harmonic analysis on nilpotent groups and singular integrals iii fractional integration along manifolds in this paper we continue the analysis of integral operators on nilpotent lie groups with kernels supported on submanifolds and or containing an oscillatory factor of polynomial type in our previous articles 13 143 we have essentially considered singular integral operators i e those defined by kernels having the critical degree of homogeneity here we focus our attention on kernels having a higher degree of homogeneity so that locally they have a finite mass but do not have enough decay at infinity more precisely if q is the homogeneous dimension of the group g which in this article we always assume endowed with automorphic dilations we consider kernels k having the same homogeneity as 1x1 q l a where 1x1 denotes a homogeneous gauge and re a 0 these kernels are assumed to be supported on analytic homogeneous submanifolds v we call them fractional integration kernels along v we discuss three interrelated problems concerning fractional integration kernels

renaming in an asynchronous environment this paper is concerned with the solvability of the problem of processor renaming in unreliable completely asynchronous distributed systems fischer et al prove in 8 that nontrivial consensus cannot be attained in such systems even when only a single benign processor failure is possible in contrast this paper shows that problems of processor renaming can be solved even in the presence of up to t n 2 faulty processors contradicting the widely held belief that no nontrivial problem can be solved in such a system the problems deal with renaming processors so as to reduce the size of the initial name space when only uniqueness of the new names is required we present a lower bound of n 1 on the size of the new name space and a renaming algorithm that establishes an upper bound on n t if the new names are required also to preserve the original order a tight bound of 2 n t 1 1 is obtained

fast wave computation via fourier integral operators this paper presents a numerical method for time upscaling wave equations i e performing time steps not limited by the courant friedrichs lewy cfl condition the proposed method leverages recent work on fast algorithms for pseudodierential and fourier integral operators fio this algorithmic approach is not asymptotic it is shown how to construct an exact fio propagator by 1 solving hamilton jacobi equations for the phases and 2 sampling rows and columns of low rank matrices at random for the amplitudes the setting of interest is that of scalar waves in two dimensional smooth periodic media of class c 1 over the torus where the bandlimit n of the waves goes to innity in this setting it is demonstrated that the algorithmic complexity for solving the wave equation to xed time t 1 can be as low as o n 2 logn with controlled accuracy numerical experiments show that the time complexity can be lower than that of a spectral method in certain situations of physical interest

harmonic analysis on nilpotent groups and singular integrals ii singular kernels supported on submanifolds on etudie des operateurs de convolution singuliers sur des groupes de lie nilpotents generaux et plus specifiquement des operateurs dont les noyaux ont pour supports des varietes de faible dimension et qui peuvent aussi contenir des facteurs polynomiaux exponentiels oscillatoires

damped oscillatory integrals and boundedness of maximal operators associated to mixed homogeneous hypersurfaces uller abstract we study the boundedness problem for maximal operators in 3 dimen sional euclidean space associated to hypersurfaces given as the graph of c f where f is a mixed homogeneous function which is smooth away from the origin and c is a constant assuming that the gaussian curvature of this surface nowhere vanishes of infinite order we prove that the associated maximal operator is bounded on l p r 3 whenever p h 2 here h denotes a height of the function f defined in terms of its maximum order of vanishing and the weights of homogeneity this result generalizes a corresponding theorem on mixed homogeneous polynomial functions by a iosevich and e sawyer in particular it shows that a certain ellipticity conditon used by these authors is not necessary if c 6 0 our result is sharp

estimates for maximal functions associated with hypersurfaces in ℝ3 and related problems of harmonic analysis we study the boundedness problem for maximal operators mathcal m associated with averages along smooth hypersurfaces s of finite type in 3 dimensional euclidean space for p 2 we prove that if no affine tangent plane to s passes through the origin and s is analytic then the associated maximal operator is bounded on l p left mathbb r 3 right if and only if p h s where h s denotes the so called height of the surface s defined in terms of certain newton diagrams for non analytic s we obtain the same statement with the exception of the exponent p h s our notion of height h s is closely related to a n varchenko s notion of height h for functions such that s can be locally represented as the graph of after a rotation of coordinates

