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

averages and higher moments for the ell torsion in class groups we prove upper bounds for the average size of the ell torsion mathrm cl k ell of the class group of k as k runs through certain natural families of number fields and ell is a positive integer we refine a key argument used in almost all results of this type which links upper bounds for mathrm cl k ell to the existence of many primes splitting completely in k that are small compared to the discriminant of k our improvements are achieved through the introduction of a new family of specialised invariants of number fields to replace the discriminant in this argument in conjunction with new counting results for these invariants this leads to significantly improved upper bounds for the average and sometimes even higher moments of mathrm cl k ell for many families of number fields k considered in the literature for example for the families of all degree d fields for d in 2 3 4 5 and non d 4 if d 4 as an application of the case d 2 we obtain the best upper bounds for the number of d p fields of bounded discriminant for primes p 3

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

harmonic analysis as spectral theory of laplacians by writing the fourier inversion formula on euclidean space in polar coordinates we obtain f x 0 fgl x dx where fgl x 2fgl x which is the spectral theory of the laplacian how do properties of f relate to properties of the family of eigenfunctions fgl answers are provided for the following spaces l2 s s d in odd dimensions analogous results are obtained for harmonic analysis on hyperbolic space constant curvature semi riemannian spaces the heisenberg group and for differential forms on hyperbolic space for f l2 there is a plancherel formula f 22 limt o 1t bt z f x 2dx d where bt z denotes the ball of radius t about an arbitrary point z which is independent of the dimension and the identical formula holds in hyperbolic space if the parametrization of the eigenvalues is shifted for certain semi riemannian symmetric spaces we obtain a paley wiener theorem that explains the role of discrete series in producing functions of compact support and also involve certain non unitary representations that contain the discrete series representations for the heisenberg group the plancherel formula is of a different nature requiring that the eigenfunctions be almost periodic in one variable

inversion of the radon transform on the laguerre hypergroup by using generalized wavelets abstract we consider the radon transform r 0 on the laguerre hypergroup k 0 r we characterize a space of infinitely differentiable and rapidly decreasing functions together with their derivatives such that r is a bijection from this space onto itself we establish an inversion formula and a plancherel theorem for the operator r finally by using the continuous wavelet transform on the laguerre hypergroup k we deduce another expression for the inverse r 1 of the operator r

characterization of eigenfunctions of the laplacian by boundedness conditions if f k x k z is a doubly infinite sequence of functions on r n which are uniformly bounded and such that f k f k 1 then f 0 f 0 this generalizes a therem of roe n 1 the analogous statement is true on the heisenberg group but false in hyperbolic space

metric differentiation monotonicity and maps to l1 this is one of a series of papers on lipschitz maps from metric spaces to l1 here we present the details of results which were announced in cheeger and kleiner ann math 2006 to appear arxiv math mg 0611954 sect 1 8 a new approach to the infinitesimal structure of lipschitz maps into l1 and as a first application an alternative proof of the main theorem of cheeger and kleiner ann math 2006 to appear arxiv math mg 0611954 that the heisenberg group does not admit a bi lipschitz embedding in l1 the proof uses the metric differentiation theorem of pauls commun anal geom 9 5 951 982 2001 and the cut metric description in cheeger and kleiner ann math 2006 to appear arxiv math mg 0611954 to reduce the nonembedding argument to a classification of monotone subsets of the heisenberg group a quantitative version of this classification argument is used in our forthcoming joint paper with assaf naor cheeger et al in arxiv 0910 2026 2009

integral points on elliptic curves and 3 torsion in class groups we give new bounds for the number of integral points on elliptic curves the method may be said to interpolate between approaches via diophantine techniques bp hbr and methods based on quasi orthogonality in the mordell weil lattice sil6 gs he we apply our results to break previous bounds on the number of elliptic curves of given conductor and the size of the 3 torsion part of the class group of a quadratic field the same ideas can be used to count rational points on curves of higher genus

