limiting point of the image set is established the usefulness of the open level subgroups of a fuzzy group is demonstrated by factorizing an open level subgroup of the product of two fuzzy subgroups of a group as defined by liu by the corresponding open level subgroups of the individual fuzzy subgroups it is shown that every hidden subgroup of the product fuzzy subgroup is the product of a hidden subgroup and an open level subgroup of the individual fuzzy subgroups lastly the adjustible and the non adjustible subgroups which appear like ghost images all along the chain of level subgroups are detected and analyzed

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

continuous fuzzy groups a continuous fuzzy group is a fuzzy group with continuous membership function on a topological group we show that if a nontrivial continuous fuzzy group exists then the underlying topological group must be totally disconnected it is also shown that all locally compact totally disconnected groups are carriers for nontrivial fuzzy groups more detailed results are obtained for compact groups

schur complement systems in the mixed hybrid finite element approximation of the potential fluid flow problem the mixed hybrid finite element discretization of darcy s law and continuity equation describing the potential fluid flow problem in porous media leads to a symmetric indefinite linear system for the pressure and velocity vector components as a method of solution the reduction to three schur complement systems based on successive block elimination is considered the first and second schur complement matrices are formed eliminating the velocity and pressure variables respectively and the third schur complement matrix is obtained by elimination of a part of lagrange multipliers that come from the hybridization of a mixed method the structural properties of these consecutive schur complement matrices in terms of the discretization parameters are studied in detail based on these results the computational complexity of a direct solution method is estimated and compared to the computational cost of the iterative conjugate gradient method applied to schur complement systems it is shown that due to special block structure the spectral properties of successive schur complement matrices do not deteriorate and the approach based on the block elimination and subsequent iterative solution is well justified theoretical results are illustrated by numerical experiments

uniform estimates for oscillatory integral operators abstract in this paper we prove the uniform l p boundedness of oscillatory singular integral operators with smooth phase functions which satisfy a certain finite type condition these phase functions include the class of real analytic functions the uniform boundedness does not hold if the finite type condition is removed we also obtained estimates for related operators including an optimal l 2 estimate for operators with singular kernels whose phase functions have non singular hessians

mixed hybrid finite element approximation of the potential fluid flow problem abstract in the paper a mixed hybrid approximation of the potential fluid flow problem based on prismatic discretization of the domain is presented trilateral prismatic elements with vertical faces and nonparallel bases suitable for the modelling of real geological circumstances are considered the set of linearly independent vector basis functions is defined and existence and uniqueness of the approximate solution from the resulting symmetric indefinite system are examined possible approaches to the solution of the discretized system are discussed

weak 1 1 bounds for oscillatory singular integrals pour un polynome p r n r n r et un noyau de calderon zygmund k l operateur t f x pv exp ip x y k x y f y dy est de type faible sur l 1 avec une borne dependant seulement de k cz et le degre de p

schur complement reduction in the mixed hybrid approximation of darcy s law rounding error analysis abstract mixed hybrid finite element approximation of the potential fluid flow problem leads to the solution of a large symmetric indefinite system for the velocity and potential head vector components such discretization gives rise to a very accurate approximation of the continuity equation in every element and for low order discretizations the structural properties of the discrete matrix blocks allow cheap block elimination of the positive definite diagonal block and subsequent reduction to the schur complement system for the pressure and lagrangian vector components this system is then frequently solved by the iterative conjugate gradient type method whereas this approach is well known considerably less attention has been paid to the numerical stability aspects of such transformation it was shown in 5 that block lu factorization can be unstable even when the system matrix is symmetric positive definite in this paper we examine this type of conditional stability for a particular application in the underground water flow modelling we show that the actual error of the computed approximate solution depends not only on the user defined tolerance in the conjugate gradient process but also on the spectral properties of the corresponding matrix blocks eliminated during the schur complement reduction it is often observed that although the backward error of the approximate solution in the iterative part is reduced to the level of machine accuracy the total residual norm after the back substitution process remains at certain accuracy level we give a bound for this maximal attainable accuracy and illustrate our theoretical results on a model example

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

zeta functions of discrete self similar sets abstract in this paper we study a class of countable and discrete subsets of a euclidean space that are self similar with respect to a finite set of affine similarities any such set can be interpreted as having a fractal structure we introduce a zeta function for these sets and derive basic analytic properties of this fractal zeta function motivating examples that come from combinatorial geometry and arithmetic are given particular attention

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

applications of fourier analysis in homogenization of dirichlet problem i pointwise estimates in this paper we prove convergence results for homogenization problem for solutions of partial differential system with rapidly oscillating dirichlet data our method is based on analysis of oscill

nonlocal inversion formulas for the x ray transform etude de la reconstruction d une fonction f x sur r n a partir de ses integrales sur des plans k dimensionnels cas de k 1

generalisations of some results of das abstract in this paper we present generalisations of some results of p s das 1981 to the case of an arbitrary fuzzy group

on falconer s distance set problem in the plane if e subset mathbb r 2 is a compact set of hausdorff dimension greater than 5 4 we prove that there is a point x in e so that the set of distances x y y in e has positive lebesgue measure

hilbert transforms and maximal functions along variable flat curves we study certain hilbert transforms and maximal functions along variable flat curves in the plane we obtain their l 2 r 2 boundedness by considering the oscillatory singular integrals which arise from an application of a partial fourier transform

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

on continuum incidence problems related to harmonic analysis abstract we consider averaging operators over curves and surfaces satisfying the rotational curvature condition of phong and stein using combinatorial arguments restricted weak type versions of the standard estimates are obtained under weaker assumptions on the regularity of these surfaces we also consider maximal functions over circles in the plane and obtain a simpler proof of wolff s endpoint result

on the image of an l fuzzy group in this short communication we show that if any subset of a lattice l containing its lowest upper bound is an image of an l fuzzy group then l must be a chain

l2 boundedness of hilbert transforms along variable curves abstract for the l 2 boundedness of the hilbert transforms along variable curves h f x 1 x 2 p v f x 1 t x 2 x 1 t d t t where c 2 r 1 odd or even 0 0 0 convex on 0 if 1 a nagel j vance s wainger and d weinberg got a necessary and sufficient condition on if is a polynomial j m bennett got a sufficient condition on in this paper we shall first give a counter example to show that under the condition of nagel vance wainger weinberg on the l 2 boundedness of h may fail even if c r 1 on the other hand we improve bennett s result by relaxing the condition on and simplifying the proof

unit distance problems we study some discrete and continuous variants of the following problem of erd h os given a finite subset p of bbb r 2 or bbb r 3 what is the maximum number of pairs p 1 p 2 with p 1 p 2 in p and p 1 p 2 1

weak type 1 1 bounds for rough operators ii in this paper we consider four operators arising in harmonic analysis in r n 1 they share one feature each is known to be bounded on lp for all 1 p o but for each it has been an open question whether a weak type 1 1 bound also holds in two cases we show that the weak 1 1 bound does hold and for the others we establish weaker estimates which are still sharper than the lp boundedness although the four operators are quite distinct they all involve convolution kernels which are related to the singular integrals of calderon and zygmund but which lack the smoothness required for the usual analysis in each case this defect is mitigated by the presence of some form of curvature consider the bochner riesz means

harmonic analysis on nilpotent groups and singular integrals i oscillatory integrals abstract this paper is devoted to the study of the operator given by where k is a standard calderon zygmund kernel and p is a real polynomial on r n r n we show t is bounded on lp to itself when 1

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