the triangle averaging operator abstract we examine the averaging operator t corresponding to the manifold in r 2 d of pairs of points u v satisfying u v u v 1 so that 0 u v is the set of vertices of an equilateral triangle we establish l p l q l r boundedness for t for 1 p 1 q 1 r in the convex hull of the set of points 0 0 0 1 0 1 0 1 1 1 p d 1 p d 2 p d where p d 5 d 3 d 2

damping oscillatory integrals let s be a smooth compact convex hypersurface of finite type in r n l with surface measure and let u be a nonnegative real valued function on s we shall study the behaviour of the fourier transform of the measure u in particular we are interested in obtaining inequalities for the decay of u 4 as 4 goes to inifinty this problem has been considered by various authors including hlawka 5 herz 4 littman 10 randol 12 and svensson 15 for an overview of such results and their significance in harmonic analysis we refer our readers to stein 14 by taking a partition of unity and using suitable coordinate systems the study of the fourier transform integral may be reduced to that of an oscillatory integral i of the form 1 2 v x exp i2 a x dx v2 r r

oscillatory integrals and fourier transforms of surface carried measures we suppose that s is a smooth hypersurface in rn 1 with gaussian curvature re and surface measure ds it is a compactly supported cut off function and we let pa be the surface measure with dsa u ka ds in this paper we consider the case where s is the graph of a suitably convex function homogeneous of degree d and estimate the fourier transform sa we also show that if s is convex with no tangent lines of infinite order then decays as i n 2 provided a n 3 2 the techniques involved are the estimation of oscillatory integrals we give applications involving maximal functions 1 introduction the purpose of this paper is to obtain estimates for the decay at infinity of certain oscillatory integrals related to the fourier transform of surface carried measures let s denote a smooth hypersurface in rn 1 with gaussian curvature n and element of surface measure ds induced by the lebesgue measure of rn 1 we fix a smooth function w with compact support in rn 1 and a nonnegative number a and consider the finite borel measure p a with dp a k awds which is carried by s we seek conditions on s and a that guarantee that the fourier transform p a of p a satisfies the estimate

hardy space estimates for the wave equation on compact lie groups abstract let l be the laplacian in this paper we prove that on a compact lie group g of dimension n the multiplier operator e i s l 1 l 2 s 0 1 extends to a bounded operator on the hardy space h p g 0 p if and only if 1 p 1 2 n 1 the result is an analogue of a well known theorem in euclidean space

topology recognition and leader election in colored networks topology recognition and leader election are fundamental tasks in distributed computing in networks the first of them requires each node to find a labeled isomorphic copy of the network while the result of the second one consists in a single node adopting the label 1 leader with all other nodes adopting the label 0 and learning a path to the leader we consider both these problems in networks whose nodes are equipped with not necessarily distinct labels called colors and ports at each node of degree d are arbitrarily numbered 0 1 d 1 colored networks are generalizations both of labeled networks in which nodes have distinct labels and of anonymous networks in which nodes do not have labels all nodes have the same color in colored networks topology recognition and leader election are not always feasible hence we study two more general problems consider a colored network and an input i given to its nodes the aim of the problem top for this colored network and for i is to solve topology recognition in this network if this is possible under input i and to have all nodes answer unsolvable otherwise likewise the aim of the problem le is to solve leader election in this network if this is possible under input i and to have all nodes answer unsolvable otherwise we show that nodes of a network can solve problems top and le if they are given as input i an upper bound k on the number of nodes of a given color called the size of this color on the other hand we show that if the nodes are given an input that does not bound the size of any color then the answer to top and le must be unsolvable even for the class of rings under the assumption that nodes are given an upper bound k on the size of a given color we study the time of solving problems top and le in the local model in which during each round each node can exchange arbitrary messages with all its neighbors and perform arbitrary local computations we give an algorithm to solve each of these problems in arbitrary networks in time o k d d log n d where d is the diameter of the network and n is its size we also show that this time is optimal by exhibiting classes of networks in which every algorithm solving problems top or le must use time k d d log n d

