higher infinitesimal neighbourhoods we study the hilbert function of thekth order neighbourhood of a collection of points in linearly general position in pn writtenhpn k m our main results are the following bounds onhpn k k 1 if deg 2n k 1 thenhpn k k 1 hpn ck k 1 wherec pnis a rational normal curve of degreen if is sufficiently general ands deg then and equality holds whenk 3

bmo from dyadic bmo via expectations on product spaces of homogeneous type using the random dyadic lattices developed by hytonen and kairema we build up a bridge between bmo and dyadic bmo and hence one between vmo and dyadic vmo via expectations over dyadic lattices on spaces of homogeneous type including both the one parameter and product cases we also obtain a similar relationship between ap and dyadic ap as well as one between the reverse holder class rhp and dyadic rhp via geometric arithmetic expectations these results extend the earlier theory along this line developed by garnett jones pipher ward xiao and treil to the more general setting of spaces of homogeneous type in the sense of coifman and weiss

a characterization of hajlasz sobolev and triebel lizorkin spaces via grand littlewood paley functions abstract in this paper we establish the equivalence between the hajlasz sobolev spaces or classical triebel lizorkin spaces and a class of grand triebel lizorkin spaces on euclidean spaces and also on metric spaces that are both doubling and reverse doubling in particular when p n n 1 we give a new characterization of the hajlasz sobolev spaces m 1 p r n via a grand littlewood paley function

holder estimates on cr manifolds with a diagonalizable levi form a for all e 0 the operator cl maps lip s po into lip s 2 m e po here lip s p denotes the space of square integrable forms of degree p q with 1 g q otheoperators o l o l q and0 map lip s p into lip s l m e po c for all e o the operators 8b8zo 8z8ho 1 q a aa cl 8 8 8 q 8 and 8 0 ab map lip s p into lip s p hebre aiag web asm acts on forms of type p q with 1 q 0 e zf f is a square integrable function on a with f e lip p then s f e lip s e p for all e 0 where s is the orthogonal projection on square integrable cr functions

pointwise multipliers of besov spaces on carnot caratheodory spaces abstract let x d be a carnot caratheodory space namely x is a smooth manifold d is a control or carnot caratheodory metric induced by a collection of vector fields of finite type is a nonnegative borel regular measure on x satisfying that there exists constant c 0 1 such that for all x x and 0 r diam x b x 2 r y x d x y 2 r c 0 b x r doubling property using the discrete calderon reproducing formula and the plancherel polya characterization of the inhomogeneous besov spaces developed by han et al 12 and han et al 2008 10 pointwise multipliers of inhomogeneous besov spaces are obtained

on maximal sobolev and holder estimates for the tangential cauchy riemann operator and boundary laplacian let m be the boundary of a smoothly bounded pseudoconvex domain in c n n 3 or more generally any compact pseudoconvex cr manifold of dimension 2 n 1 for which the range of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is closed in l 2 in this article we study the l p sobolev and holder regularity properties of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i and b near a point of finite type under a comparable eigenvalues condition on the levi form we show that if all possible sums of q 0 eigenvalues of the levi matrix are comparable to its trace near a point of finite commutator type condition d q 0 then the inverse k q of b on 0 q forms for q 0 q n 1 q 0 satisfies sharp kernel estimates in terms of the quasi distance associated to the hormander sum of squares operator in particular we obtain the maximal l p estimates for b which were conjectured in the 1980s we also prove sharp estimates for certain parts of the kernels of k q 0 1 and k n q 0 and give some applications concerning domains with at most one degenerate eigenvalue finally we establish the composition and mapping properties of a class of singular integral nonisotropic smoothing operators that arises naturally in complex analysis these results yield optimal regularity of k q and related operators in the sobolev and lipschitz norms both isotropic and nonisotropic