the laguerre calculus on the nilpotent lie groups of step two abstract the laguerre calculus is widely used for the inversion of differential operators on the heisenberg group we extend the laguerre calculus for nilpotent groups of step two and test it in the determining of the fundamental solution of the sub laplace operator we also apply it to find the szego kernels of the projection operators to a kind of regular functions on the quaternion heisenberg group

counting cubic extensions with given quadratic resolvent abstract given a number field k and a quadratic extension k 2 we give an explicit asymptotic formula for the number of isomorphism classes of cubic extensions of k whose galois closure contains k 2 as quadratic subextension ordered by the norm of their relative discriminant ideal the main tool is kummer theory we also study in detail the error term of the asymptotics and show that it is o x for an explicit 1

the radon transform on the heisenberg group and the transversal radon transform abstract the radon transform on the heisenberg group was introduced by r strichartz we regard it as a particular case of a more general transversal radon transform that integrates functions on r m over hyperplanes meeting the last coordinate axis the paper contains new boundedness results and explicit inversion formulas for both transforms of l p functions in the full range of the parameter p we also show that these transforms are isomorphisms of the corresponding semyanistyi lizorkin spaces of smooth functions in the framework of these spaces we obtain inversion formulas which are pointwise analogues of the corresponding formulas by r strichartz

image of the schwartz space under spectral projection let x g k be a riemannian symmetric space of non compact type and rank one the spectral projection p f of a function f on x can be written p f f where is the elementary spherical function corresponding to the complex parameter we characterize the image of the schwartz space mathcal s p x under the spectral projection for 0 p 2

singular integrals with flag kernels and analysis on quadratic cr manifolds abstract we study a class of operators on nilpotent lie groups g given by convolution with flag kernels these are special kinds of product type distributions whose singularities are supported on an increasing subspace 0 v1 vk g we show that product kernels can be written as finite sums of flag kernels that flag kernels can be characterized in terms of their fourier transforms and that flag kernels have good regularity restriction and composition properties we then apply this theory to the study of the b complex on certain quadratic cr submanifolds of c n we obtain lp regularity for certain derivatives of the relative fundamental solution of b and for the corresponding szego projections onto the null space of b by showing that the distribution kernels of these operators are finite sums of flag kernels

lp harmonic analysis and radon transforms on the heisenberg group in this paper harmonic analysis in the heisenberg group heisn cn r with group law z t z t z z t t 12im z z is taken to mean the joint spectral theory of the operators l heisenberg laplacian and t t the spectral decomposition of e l2 is given explicitly as e 0 e d where e 1 k 0 1 2 and e z t n 2 n 1 n 2k n 1 exp ie tn 2k exp z 24 n 2k lkn 1 z 22 n 2k where lkn 1 denotes the laguerre polynomial the eigenvalues are it e e n 2k e and l k e k e this decomposition is essentially well known but has not previously been described in this light it is also the decomposition into irreducibles of the representation of the heisenberg motion group semi direct product of heisn and u n on l2 heisn the main result is a summability in lp of the decomposition for e lp 1 p limr 1 rk 0 ed for p 1 only a weaker substitute is available and for p the decomposition is false there is also a plancherel formula for l2 2 2 k e n 2k 0 k e z 0 2dzd a number of explicit examples of the decomposition are computed including part of the schrodinger propagator eisl the heisenberg radon transform is defined by r z t n z t w 0 dw this operator was studied by geller and stein it is shown to be bounded from lp to lq if and only if p 2n 2 2n 1 and q 2n 2 some generalizations of these results to the free step two nilpotent lie groups are given but the results are not as complete

bounded k spherical functions on heisenberg groups abstract let hn be the 2n 1 dimensional heisenberg group and let k be a compact subgroup of aut hn the group of automorphisms of hn the pair k hn is called a gelfand pair if lk1 hn the subalgebra of elements of l1 hn that are invariant under the action of k is commutative in this case the continuous homomorphisms on lk1 hn are given by integrating against certain k invariant functions on hn these functions are the k spherical functions associated to the gelfand pair k hn in this paper we show how to compute the bounded k spherical functions on hn