weighted strichartz estimates and global existence for semilinear wave equations in this paper we prove a sharp global existence theorem in all dimensions for nonlinear wave equations with power type nonlinearities the proof is based on a weighted strichartz estimate involving powers of the lorentz distance

resolution of singularities in two dimensions and the stability of integrals abstract for a small disk d centered at the origin in r 2 a smooth real valued function s x y on d and a positive epsilon we consider the measure of the points in d where s x y as well as oscillatory integral analogues specifically we consider the effect of perturbing s x y on these quantities besides being of intrinsic interest these questions are important in the analysis of fourier transforms of surface supported measures complex and higher dimensional analogues of these questions are also connected to various issues in algebraic and complex geometry for real analytic s x y this question has been investigated for example by karpushkin using versal deformation theory and by phong stein sturm who developed a method often referred to as the method of algebraic estimates in this paper we show how the use of resolution of singularities algorithms in two dimensions along with some one dimensional van der corput type lemmas provides another method for dealing with such questions as a result we prove new estimates and theorems for these and related quantities furthermore since these algorithms apply to all smooth functions the theorems will hold for all smooth functions as opposed to the earlier real analytic results

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

uniqueness in cauchy problems for hyperbolic differential operators in this paper we prove a unique continuation theorem for second order strictly hyperbolic differential operators results also hold for higher order operators if the hyperbolic cones are strictly convex these results are proved via certain carleman inequalities unlike 6 the parametrices involved only have real phase functions but they also have gaussian factors we estimate the parametrices and associated remainders using sharp l p estimates for fourier integral operators which are due to brenner 1 and seeger stein and the author 5

electing a leader in a synchronous ring the problem of electing a leader in a synchronous ring of n processors is considered both positive and negative results are obtained on the one hand if processor ids are chosen from some countable set then there is an algorithm that uses only o n messages in the worst case on the other hand any algorithm that is restricted to use only comparisons of ids requires o n log n messages in the worst case alternatively if the number of rounds is required to be bounded by some t in the worst case and ids are chosen from any set having at least n t elements for a certain very fast growing function then any algorithm requires o n log n messages in the worst case

wave atoms and time upscaling of wave equations we present a new geometric strategy for the numerical solution of hyperbolic wave equations in smoothly varying two dimensional time independent periodic media the method consists in representing the time dependent green s function in wave atoms a tight frame of multiscale directional wave packets obeying a precise parabolic balance between oscillations and support size namely wavelength diameter 2 wave atoms offer a uniquely structured representation of the green s function in the sense that the resulting matrix is universally sparse over the class of c coefficients even for large times the matrix has a natural low rank block structure after separation of the spatial indices the parabolic scaling is essential for these properties to hold as a result it becomes realistic to accurately build the full matrix exponential in the wave atom frame using repeated squaring up to some time typically of the form delta t sim sqrt delta x which is bigger than the standard cfl timestep once the expensive precomputation of the green s function has been carried out it can be used to perform unusually large upscaled cheap time steps the algorithm is relatively simple in that it does not require an underlying geometric optics solver we prove accuracy and complexity results based on a priori estimates of sparsity and separation ranks on a n by n grid the expensive precomputation takes somewhere between o n 3log n and o n 4log n steps depending on the separability of the acoustic medium the complexity of upscaled timestepping however beats the o n 3log n bottleneck of pseudospectral methods on an n by n grid for a wide range of physically relevant situations in particular we show that a naive version of the wave atom algorithm provably runs in o n 2 operations for arbitrarily small but for the final algorithm we had to slightly increase the exponent in order to reduce the large constant as a result we get estimates between o n 2 5 log n and o n 3 log n for upscaled timestepping we also show several numerical examples in practice the current wave atom solver becomes competitive over a pseudospectral method in regimes where the wave equation should be solved hundreds of times with different initial conditions as in reflection seismology in academic examples of accurate propagation of bandlimited wavefronts if the precomputation step is factored out then the wave atom solver is indeed faster than a pseudospectral method by a factor of about 3 5 at n 512 and a factor 10 20 at n 1024 for the same accuracy very similar gains are obtained in comparison versus a finite difference method

