on a theorem of muckenhoupt and wheeden and a weighted inequality related to schrodinger operators we extend in several directions a theorem of b muckenhoupt and r wheeden relating the l p norms of riesz potentials and fractional maximal operators we apply these results to give a simple proof and sharpen a weighted inequality for schrodinger operators of chang wilson and wolff

harnack s inequality for degenerate schrodinger operators on etablit une inegalite d harnack pour les solutions faibles non negatives de certaines equations de schrodinger de la forme lu vu 0 ou l est un operateur elliptique degenere du second ordre sous forme de divergence et v un potentiel d une certaine classe

regularity of quasi minimizers for non uniformly elliptic integrals abstract in this paper we consider a class of non uniformly elliptic integral functionals and we prove the local boundedness of the quasi minimizers our approach is based on a suitable adaptation of the celebrated de giorgi proof and it relies on an appropriate caccioppoli type inequality

the ring of evenly weighted points on the line let m w mathbb p 1 n hbox sl 2 denote the geometric invariant theory quotient of mathbb p 1 n by the diagonal action of hbox sl 2 using the line bundle mathcal o w 1 w 2 ldots w n on mathbb p 1 n let r w be the coordinate ring of m w we give a closed formula for the hilbert function of r w which allows us to compute the degree of m w the graded parts of r w are certain kostka numbers so this hilbert function computes stretched kostka numbers if all the weights w i are even we find a presentation of r w so that the ideal i w of this presentation has a quadratic grobner basis in particular r w is koszul we obtain this result by studying the homogeneous coordinate ring of a projective toric variety arising as a degeneration of m w

multigraded regularity and the koszul property abstract we give a criterion for the section ring of an ample line bundle to be koszul in terms of multigraded regularity we discuss applications to adjoint bundles on toric varieties as well as to polytopal semigroup rings

unique continuation for δ and the c fefferman phong class we show that the strong unique continuation property holds for the inequality u v u where the potential v x satisfies the c fefferman phong condition in a certain range of p values we also deal with the situation of u x vanishing at infinity these are all consequences of appropriate carleman inequalities

a sobolev inequality with reciprocal weights abstract we give a sobolev inequality with the weight k x belonging to the class a 2 g n for the function u t and the weight k x 1 for u 2 the constant in the relevant inequality is seen to depend on the g n and a 2 constants of the weight

corrigendum to on norms of composition operators acting on bergman spaces j math anal appl 291 2004 189 202 abstract a theorem and a corollary in the paper cited in the title were stated incorrectly as was pointed out by christopher hammond we now state correctly and prove both of them these results still generalize and explain the geometric meaning of the cowen hurst norm formula we also include additional references and provide an example relevant for further study

estimates for operator norms on weighted spaces and reverse jensen inequalities we examine the dependence on the a p norm of w of the operator norms of singular integrals maximal functions and other operators in l p w we also examine connections between some fairly general reverse jensen inequalities and the a p and rh p weight conditions

the norm of a composition operator with linear symbol acting on the dirichlet space abstract we obtain a representation for the norm of a composition operator on the dirichlet space induced by a map of the form z a z b we compare this result to an upper bound for c that is valid whenever is univalent our work relies heavily on an adjoint formula recently discovered by gallardo gutierrez and montes rodriguez

initial ideals veronese subrings and rates of algebras abstract let s be a polynomial ring over an infinite field and let i be a homogeneous ideal of s let t d be a polynomial ring whose variables correspond to the monomials of degree d in s we study the initial ideals of the ideals v d i t d that define the veronese subrings of s i in suitable orders they are easily deduced from the initial ideal of i we show that in v d i is generated in degree max reg i d 2 where reg i is the regularity of the ideal i in other words the d th veronese subrings of any commutative graded ring s i has a grobner basis of degree max i d 2 we also give bounds on the regularity of i in terms of the degrees of the generators of in i and some combinatorial data this implies a version of backelin s theorem that high veronese subrings of any ring are homogeneous koszul algebras in the sense of priddy trans amer math soc 152 1970 39 60 we also give a general obstruction for a homogeneous ideal i s to have an initial ideal in i that is generated by quadrics beyond the obvious requirement that i itself should be generated by quadrics and the stronger statement that s i is koszul we use the obstruction to show that in certain dimensions a generic complete intersection of quadrics cannot have an initial ideal that is generated by quadrics for the application to backelin s theorem we require a result of backelin whose proof has never appeared we give a simple proof of a sharpened version bounding the rate of growth of the degrees of generators for syzygies of any multihomogenous module over a polynomial ring modulo an ideal generated by monomials following a method of bruns and herzog