boundedness of multi parameter fourier multiplier operators on triebel lizorkin and besov lipschitz spaces abstract the main purpose of this paper is three fold first we prove that under the limited smoothness conditions multi parameter fourier multiplier operators are bounded on multi parameter triebel lizorkin and besov lipschitz spaces by the littlewood paley decomposition and the strong maximal operator second we offer a different and more direct method to deal with the boundedness instead of transforming fourier multiplier operators into multi parameter calderon zygmund operators third we also prove the boundedness of multi parameter fourier multiplier operators on weighted multi parameter triebel lizorkin and besov lipschitz spaces when the fourier multiplier is only assumed with limited smoothness

besov and triebel lizorkin spaces on metric spaces embeddings and pointwise multipliers abstract in this paper we obtain the franke jawerth embedding property of hajlasz besov and hajlasz triebel lizorkin spaces on a measure metric space x d which is ahlfors regular with dimension q as applications we show that when x d is doubling and satisfies an ahlfors lower bound condition with q then the hajlasz besov space n p q s x with p q s q p 1 and q 0 and the hajlasz triebel lizorkin space m p q s x with p q s q p 1 and q q q s are algebras under pointwise multiplication and moreover when x is ahlfors q regular we characterize the class of all pointwise multipliers on the hajlasz triebel lizorkin space m p q s x for p q s q p 1 and q q q s by its related uniform space

an algebra containing the two sided convolution operators abstract we present an intrinsically defined algebra of operators containing the right and left invariant calderon zygmund operators on a stratified group the operators in our algebra are pseudolocal and bounded on l p 1 p this algebra provides an example of an algebra of singular integrals that falls outside of the classical calderon zygmund theory

on the covering radius of codes the covering radius r of a code is the maximal distance of any vector from the code this work gives a number of new results concerning t n k the minimal covering radius of any binary code of length n and dimension k for example t n 4 and t n 5 are determined exactly and reasonably tight bounds on t n k are obtained for any k when n is large these results are found by using several new constructions for codes with small covering radius one construction the amalgamated direct sum involves a quantity called the norm of a code codes with norm leq 2 r 1 are called normal and may be combined efficiently the paper concludes with a table giving bounds on t n k for n leq 64

the geometric interpretation of froberg iarrobino conjectures on infinitesimal neighbourhoods of points in projective space abstract the study of infinitesimal deformations of a variety embedded in projective space requires at ground level that of deformation of a collection of points as specified by a zero dimensional scheme further basic problems in infinitesimal interpolation correspond directly to the analysis of such schemes an optimal hilbert function of a collection of infinitesimal neighbourhoods of points in projective space is suggested by algebraic conjectures of r froberg and a iarrobino we discuss these conjectures from a geometric point of view they give for each such collection a function based on dimension number of points and order of each neighbourhood which should serve as an upper bound to its hilbert function weak conjecture the strong conjecture predicts when the upper bound is sharp in the case of equal order throughout in general we refer to the equality of the hilbert function of a collection of infinitesimal neighbourhoods with that of the corresponding conjectural function as the strong hypothesis we interpret these conjectures and hypotheses as accounting for the infinitesimal neighbourhoods of projective subspaces naturally occurring in the base locus of a linear system with prescribed singularities at fixed points we develop techniques and insight toward the conjectures verification and refinement the main result gives an upper bound on the hilbert function of a collection of infinitesimal neighbourhoods in p n based on hilbert functions of certain such subschemes of p n 1 further equality occurs exactly when the scheme has only the expected linear obstructions to the linear system at hand it follows that an infinitesimal neighbourhood scheme obeys the weak conjecture provided that the schemes identified in codimension one satisfy the strong hypothesis this observation is then applied to show that the weak conjecture does hold valid in p n for n 3 the main feature here is that the result is obtained although the strong hypothesis is not known to hold generally in p 2 and further p 2 presents special exceptional cases consequences of the main result in higher dimension are then examined we note then that the full weight of the strong conjecture and validity of the strong hypothesis are not necessary toward using the main theorem in the next dimension we end with the observation of how our viewpoint on the strong hypothesis pertains to extra algebraic information namely on the structure of the minimal free resolution of an ideal generated by linear forms