quantum stochastics the passage from a relativistic to a non relativistic path integral feynman s path integral for the one dimensional dirac particle uses paths for which delta x approximately delta t for non relativistic path integrals typical paths satisfy delta x 2 approximately delta t as in brownian motion the authors demonstrate the consistency of these two stochastic schemes and show how the relativistic formalism contains within itself the scale of the transition regime namely delta x approximately h cross mc the compton wavelength

decay of the fourier transform of surfaces with vanishing curvature we prove lp bounds on the fourier transform of measures supported on two dimensional surfaces our method allows to consider surfaces whose gauss curvature vanishes on a one dimensional submanifold under a certain non degeneracy condition we prove that hat mu in l 4 beta 0 and we give a logarithmically divergent bound on the l4 norm we use this latter bound to estimate almost singular integrals involving the dispersion relation e p sum 1 3 1 cos p j of the discrete laplace operator on the cubic lattice we briefly explain our motivation for this bound originating in the theory of random schrodinger operators

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

a local to global boundedness argument and fourier integral operators abstract we give a criterion for the global boundedness of integral operators which are known to be locally bounded as an application we discuss the global l p boundedness for a class of fourier integral operators while the local l p boundedness of fourier integral operators is known from the work of seeger sogge and stein 26 not so many results are available for the global boundedness on l p r n we give several natural sufficient conditions for them

maximal functions over hypersurfaces with flat points abstract for n 4 let s be a flat hypersurface in r n and let d d where c 0 r n and is the surface area measure on s then the maximal functions mf associated to s and by mf x sup t 0 s f x t d f s r n are bounded on certain orlicz spaces l r n

estimates for rough fourier integral and pseudodifferential operators and applications to the boundedness of multilinear operators we study the boundedness of rough fourier integral and pseudodifferential operators defined by general rough hormander class amplitudes on banach and quasi banach lplp spaces thereafter we apply the aforementioned boundedness in order to improve on some of the existing boundedness results for hormander class bilinear pseudodifferential operators and certain classes of bilinear as well as multilinear fourier integral operators for these classes of amplitudes the boundedness of the aforementioned fourier integral operators turn out to be sharp furthermore we also obtain results for rough multilinear operators

optimal local well posedness theory for the kinetic wave equation abstract we prove local existence and uniqueness results for the space homogeneous 4 wave kinetic equation in wave turbulence theory we consider collision operators defined by radial but general dispersion relations satisfying suitable bounds and we prove two local well posedness theorems in nearly critical weighted spaces

maximal averages over hypersurfaces and the newton polyhedron abstract using some resolution of singularities and oscillatory integral methods in conjunction with appropriate damping and interpolation techniques l p boundedness theorems for p 2 are obtained for maximal averages over hypersurfaces in r n for n 2 these estimates are sharp in various situations including the convex hypersurfaces of finite line type considered by several authors as a corollary we also give a generalization of the result of sogge and stein that for some finite p the maximal operator corresponding to a hypersurface whose gaussian curvature does not vanish to infinite order is bounded on l p for some finite p analogous estimates are proven for fourier transforms of surface measures and these are sharp for the same hypersurfaces as the maximal operators

wave equation for sums of squares on compact lie groups in this paper we investigate the well posedness of the cauchy problem for the wave equation for sums of squares of vector fields on compact lie groups we obtain the loss of regularity for solutions to the cauchy problem in local sobolev spaces depending on the order to which the hormander condition is satisfied but no loss in globally defined spaces we also establish gevrey well posedness for equations with irregular coefficients and or multiple characteristics as in the sobolev spaces if formulated in local coordinates we observe well posedness with the loss of local gevrey order depending on the order to which the hormander condition is satisfied

wave front sets local smoothing and bourgain s circular maximal theorem the purpose of this paper is to improve certain known regularity results for the wave equation and to give a simple proof of bourgain s circular maximal theorem 1 we use easy wave front analysis along with techniques previously used in proofs of the carleson sj6lin theorem see 3 5 7 and in the proof of sharp regularity properties of fourier integral operators 13 the circular means operators are defined by

