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

on wave boundary elements for radiation and scattering problems with piecewise constant impedance discrete methods of numerical analysis have been used successfully for decades for the solution of problems involving wave diffraction etc however these methods including the finite element and boundary element methods can require a prohibitively large number of elements as the wavelength becomes progressively shorter in this paper a new type of interpolation for the wave field is described in which the usual conventional shape functions are modified by the inclusion of a set of plane waves propagating in multiple directions including such a plane wave basis in a boundary element formulation is found in this paper to be highly successful results are shown for a variety of scattering radiating problems from convex and nonconvex obstacles on which are prescribed piecewise constant robin conditions notable results include a conclusion that using this new formulation only approximately three degrees of freedom per wavelength are required

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

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

the green formula and h p spaces on trees a theory o f h e spaces on not necessarily homogeneous trees was developed in kpt these spaces are defined by means of certain maximal or square function operators associated with a nearest neighbour transition operator p which is very regular i e for some 6o with 0 60 89 we assume the i 60 for every pair of neighbours x y the regularity bounds 60 p x y condition yields uniform estimates for transience o f p and therefore gives a good control over several operators defined on the boundary f2 of t by means of which one defines the h p spaces as observed long ago in the homogeneous case ta a nearest neighbour transition operator on an arbitrary tree gives rise to a natural martingale structure on its boundary see kpt for details this martingale inherits the strong regularity property which is heavily used in kpt to prove that the different definitions of h p are equivalent the approach of kpt is in the same spirit o f the construction o f h p spaces by means of martingales developed in bgs in a very general setting on the other hand another independent approach to h p theory in smooth domains say the disc or the upper half plane was established in fs some of its operators as the lusin area integral are defined in terms of differential operators and the equivalence between different constructions o f h p is based upon good lambda inequalities an important tool is the green formula which is used to transform the area integral of a harmonic function over smooth compact domains to a line integral over their boundary curves the method of fs has been extended to semisimple symmetric spaces of rank one ci

plane wave interpolation in direct collocation boundary element method for radiation and wave scattering numerical aspects and applications the classical boundary element formulation for the helmholtz equation is rehearsed and its limitations with respect to the number of variables needed to model a wavelength are explained a new type of interpolation for the potential is then described in which the usual boundary element shape functions are modified by the inclusion of a set of plane waves propagating in a range of directions this is termed the plane wave basis boundary element method the modifications needed to the classical procedures in terms of integration of the element matrices and location of collocation points are described the well known singular value decomposition solution technique which is adopted here for the solution of the system matrix equation in its complex form is briefly outlined the conditioning of the system matrix is analysed for a simple radiation problem the corresponding diffraction problem is also analysed and results are compared with analytical and classical boundary element solutions the chief method is adopted to enhance the quality of the solution particularly in the vicinity of irregular frequencies the plane wave basis boundary element method is then applied to two problems scattering of plane waves by an elliptical cylinder and the multiple circular cylinder plane wave scattering problem in both cases results are compared with analytical solutions the results clearly demonstrate that the new method is considerably more efficient than the classical approach for a given number of degrees of freedom the frequency for which accurate results can be obtained using the new technique can be up to three or four times higher than that of the classical method this makes the method a powerful new addition to our tools for tackling high frequency radiation and scattering problems

a contribution to scattering calculation for small wavelengths the high frequency panel method numerical treatment of electromagnetic scattering in the case of small wavelengths too small for reasonable matrix sizes when using discretization methods and too large for traditional asymptotic methods is considered an approach which takes advantage of the user friendly nature of matrix discretization methods while avoiding the explosive increase in computing effort as the wavelength decreases is suggested it appears that no sensible increase of matrix size is required as the wavelength decreases since this is an untried technique applied in an inaccessible computational domain the preliminary work described is confined to two dimensional conducting circular cylinders where the answers are well known and well understood the overall idea however appears to be quite general