a restriction theorem for the heisenberg group since the discovery by e m stein see fefferman 5 that the fourier transform of an lp function on r has a well defined restriction to the unit sphere s which is square integrable on sn 1 if p is close enough to 1 various new restriction theorems have been proved see e g 10 22 27 29 the importance of such theorems in harmonic analysis see e g 5 6 24 28 as well as in the theory of partial differential equations 18 19 25 has become evident it has turned out 29 that the crucial step in proving the stein tomas restriction theorem is the following let a denote the surface measure of s n 1 and let k denote the fourier transform of a then the operator af f k is bounded from lp to lp if p is sufficiently close to 1 9 can be interpreted in terms of the laplacian a on rn let a fox dex be the spectral decomposition of a then formally 9 const f106 1 x dex where 6 denotes dirac s point measure therefore it is natural to study analogues for 9 where a is replaced by a more general positive differential operator l in the case of second order elliptic operators on compact manifolds this has been done by sogge 25

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

the number of integral points on arcs and ovals integral lattice points and that the exponent and constant are best possible however swinnerton dyer 10 showed that the preceding result can be substantially improved if we start with a fixed c strictly convex arc and consider the number of lattice points on t the dilation of by a factor t t 1 this of course is the same as asking for rational points mn n n on as n in fact swinnerton dyer proves a bound of type t zz c e t 3 5 e

vertical versus horizontal sobolev spaces abstract let 0 1 p and let h n be the heisenberg group folland in 1975 showed that if f h n r is a function in the horizontal sobolev space s 2 p h n then f belongs to the euclidean sobolev space s p r 2 n 1 for any test function in short s 2 p h n s loc p r 2 n 1 we show that the localisation can be omitted if one only cares for sobolev regularity in the vertical direction the horizontal sobolev space s 2 p h n is continuously contained in the vertical sobolev space v p h n our search for the sharper result was motivated by the following two applications first combined with a short additional argument it implies that bounded lipschitz functions on h n have a 1 2 order vertical derivative in bmo h n second it yields a fractional order generalisation of the non endpoint vertical versus horizontal poincare inequalities of v lafforgue and a naor

the spherical transform of a schwartz function on the heisenberg group suppose thatk u n is a compact lie group acting on the 2n 1 dimensional heisenberg grouphn we say that k hn is a gelfand pair if the convolution algebral1k hn of integrablek invariant functions onhnis commutative in this case the gelfand space k hn is equipped with the godement plancherel measure and the spherical transform l2k hn l2 k hn is an isometry the main result in this paper provides a complete characterization of the set sk hn f f sk hn of spherical transforms ofk invariant schwartz functions onhn we show that a functionfon k hn belongs to sk hn if and only if the functions obtained fromfvia application of certain derivatives and difference operators satisfy decay conditions we also consider spherical series expansions fork invariant schwartz functions onhnmodulo its center

analysis of the kohn laplacian on quadratic cr manifolds abstract we study the kohn laplacian b q acting on 0 q forms on quadratic cr manifolds we characterize the operators b q that are locally solvable and hypoelliptic respectively in terms of the signatures of the scalar components of the levi form

poincare hopf theorems on singular spaces in 1991 m h schwartz proved a poincare hopf theorem for stratified radial vector fields on whitney stratified analytic varieties with boundary in this paper we extend her theory and results to a wide class of stratified sets with boundary called radial manifold complexes which include all thom mather stratified sets but also arbitrary closure orderable finite partitions of subanalytic sets into subanalytic submanifolds in the process we give a short proof of a generalization of the poincare hopf theorem for radial vector fields of m h schwartz our theorem 3 2 we then study more general stratified vector fields which need not be radial and may further be tangent to the boundary of the manifold complex and prove the corresponding poincarehopf theorems in particular a notion is given of the virtual index of a vector field on a radial manifold complex and the euler characteristic of the complex is shown to be the sum of the virtual indices

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

on riesz means of eigenfunction expansions for the kohn laplacian in a recent article c sogge improved some classical results by l hormander about the l p convergence of the riesz means of eigenfunction expansions associated to elliptic differential operators on compact manifolds in this paper we shall deal with the analogous problem for the model operator of a non elliptic but subelliptic differential operator the kohn laplacian on the heisenberg group

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