long time existence for small amplitude semilinear wave equations we prove that a certain class of semilinear wave equations has global solutions if the initial data is small these existence results rely on mixed norm angular radial space time estimates more specifically we consider power nonlinearities u f p u where f p u u p and 2 t denotes the d alembertian on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i in 1979 john investigated this equation for small initial data in dimension n 3 he proved that if p 1 2 then there is a global solution whereas for p n dimensions and the critical power p c above which global solutions exist should be the positive root of n 1 p 2 c n 1 p c 2 0 sideris proved that for all n the solutions can blow up if p p c recently zhou proved that for n 4 one has global solutions if p p c 2 here we prove that for n 8 one has global solutions if p p c furthermore we proved that for all n one has global solutions if p p c and the initial data are spherically symmetric in the radial case we use l p estimates derived from the explicit form of the fundamental solution using inequalities from classical analysis in the nonradial case we write the fundamental solution in polar coordinates and use fourier integral and special function estimates to handle the angular part

estimates for maximal functions associated to hypersurfaces in ℝ with height 2 part i deutsche forschungsgemeinschaft dfg grant mu 761 11 1 national science foundation grant no dms 1440140

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

differential structure on the dual of a compact lie group abstract in this paper we define difference operators and homogeneous sobolev type spaces on the dual of a compact lie group as an application and to show that this defines a relevant differential structure we state and prove multiplier theorems of hormander mihlin and marcinkiewicz types together with the sharpness in the sobolev exponent for the result of hormander type

a seeger sogge stein theorem for bilinear fourier integral operators abstract we establish the regularity of bilinear fourier integral operators with bilinear amplitudes in s 1 0 m n 2 and non degenerate phase functions from l p l q l r under the assumptions that m n 1 1 p 1 2 1 q 1 2 and 1 p 1 q 1 r this is a bilinear version of the classical theorem of seeger sogge stein concerning the l p boundedness of linear fourier integral operators moreover our result goes beyond the aforementioned theorem in that it also includes the case of quasi banach target spaces

weighted l p estimates for rough bi parameter fourier integral operators abstract fourier integral operators play an important role in applications to partial differential equations the main purpose of this paper is to investigate the weighted l p boundedness of the bi parameter fourier integral operators or fios for short of the following form t f x r n 1 r n 2 e i x a x f d d where for x x 1 x 2 r n 1 r n 2 r n 1 0 and r n 2 0 the amplitude function a x l b s m and the phase function is of the form x 1 x 1 2 x 2 when 1 l 2 r n 1 r n 1 0 2 l 2 r n 2 r n 2 0 satisfy certain differential conditions we will establish the weighted l w p boundedness of t under the assumption that the weight w is in the class of the product muckenhoupt a p r n 1 r n 2 weights

convex hypersurfaces and fourier transforms hilbert transforms and maximal functions along curves and surfaces spectral synthesis problems and the study of certain operators related to hyperbolic partial differential and pseudodifferential operators the problem of estimating such fourier transforms has a long history see for example hlawka 3 herz 2 littman 4 randol 9 10 svensson 15 sogge and stein 12 13 stein 14 marshall 5 etc the results of these investigations suggest a strong relation between the local curvature properties of the surface s and the decay of the function a we are mainly interested in how the geometry of s near a fixed point xo affects the decay of a6 for by using a partition of unity on s it is clearly enough to study integrals of the type

regularity of fourier integral operators with caustics the caustics of fourier integral operators are defined as caustics of the corresponding schwartz kernels lagrangian distributions on x y the caustic set c of the canonical relation is characterized as the set of points where the rank of the projection c x x y is smaller than its maximal value dim x x y 1 we derive the l p y l q x estimates on fourier integral operators with caustics of corank 1 such as caustics of type a m 1 m n for the values of p and q outside of a certain neighborhood of the line of duality q p the l p l q estimates are proved to be caustics insensitive we apply our results to the analysis of the blow up of the estimates on the half wave operator just before the geodesic flow forms caustics