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

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

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

solution of helmholtz equation in the exterior domain by elementary boundary integral methods in this paper elementary boundary integral equations for the helmholtz equation in the exterior domain based on green s formula or through representation of the solution by layer potentials are considered even when the partial differential equation has a unique solution for any given closed boundary i these elementary boundary integral equations can be shown to be singular at a countable set of characteristic wavenumbers spectral properties and conditioning of the boundary integral operators and their discrete boundary element counterparts are studied near characteristic wavenumbers with a view to assessing the suitability of these formulations for the solution of the exterior helmholtz equation collocation methods are used for the discretisation of the boundary integral equations which are either of the fredholm first kind second kind or hyper singular type the effect of quadrature errors on the accuracy of the discrete collocation methods is systematically investigated

an improved partition of unity finite element model for diffraction problems the partition of unity finite element method pufem is explored and improved to deal with practical diffraction problems efficiently the use of plane waves as an external function space allows an efficient implementation of an approximate exterior non reflective boundary condition improving the original proposed by higdon for general diffraction problems a virtually analytical integration procedure is introduced for multi dimensional high frequency problems which exhibits a dramatic decrease in the number of operations for a given error compared with standard integration methods suitable conjugate gradient type solvers for the whole range of wavenumbers are used including such cases in which pufem can produce nearly singular matrices caused by round off limits copyright 2001 john wiley sons ltd

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

modelling of short wave diffraction problems using approximating systems of plane waves this paper describes a finite element model for the solution of helmholtz problems at higher frequencies that offers the possibility of computing many wavelengths in a single finite element the approach is based on partition of unity isoparametric elements at each finite element node the potential is expanded in a discrete series of planar waves each propagating at a specified angle these angles can be uniformly distributed or may be carefully chosen they can also be the same for all nodes of the studied mesh or may vary from one node to another the implemented approach is used to solve a few practical problems such as the diffraction of plane waves by cylinders and spheres the wave number is increased and the mesh remains unchanged until a single finite element contains many wavelengths in each spatial direction and therefore the dimension of the whole problem is greatly reduced issues related to the integration and the conditioning are also discussed

application of the integral equation asymptotic phase method to two dimensional scattering a hybrid procedure called the integral equation asymptotic phase ie ap method is investigated for scattering from perfectly conducting cylinders of arbitrary cross section shape the ie ap approach employs an asymptotic solution to predict the relatively rapid phase dependence of the unknown current distribution to leave a slowly varying residual function that can be represented by a coarse density of unknowns in the present investigation the current density appearing within the combined field integral equation is replaced by the product of a rapidly varying phase function obtained from the physical optics current and a residual function the resulting equation is discretized by the method of moments using subsectional quadratic polynomial basis functions defined on curved cells to represent the residual function results show that the required density of unknowns can often be as few as one per wavelength on average without a significant loss of accuracy in the computed current density even for scatterers with corners

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

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

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

a wavenumber independent boundary element method for an acoustic scattering problem in this paper we consider the impedance boundary value problem for the helmholtz equation in a half plane with piecewise constant boundary data a problem which models for example outdoor sound propagation over inhomogeneous flat terrain to achieve good approximation at high frequencies with a relatively low number of degrees of freedom we propose a novel galerkin boundary element method using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that on each element the approximation space contains polynomials of degree nu multiplied by traces of plane waves on the boundary we prove stability and convergence and show that the error in computing the total acoustic field is cal o n nu 1 log 1 2 n where the number of degrees of freedom is proportional to n log n this error estimate is independent of the wavenumber and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity

carleson measures on a homogeneous tree abstract we introduce the notion of s carleson measure s 1 on a homogeneous tree t and give several characterizations of such measures in particular we prove the following discrete version of the extension of carleson s theorem due to duren for p 1 and s 1 a finite measure on t is s carleson if and only if there exists c 0 such that for all f l p t p f l s p c f l p t where p f denotes the poisson integral of f here l p is the space of functions g defined on t such that g p is integrable with respect to and l p t is the space of functions f defined on the boundary of t such that f p is integrable with respect to the representing measure of the harmonic function 1

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