a degenerate sobolev inequality for a large open set in a homogeneous space in current literature existence results for degenerate elliptic equations with rough coefficients on a large open set of a homogeneous space d have been demonstrated see the paper by gutierrez and lanconelli 2003 these results require the assumption of a sobolev inequality on of the form 1 x 2 dx 1 2 c q x x dx holding for lip 0 and some 1 2 however it is unclear when such an inequality is valid as techniques often yield only a local version of 1 2 1 br br x 2 1 2 cr 1 br br q x x dx 1 br br x 2 dx b r br b r b r b r b r holding for lip 0 b r with as above the main result of this work shows that the global sobolev inequality 1 can be obtained from the local sobolev inequality 2 provided standard regularity hypotheses are assumed with minimal restrictions on the quadratic form q x this is achieved via a new technique involving existence of weak solutions with global estimates to a 1 parameter family of dirichlet problems on and a maximum principle

weighted poincare and sobolev inequalities and estimates for weighted peano maximal functions 1 introduction in this paper we derive local poincare and sobolev inequalities involving two weight functions the estimates we obtain include those proved by fabes kenig and serapioni in 8 for the one weight function case we also consider the notion of weighted peano derivatives and show that the peano derivative formed with a given weight function can be estimated in terms of a maximal function involving a second weight function in the special case of ordinary lebesgue measure results of this kind lead to the theorems of a p calderon and a zygmund about local differentiability of functions in sobolev spaces and of functions which are of bounded variation in the sense of tonelli see 3 and 4 all our results are based on variants of the basic estimate in 5 but we do not presuppose any facts from 5 let v x be a weight function on r i e let v x be nonnegative and locally integrable with respect to lebesgue measure we shall use the notation v e se v x dx ordinary lebesgue measure will be denoted ie l if bh x is the open ball with center x and radius h we say v satisfies the doubling condition with respect to lebesgue measure if

norm attaining integral operators on analytic function spaces abstract if f and g are analytic functions in the unit disc d we define s g f z 0 z f w g w d w z d if g is bounded then the integral operator s g is bounded on the bloch space on the dirichlet space and on b m o a we show that s g is norm attaining on the bloch space and on b m o a for any bounded analytic function g but does not attain its norm on the dirichlet space for non constant g some results are also obtained for s g on the little bloch space and for another integral operator t g from the dirichlet space to the bergman space

subnormality and composition operators on h2 if is a non constant analytic function defined on the unit disk d such that d d the composition operator c is the operator on the hardy space h2 of the unit disk defined by c f f in this paper we investigate the relationship between properties of the symbol and the subnormality of the operators c and c the main theorem shows under a regularity hypothesis that c is subnormal if and only if is in a restricted class of linear fractional transformations the two previously known special cases the affine transformations associated with the cesaro operator and the automorphisms are included as the extreme examples the proof of the main theorem consists of verifying a moment condition and yields in addition the construction of a measure that exhibits cgj as multiplication by an analytic function on p2 and the identification of the minimal normal extension as a weighted sum of shifts

compact composition operators on the bloch space necessary and sufficient conditions are given for a composition operator cof f o q to be compact on the bloch space w and on the little bloch space o weakly compact composition operators on o are shown to be compact if q e o is a conformal mapping of the unit disk d into itself whose image q d approaches the unit circle t only in a finite number of nontangential cusps then co is compact on 7o on the other hand if there is a point of t n q d at which q d does not have a cusp then co is not compact

geometry in grassmannians and a generalization of the dilogarithm abstract in this paper we introduce some polyhedra in grassman manifolds which we call grassmannian simplices we study two aspects of these polyhedra their combinatorial structure section 2 and their relation to harmonic differential forms on the grassmannian section 3 using this we obtain results about some new differential forms one of which is the classical dilogarithm section 1 the results here unite two threads of mathematics that were much studied in the 19th century the analytic one concerning the dilogarithm goes back to leibnitz 1696 and euler 1779 and the geometric one concerning grassmannian simplices can be traced to binet 1811 in section 4 we give some of this history along with some recent related results and open problems in section 0 we give as an introduction an account in geometric terms of the simplest cases

on the continuity of solutions to degenerate elliptic equations abstract the local behavior of solutions to a degenerate elliptic equation div a x u 0 in r n where a x a t x and w x 2 a x v x 2 for weights w x 0 and v x has been studied by chanillo and wheeden in chanillo and wheeden 1986 7 they generalize the results of fabes kenig and serapioni 1961 8 relative to the case v x w x we consider the case where w x 1 k x and v x k x the assumption that v a 2 the muckenhoupt class is not sufficient as it was in the case v x w x to obtain the continuity of local solutions however if v g n the gehring class and if s v is the domain of the maximal function of v s v x m v x then the restriction to s v of the precise representative u of any non negative solution u is continuous