on maximal functions associated to hypersurfaces and the cauchy problem for strictly hyperbolic operators in this paper we prove a maximal fourier integral theorem for the types of operators which arise in the study of maximal functions associated to averaging over hypersurfaces and also the cauchy problem for hyperbolic operators we apply the fourier integral theorem to generalize stein s spheri cal maximal theorem see 8 and also to prove a sharp theorem for the almost everywhere convergence to ip initial data of solutions to the cauchy problem for second order strictly hyperbolic operators our results improve those of greenleaf 3 and ruiz 6 we also can prove almost everywhere convergence to l2 initial data for operators of order m 3

boundedness of bi parameter pseudo differential operators on bi parameter α modulation spaces abstract though the one parameter modulation spaces have been extensively studied since it was first introduced by grobner not much is known about the bi parameter modulation spaces this paper is to introduce the bi parameter modulation spaces and establish the boundedness of bi parameter pseudo differential operators on the bi parameter modulation spaces by the special relationship between modulation spaces and modulation spaces we also obtain the boundedness of bi parameter pseudo differential operators on the bi parameter modulation spaces

maximal operators and hilbert transforms along variable non flat homogeneous curves we prove that the maximal operator associated with variable homogeneous planar curves t ut t r 1 positive is bounded on lp r2 for each p 1 under the assumption that u r2 r is a lipschitz function furthermore we prove that the hilbert transform associated with t ut t r 1 positive is bounded on lp r2 for each p 1 under the assumption that u r2 r is a measurable function and is constant in the second variable our proofs rely on stationary phase methods tt arguments local smoothing estimates and a pointwise estimate for taking averages along curves

the weak type l1 convergence of eigenfunction expansions for pseudodifferential operators soit m une variete compacte connexe lisse sans bord de dimension n 2 soit p d un operateur pseudodifferentiel elliptique d ordre 1 sur m qui est auto adjoint par rapport a une densite positive lisse dx on demontre la convergence faible de type l 1 pour les developpements en fonctions propres

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

amplitudes and the universal influence function following ideas of hemion a method for deriving the complex amplitudes of quantum mechanics from physically motivated principles is presented first a generalized action and a universal influence function is introduced it is postulated that the probability of a measurement outcome is given in terms of the sum of influences between alternatives that result in that outcome

fourier integral operators on noncompact symmetric spaces of real rank one let x g k be a noncompact symmetric space of real rank one the purpose of this paper is to investigate lp boundedness properties of a certain class of radial fourier integral operators on the space x we will prove that if u is the solution at some fixed time of the natural wave equation on x with initial data f and g and 1 p then u lp x cp f lpbp x 1 g lpbp 1 x we will obtain both the precise behavior of the norm cp and the sharp regularity assumptions on the functions f and g i e the exponent bp that make this inequality possible in the second part of the paper we deal with the analog of e m stein s maximal spherical averages and prove exponential decay estimates of a highly non euclidean nature on the lp norm of supt t 1 f d z where d is a normalized spherical measure

near peak scattering and the corrected kirchhoff approximation for a convex obstacle abstract the uniform asympotic behavior of the scattering amplitude near the forward peak in the case of classical scattering of waves by a convex obstacle is derived a microlocal model is obtained for the scattering operator this is achieved by use of a parametrix for diffractive boundary problems and by a new study of a class of fourier integral operators those with folding canonical relations a crucial ingredient consists of putting a fourier integral operator with folding canonical relation into a normal form the analysis also gives the asymptotic behavior of the normal derivative of the scattered wave on a neighbourhood of the shadow boundary thus providing a corrected version of the kirchhoff approximation

maximal averages over surfaces letmf x supt 0 f t d x denote the maximal operator associated with surface measured on a smooth surfaces we prove that ifsis convex and has finite order contact with its tangent lines then m is bounded onlp rn p 2 if and only ifd x h 1 l1 loc s for all tangent planeshnot passing through the origin letm f x supt 0 f t d x be the maximal operator associated with a nonisotropic dilation tof surface measured we prove that m often behaves far better than m due to a rotational curvature in the time parametert