harmonic analysis on product spaces rn x r and commute with the dilations p33 8 x2 x2 61x1 62x2 for x1 e rn x2 e rm and 61 82 0 then the corresponding operators are related to many problems in the theory of multiple fourier integrals see zygmund 21 the maximal operator which commutes with the p31382 is the jessen marcinkiewicz and zygmund 14 strong maximal operator ms defined by 1 ms f x1 x2 sup ri a y1 y2 1 dy1 dy2 xl x2 gr jr1

subcodes and covering radius corresp a sharp lower bound on the covering radius of a linear binary code is given in terms of the covering radius of any subcode of index two an upper bound on the norm of a code is then derived these results hold for all linear codes over gf 2

estimates for the bergman and szego kernels in mathbf c 2 the purpose of this paper some of whose conclusions were announced in nrsw is to study the bergman and szegd projection operators on pseudoconvex domains q of finite type in c2 the results we obtain are of three kinds i the size estimates of the bergman and szeg6 kernels and their derivatives ii the cancellation properties of those kernels expressed in terms of the actions of these operators on suitable bump functions these properties have an interest in their own right but are also crucial in iii below iii the sharp mapping properties of these operators on functions spaces such as lp the nonisotropic sobolev spaces and the nonisotropic lipschitz spaces which are naturally attached to the geometry of the boundary this also leads ultimately to sharp results in the regularity properties of solutions of the equation du f in q and 9bu f on the boundary let us describe these matters in more detail our starting points are certain basic geometric constructs associated to q a metric p defined in terms of the vector fields x1 and x2 which are the real and imaginary parts of the tangential levi vector field on d 2 and a function a p 8 a polynomial in 8 for p e d 2 which represents the higher levi invariant attached to d 9 the significance of these to the geometry of d q can be understood from the following facts the ball b p 8 c d q centered at p of radius 8 in the metric p when viewed in the appropriate coordinates has width 8 in the complex tangential directions but its width is a p 8 in the complementary direction thus the volume of the ball is 82a p 8 among other crucial properties

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

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

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

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

some results on the norm of codes two upper bounds for the norm n c of a binary linear code c with minimal weight d and covering radius r are given the second of these bounds implies that c is normal if r 3

inverse systems of zero dimensional schemes in p n abstract the authors construct the global macaulay inverse system l z for a zero dimensional subscheme z of projective n space p n over an algebraically closed field k from the local inverse systems of the irreducible components of z they show that when z is locally gorenstein a generic element f of degree d apolar to z determines z if d is larger than an invariant z as a consequence of this globalization they show that a natural upper bound for the hilbert function of gorenstein artin quotients of the coordinate ring of z is achieved for large socle degree they also show the uniqueness of generalized additive decompositions of a homogeneous form into powers of linear forms under suitable hypotheses the main tools are elementary but delicate they involve a careful study of how to homogenize a local inverse system and of the behavior of the homogenization under a change of coordinates

the hull of holomorphy of a nonisotropic ball in a real hypersurface of finite type we show that the hull of holomorphy of a nonisotropic ball in a real hypersurface of finite type in cn contains an open set in cn which emanates from the hypersurface a distance which is proportional to the length of the minor axis of the nonisotropic ball in addition we prove a maximal function estimate for plurisubharmonic functions which is important in the study of boundary values of holomorphic functions

and variants of multiparameter calderon zygmund theory we consider calderon zygmund operators on product domains under certain weak conditions on the kernel a singular integral operator can be proved to be bounded on h p r r r 0 p 1 if its behaviour on l 2 and on certain scalar valued and vector valued rectangle atoms is known another result concerns an extension of the authors results on l p variants of calderon zygmund theory 1 23 to the product domain setting as an application one obtains estimates for fourier multipliers and pseudo differential operators