regularity of minimizers under limit growth conditions abstract it is well known that an integral of the calculus of variations satisfying anisotropic growth conditions may have unbounded minimizers if the growth exponents are too far apart under sharp assumptions on the exponents we prove the local boundedness of minimizers of functionals with anisotropic p q growth via the de giorgi method as a by product regularity of minimizers of some non coercive functionals is obtained by reduction to coercive ones

norm attaining composition operators on the bloch spaces abstract we prove that every composition operator c on the bloch space modulo constant functions attains its norm and characterize the norm attaining composition operators on the little bloch space modulo constant functions we also identify the extremal functions for c in both cases

regularity of solutions to degenerate p laplacian equations abstract we prove regularity results for solutions of the equation div a x u x u p 2 2 a x u 0 1 p where x x 1 x m is a family of vector fields satisfying hormander s condition and a is an m m symmetric matrix that satisfies degenerate ellipticity conditions if the degeneracy is of the form w x 2 p 2 a x w x 2 p 2 w a p then we show that solutions are locally holder continuous if the degeneracy is of the form k x 2 p 2 a x k x 2 p 2 k a p r h where depends on the homogeneous dimension then the solutions are continuous almost everywhere and we give examples to show that this is the best result possible we give an application to maps of finite distortion

unique continuation and absence of positive eigenvalues for schrodinger operators on considere l operateur de laplace sur r n et une fonction v x sur un sous ensemble connexe ouvert de r n on montre que si n 3 une propriete de prolongement unique est vraie pour v l loc n 2 r n dans l espace de sobolev h loc 2 q r n avec q 2n n 2

existence of extremal functions for the stein weiss inequalities on the heisenberg group abstract in this paper we establish the existence of extremals for two kinds of stein weiss inequalities on the heisenberg group more precisely we prove the existence of extremals for the stein weiss inequalities with full weights in theorem 1 1 and the stein weiss inequalities with horizontal weights in theorem 1 4 different from the proof of the analogous inequality in euclidean spaces given by lieb 26 using riesz rearrangement inequality which is not available on the heisenberg group we employ the concentration compactness principle to obtain the existence of the maximizers on the heisenberg group our result is also new even in the euclidean case because we don t assume that the exponents of the double weights in the stein weiss inequality 1 1 are both nonnegative see theorem 1 3 and more generally theorem 1 5 therefore we extend lieb s celebrated result of the existence of extremal functions of the stein weiss inequality in the euclidean space to the case where the exponents are not necessarily both nonnegative see theorem 1 3 furthermore since the absence of translation invariance of the stein weiss inequalities additional difficulty presents and one cannot simply follow the same line of lions idea to obtain our desired result our methods can also be used to obtain the existence of optimizers for several other weighted integral inequalities theorem 1 5

rectifiability and perimeter in the heisenberg group in this paper we fully extend to the heisenberg group endowed with its intrinsic carnot caratheodory metric and perimeter the classical de giorgi s rectifiability divergence theorems

fractional integrals potential operators and two weight weak type norm inequalities on spaces of homogeneous type we prove two weight weak type norm inequalities for potential operators and fractional integrals defined on spaces of homogeneous type we show that the operators in question are bounded from lp v to lq u 1 p q provided the pair of weights u v verifies a muckenhoupt condition with a power bump on the weight u

nonuniformly elliptic energy integrals with p q growth abstract we study the local boundedness of minimizers of a nonuniformly energy integral of the form f x d v d x under p q growth conditions of the type x p f x x 1 q for some exponents q p 1 and with nonnegative functions satisfying some summability conditions we use here the original notation introduced in 1971 by trudinger 26 where x and x had the role of the minimum and the maximum eigenvalues of an n n symmetric matrix a i j x and f x i j 1 n a i j x i j was the energy integrand associated to a linear nonuniformly elliptic equation in divergence form in this paper we consider a class of energy integrals associated to nonlinear nonuniformly elliptic equations and systems with integrands f x satisfying the general growth conditions above

moduli spaces of weighted pointed stable curves abstract a weighted pointed curve consists of a nodal curve and a sequence of marked smooth points each assigned a number between zero and one a subset of the marked points may coincide if the sum of the corresponding weights is no greater than one we construct moduli spaces for these objects using methods of the log minimal model program and describe the induced birational morphisms between moduli spaces as the weights are varied in the genus zero case we explain the connection to geometric invariant theory quotients of points in the projective line and to compactifications of moduli spaces studied by kapranov keel and losev manin

a characterization of two weight norm inequalities for fractional and poisson integrals caracterisation de deux inegalites de normes ponderees pour les integrales fractionnaires et de poisson applications aux operateurs differentiels elliptiques degeneres