fast computation of fourier integral operators we introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation general hyperbolic equations and curvilinear tomography the problem is to numerically evaluate a so called fourier integral operator fio of the form int e 2 pi i phi x xi a x xi hat f xi d xi at points given on a cartesian grid here xi is a frequency variable hat f xi is the fourier transform of the input f a x xi is an amplitude and phi x xi is a phase function which is typically as large as xi hence the integral is highly oscillatory because a fio is a dense matrix a naive matrix vector product with an input given on a cartesian grid of size n by n would require o n 4 operations this paper develops a new numerical algorithm which requires o n 2 5 log n operations and as low as o sqrt n in storage space the constants in front of these estimates are small it operates by localizing the integral over polar wedges with small angular aperture in the frequency plane on each wedge the algorithm factorizes the kernel e 2 pi i phi x xi a x xi into two components 1 a diffeomorphism which is handled by means of a nonuniform fft and 2 a residual factor which is handled by numerical separation of the spatial and frequency variables the key to the complexity and accuracy estimates is the fact that the separation rank of the residual kernel is provably independent of the problem size several numerical examples demonstrate the numerical accuracy and low computational complexity of the proposed methodology we also discuss the potential of our ideas for various applications such as reflection seismology

convergence in lp for feynman path integrals abstract we consider a class of schrodinger equations with time dependent smooth magnetic and electric potentials having a growth at infinity at most linear and quadratic respectively we study the convergence in l p with loss of derivatives 1 p of the time slicing approximations of the corresponding feynman path integral the results are completely sharp and hold for long time where no smoothing effect is available the techniques are based on the decomposition and reconstruction of functions and operators with respect to certain wave packets in phase space

inequalities for some maximal functions ii let s be a smooth compact hypersurface in rn and let u be a measure on s absolutely continuous with respect to surface measure for t in r tt denotes the dilate of u by t normalised to have the same total variation as u for f in s rn t f denotes the maximal function supt o l t f we seek conditions on u which guarantee that the a priori estimate ll0 flip clhflip f g s rn holds this estimate entails that the sublinear operator extends to a bounded operator on the lebesgue space lp rn our methods generalise e m stein s treatment of the spherical maximal function 5 a study of riesz operators g functions and analytic families of measures reduces the problem to that of obtaining decay estimates for the fourier transform of u these depend on the geometry of s and the relation between u and surface measure on s in particular we find that there are natural geometric maximal operators limited on lp rn if and only if p e q oo q is some number in 1 oo and may be greater than 2 this answers a question of s wainger posed by stein 61 0 notation and contents the following notation will be used the fourier transform f of a function f on rn is given by f f dxf x e 2 x o e rn rn by b x r we mean the open ball of radius r and centre x the unit sphere in rn is sn 1 we often use polar coordinates p and o with p in 0 x and o in sn by expressions like c cp ck r etc we denote various constants which may vary from line to line and which depend on p k and r and so on these constants will normally also depend on other parameters like n the dimension of the ambient space but we do not mention this dependence explicitly given f in s rn we denote by t secondary 42b10

distinguishing views in symmetric networks the view of a node in a port labeled network is an infinite tree encoding all walks in the network originating from this node we prove that for any integers n d 1 there exists a port labeled network with at most n nodes and diameter at most d which contains a pair of nodes whose infinite views are different but whose views truncated to depth d log n d are identical

sobolev estimates for averaging operators over a convex hypersurface in r3 abstract we prove sharp l p sobolev estimates for averaging operators associated with a certain type of convex hypersurface in r 3

diophantine equations and ergodic theorems let x be a probability measure space and t 1 t n be a family of commuting measure preserving invertible transformations on x let q m 1 m n be a homogeneous positive polynomial with integer coefficients and let s m e z n q m a denote the set of integer solutions m m 1 m n of the diophantine equation q m a we prove that under a certain nondegeneracy condition on the polynomial q m and an ergodic condition on the family of transformations t t 1 t n the images of the solution sets q x t m1 1 t m2 2 t m n n x m s become uniformly distributed on x w r t for a e x e x as that is the pointwise ergodic theorem holds when the standard averages are replaced by the ones where the exponents satisfy a diophantine equation the proof uses a variant of the hardy littlewood method of exponential sums developed by birch and davenport and techniques from harmonic analysis a key point is the corresponding maximal theorem which is a discrete analogue of a maximal theorem on r n corresponding to the level surfaces of the polynomial q x