inverse system of a symbolic power ii the waring problem for forms abstract the classical waring problem for forms is to determine the smallest length s of an additive decomposition of a general degree d homogeneous polynomial or form f in r variables as sum of s d th powers of linear forms we show that its solution is implied by a result of j alexander and a hirschowitz concerning the hilbert functions of the ideal of functions vanishing to order two at a generic set of s points in p r 1 using macaulay s inverse systems we show that the alexander hirschowitz result is equivalent to determining the number of linear syzygies of s homogeneous forms in r variables that are d th powers of a given set of general linear forms we also determine the dimension of the family of degree d forms that have additive decompositions of length s we then study several notions of length for forms f having to do with the kind of length s zero dimensional schemes z in p r 1 whose defining ideal i z annihilates the inverse system of f when z is to consist of distinct points we obtain the above length of additive decomposition of f when z is smoothable we obtain the smoothable length of f when z is arbitrary we obtain a scheme length of f all these lengths are at least as large as the dimension of the vector space of all order i partial derivates of f for each i the above mentioned length functions are distinct using results about the existence of nonsmoothable gorenstein point singularities in codimension 4 we show that when r 5 there are forms f of scheme length s which are not in the closure of the family of forms having additive decompositions of length s finally we propose a new set of waring problems for forms using these lengths

a brief proof of a maximal rank theorem for generic double points in projective space we give a simple proof of the following theorem of j alexander and a hirschowitz given a general set of points in projective space the homogeneous ideal of polynomials that are singular at these points has the expected dimension in each degree of 4 and higher except in 3 cases

product hardy spaces associated to operators with heat kernel bounds on spaces of homogeneous type the aim of this article is to develop the theory of product hardy spaces associated with operators which possess the weak assumption of davies gaffney heat kernel estimates in the setting of spaces of homogeneous type we also establish a calderon zygmund decomposition on product spaces which is of independent and use it to study the interpolation of these product hardy spaces we then show that under the assumption of generalized gaussian estimates the product hardy spaces coincide with the lebesgue spaces for an appropriate range of p

multiparameter hardy space theory on carnot caratheodory spaces and product spaces of homogeneous type this paper is inspired by the work of nagel and stein in which the lp 1 p theory has been developed in the setting of the product carnot caratheodory spaces m m1 mn formed by vector fields satisfying hormander s finite rank condition the main purpose of this paper is to provide a unified approach to develop the multiparameter hardy space theory on product spaces of homogeneous type this theory includes the product hardy space its dual the product bmo space the boundedness of singular integral operators and calderon zygmund decomposition and interpolation of operators as a consequence we obtain the endpoint estimates for those singular integral operators considered by nagel and stein 2004 in fact we will develop most of our theory in the framework of product spaces of homogeneous type which only satisfy the doubling condition and some regularity assumption on the metric all of our results are established by introducing certain banach spaces of test functions and distributions developing discrete calderon identity and discrete littlewood paley stein theory our methods do not rely on the journe type covering lemma which was the main tool to prove the boundedness of singular integrals on the classical product hardy spaces

necessary conditions for subellipticity of the overline partial neumann problem let a be a a closed form of type p q with l2 coefficients on a smoothly bounded domain q in c the problem of determining both the existence and regularity properties of the solution u of au a where u is orthogonal to the null space of a on p q l forms is known as the a neumann problem one of the principal methods used in the investigation of this problem is the proof of certain a priori subelliptic estimates let u be a neighborhood of a point z0 in the boundary of q a subelliptic estimate is said to hold in u if the estimate

estimates for the parametrix of the kohn laplacian on certain domains soit le demi espace superieur degenere generalise dans c 3 d p w 1 w 2 w imw p w 1 w 2 2 ou p c p 0 0 p est reel p 0 p 0 0 k p 0 0 pour k 2 on etudie la taille de la solution fondamentale de b sur des formes 0 1 sur la frontiere de d p