coordinate rings for the moduli stack of sl2 c quasi parabolic principal bundles on a curve and toric fiber products abstract we continue the program started in manon 2010 m1 to understand the combinatorial commutative algebra of the projective coordinate rings of the moduli stack m c p sl 2 c of quasi parabolic sl 2 c principal bundles on a generic marked projective curve we find general bounds on the degrees of polynomials needed to present these algebras by studying their toric degenerations in particular we show that the square of any effective line bundle on this moduli stack yields a koszul projective coordinate ring this leads us to formalize the properties of the polytopes used in proving our results by constructing a category of polytopes with term orders we show that many of results on the projective coordinate rings of m c p sl 2 c follow from closure properties of this category with respect to fiber products

the equations for the moduli space of n points on the line a central question in invariant theory is that of determining the relations among invariants geometric invariant theory git quotients come with a natural ample line bundle and hence often a natural projective embedding this question translates to determining the equations of the moduli space under this embedding this article deals with one of the most classical quotients the space of ordered points on the projective line we show that under any weighting of the points this quotient is cut out scheme theoretically by a particularly simple set of quadric relations with the single exception of the segre cubic threefold the space of six points with equal weight we also show that the ideal of relations is generated in degree at most four and we give an explicit description of the generators if all the weights are even e g in the case of equal weight for odd n then we show that the ideal of relations is generated by quadrics

riesz transforms and fractional integration for orthogonal expansions on spheres balls and simplices abstract this paper studies the hardy littlewood sobolev hls inequality and the riesz transforms for fractional integration associated to weighted orthogonal polynomial expansions on spheres balls and simplexes with weights being invariant under a general finite reflection group on r d the sharp index for the validity of the hls inequality is determined and the l p boundedness of the riesz transforms is established in particular our results extend a classical inequality of muckenhoupt and stein on conjugate ultraspherical polynomial expansions our idea is based on a new decomposition of the dunkl laplace beltrami operator on the sphere and some sharp asymptotic estimates of the weighted christoffel functions

a two weight weak type inequality for fractional integrals for 1 p q oo 0 a n and w x v x nonnegative weight functions on r 1 we show that the weak type inequality f xw x dx a q if x l v x dx holds for allf 0 if and only if f ta xqw x v x i dx p 1b aw cc for all cubes q in rn here t7 denotes the fractional integral of order a ta f x f i x y i f y dy more generally we can replace ta by any suitable convolution operator with radial kernel decreasing in i x i

topological structures of the sets of composition operators on the bloch spaces we study properties of the topological sets of composition operators on bloch and little bloch spaces in the operator topology

a vector partition function for the multiplicities of slkc abstract we use gelfand tsetlin diagrams to write down the weight multiplicity function for the lie algebra sl k c type ak 1 as a single partition function this allows us to apply known results about partition functions to derive interesting properties of the weight diagrams we relate this description to that of the duistermaat heckman measure from symplectic geometry which gives a large scale limit way to look at multiplicity diagrams we also provide an explanation for why the weight polynomials in the boundary regions of the weight diagrams exhibit a number of linear factors using symplectic geometry we prove that the partition of the permutahedron into domains of polynomiality of the duistermaat heckman function is the same as that for the weight multiplicity function and give an elementary proof of this for sl 4 c a3

gorenstein semigroup algebras of weighted trees and ordered points on the projective line abstract we determine exactly which semigroup algebras of weighted trees are gorenstein these algebras arise as toric degenerations of projective coordinate rings of moduli of weighted points on the projective line as a corollary we find exactly when these families of algebras are gorenstein as well

sharp weighted bounds for fractional integral operators abstract the relationship between the operator norms of fractional integral operators acting on weighted lebesgue spaces and the constant of the weights is investigated sharp bounds are obtained for both the fractional integral operators and the associated fractional maximal functions as an application improved sobolev inequalities are obtained some of the techniques used include a sharp off diagonal version of the extrapolation theorem of rubio de francia and characterizations of two weight norm inequalities

on norms of composition operators acting on bergman spaces for arbitrary composition operators acting on a general bergman space we improve the known lower bound for the norm and also generalize a related recent theorem of d g pokorny and j e shapiro next we obtain a geometric formula for the norms of composition operators with linear fractional symbols thus extending a result of c cowen and p hurst and revealing the meaning of their computation finally we obtain a lower bound for essential norm of an arbitrary composition operator related to the well known criterion of b maccluer and j h shapiro as a corollary norms and essential norms are obtained for certain univalently induced noncompact composition operators in terms of the minimum of the angular derivative of the symbol

on a measure theoretic area formula we review some classical differentiation theorems for measures showing how they can be turned into an integral representation of a borel measure with respect to a fixed caratheodory measure we focus our attention on the case when this measure is the spherical hausdorff measure giving a metric measure area formula our aim is to use certain covering derivatives as generalized densities some consequences for the sub riemannian heisenberg group are also pointed out