l p regularity of averages over curves and bounds for associated maximal operators we prove that for a finite type curve in r 3 the maximal operator generated by dilations is bounded on if for sufficiently large p we also show the endpoint lp l p i p regularity result for the averaging operators for large p the proofs make use of a deep result of thomas wolff about decompositions of cone multipliers

a theory of amplitudes a stochastic framework for quantum mechanics is presented in which the elements are measurements and amplitudes the resulting structure provides three stochastic levels at the bottom level are the sample points these may be unobservable in general at the next level are the outcomes of measurements the amplitude of an outcome is computed by summing the amplitudes of the sample points resulting in that outcome upon a measurement at the highest level are the events of a measurement the probability of an event is computed by summing the moduli squared of the amplitudes for the outcomes comprising that event using these guidelines various probabilities conditional probabilities and expectations are defined superpositions of amplitude functions are investigated and superselection sectors are shown to occur in a natural way the framework is illustrated in various mathematical models such as spin 1 2 models the two slit experiment and phase space quantum mechanics finally a theory of discrete feynman amplitudes is presented

regularity theory for the spatially homogeneous boltzmann equation with cut off we develop the regularity theory of the spatially homogeneous boltzmann equation with cut off and hard potentials for instance hard spheres by i revisiting the lp theory to obtain constructive bounds ii establishing propagation of smoothness and singularities iii obtaining estimates on the decay of the singularities of the initial datum our proofs are based on a detailed study of the regularity of the gain operator an application to the long time behavior is presented

self similar solutions of semilinear wave equation with variable speed of propagation abstract we investigate the issue of existence of the self similar solutions of the generalized tricomi equation in the half space where the equation is hyperbolic we look for the self similar solutions via the cauchy problem an integral transformation suggested in k yagdjian a note on the fundamental solution for the tricomi type equation in the hyperbolic domain j differential equations 206 2004 227 252 is used to represent solutions of the cauchy problem for the linear tricomi type equation in terms of fundamental solutions of the classical wave equation this representation allows us to prove decay estimates for the linear tricomi type equation with a source term obtained in k yagdjian the self similar solutions of the tricomi type equations z angew math phys in press doi 10 1007 s00033 006 5099 2 estimates for the self similar solutions of the linear tricomi type equation are the key tools to prove existence of the self similar solutions

uniform lxp lx rq improving for dilated averages over polynomial curves abstract numerous authors have considered the problem of determining the lebesgue space mapping properties of the operator a given by convolution with affine arc length measure on some polynomial curve in euclidean space essentially a takes weighted averages over translates of the curve in this paper a variant of this problem is discussed where averages over both translates and dilates of a fixed curve are considered the sharp range of estimates for the resulting operator is obtained in all dimensions except for an endpoint the techniques used are redolent of those previously applied in the study of a in particular the arguments are based upon the refinement method of christ although a significant adaptation of this method is required to fully understand the additional smoothing afforded by averaging over dilates

computing boolean functions on anonymous networks abstract we study the bit complexity of computing boolean functions on anonymous networks let n be the number of nodes the diameter and d the maximal node degree of the network for arbitrary anonymous networks we give a general algorithm of polynomial bit complexity o n 3 d 2 log n for computing any boolean function which is computable on the network this improves upon the previous best known algorithm which was of exponential bit complexity o d n 2 for symmetric functions on arbitrary networks we give an algorithm with bit complexity o n 3 d 2 log 2 n this same algorithm is shown to have even lower bit complexity for a number of specific networks for example tori hypercubes and random regular graphs we also consider the class of distance regular unlabeled networks and show that on such networks symmetric functions can be computed efficiently in o n d log n bits