uniform boundary harnack principle and generalized triangle property let d be a bounded open subset in rd d 2 and let g denote the green function for d with respect to 2 0 2 d if 2 assume that d satisfies the interior corkscrew condition if 2 i e if g is the classical green function on d assume more restrictively that d is a uniform domain let g g y0 1 for some y0 d based on the uniform boundary harnack principle it is shown that g has the generalized triangle property which states that g z y g z cg x y g x when d z x d z y an intermediate step is the approximation g x y x y dg x g y g a 2 where a is an arbitrary point in a certain set b x y this is discussed in a general setting where d is a dense open subset of a compact metric space satisfying the interior corkscrew condition and g is a quasi symmetric positive numerical function on d d which has locally polynomial decay and satisfies harnack s inequality under these assumptions the uniform boundary harnack principle the approximation for g and the generalized triangle property turn out to be equivalent

conditional gauge and potential theory for the schrodinger operator on etend le theoreme de jauge conditionnelle a des operateurs plus generaux et a des domaines moins reguliers on obtient des resultats de theorie du potentiel pour l equation de schrodinger

absolutely trianalytic tori in the generalized kummer variety we prove that a generic complex deformation of a generalized kummer variety contains no complex analytic tori

a class of path valued markov processes and its applications to superprocesses let s be a continuous markov process satisfying certain regularity assumptions we introduce a path valued strong markov process associated with s which is closely related to the so called superprocess with spatial motion s in particular a subseth of the state space of s intersects the range of the superprocess if and only if the set of paths that hith is not polar for the path valued process the latter property can be investigated using the tools of the potential theory of symmetric markov processes a set is not polar if and only if it supports a measure of finite energy the same approach can be applied to study sets that are polar for the graph of the superprocess in the special case when s is a diffusion process we recover certain results recently obtained by dynkin

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

topological invariants of o grady s six dimensional irreducible symplectic variety we study o grady examples of irreducible symplectic varieties we establish that both of them can be deformed into lagrangian fibrations we analyze in detail the topology of the six dimensional example in particular we compute its euler characteristic and determine its beauville form

stationary navier stokes equations with critically singular external forces existence and stability results abstract we show the unique existence of solutions to stationary navier stokes equations with small singular external forces belonging to a critical space to the best of our knowledge this is the largest critical space that is available up until now for this kind of existence this result can be viewed as the stationary counterpart of the existence result obtained by h koch and d tataru for the free non stationary navier stokes equations with initial data in bmo 1 the stability of the stationary solutions in such spaces is also obtained by a series of sharp estimates for resolvents of a singularly perturbed operator and the corresponding semigroup

weighted norm inequalities for potentials with applications to schrodinger operators fourier transforms and carleson measures on donne une nouvelle caracterisation de l inegalite de trace pour des operateurs potentiel et on l utilise pour affiner des resultats sur la distribution des valeurs propres des operateurs de schrodinger

nonlinear equations and weighted norm inequalities we study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities in particular we obtain explicit criteria for the solvability of the dirichlet problem u v u w u 0 on u 0 on on a regular domain in rn in the superlinear case q 1 the coefficients v w are arbitrary positive measurable functions or measures on we also consider more general nonlinear differential and integral equations and study the spaces of coefficients and solutions naturally associated with these problems as well as the corresponding capacities our characterizations of the existence of positive solutions take into account the interplay between v w and the corresponding green s kernel they are not only sufficient but also necessary and are established without any a priori regularity assumptions on v and w we also obtain sharp two sided estimates of solutions up to the boundary some of our results are new even if v 1 and is a ball or half space the corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so called 3g inequality by an elementary integration by parts argument this also gives a new unified proof for some classical inequalities including the carleson measure theorem for poisson integrals and trace inequalities for riesz potentials and green potentials

the 2 factoriality of the o grady moduli spaces the aim of this work is to show that the moduli space m10 introduced by o grady is a 2 factorial variety namely m10 is the moduli space of semistable sheaves with mukai vector v 2 0 2 in h ev x mathbb z on a projective k3 surface x as a corollary to our construction we show that the donaldson morphism gives a hodge isometry between v perp sublattice of the mukai lattice of x and its image in h 2 widetilde m 10 mathbb z lattice with respect to the beauville form of the 10 dimensional irreducible symplectic manifold widetilde m 10 obtained as symplectic resolution of m10 similar results are shown for the moduli space m6 introduced by o grady to produce its 6 dimensional example of irreducible symplectic variety

weighted trace inequalities for fractional integrals and applications to semilinear equations we show that the two weight trace inequality for the riesz potentials i i f lp w c f lp v holds if i w lp loc and i i w p ci w a e here w and v are non negative weight functions on rn and v1 p the converse is also also true under some mild restrictions on w and v we also consider more general inequalities for measures which are not necessarily absolutely continuous with respect to lebesgue measure in contrast to the known characterizations of the trace inequality this pointwise condition is stated explicitly in terms of potentials of w and rather than measures of some subsets of rn applications to the problem of the existence of positive solutions for the semilinear elliptic equation u x uq w x 1 q on rn are given

superlinear elliptic inequalities on manifolds abstract let m be a complete non compact riemannian manifold and let be a radon measure on m we study the problem of existence or non existence of positive solutions to a semilinear elliptic inequality u u q in m where q 1 we obtain necessary and sufficient criteria for existence of positive solutions in terms of green function of in particular explicit necessary and sufficient conditions are given when m has nonnegative ricci curvature everywhere in m or more generally when green s function satisfies the 3g inequality

blow up and global existence of solutions to an inhomogeneous parabolic system abstract we study the critical phenomenon of a weakly coupled inhomogeneous parabolic system which is largely unexplored even though the homogeneous problem has been widely studied the conditions on the powers of the nonlinear terms are new compared with the well known homogeneous case the method appears to be different from the standard ones in the literature an application to the lame emden system is also given

deformation of the o grady moduli spaces in this paper we study moduli spaces of sheaves on an abelian or projective k3 surface if s is a k3 v 2w is a mukai vector on s where w is primitive and w 2 2 and h is a v generic polarization on s then the moduli space m v of h semistable sheaves on s whose mukai vector is v admits a symplectic resolution m over tilde v a particular case is the 10 dimensional o grady example m over tilde 10 of an irreducible symplectic manifold we show that m over tilde v is an irreducible symplectic manifold which is deformation equivalent to m over tilde 10 and that h 2 m v z is hodge isometric to the sublattice v perpendicular to of the mukai lattice of s similar results are shown when s is an abelian surface

on the euler characteristic of generalized kummer varieties we apply the yau zaslow beauville method to compute the euler characteristic of the generalized kummer varieties attached to a complex abelian surface a a calculation also done by gottsche and soergel by a different method the argument is very geometric given a nondivisible ample line bundle l of degree n on a we construct a projective symplectic variety j with a lagrangian fibration whose euler characteristic is n times the number of genus 2 curves in l to wit n 2 m n m using the mukai fourier transform a degeneration to the case when a is a product of elliptic curves and a result of huybrechts we prove that j and the generalized 2 n 2 dimensional kummer variety of a are diffeomorphic hence have the same euler characteristic

solvability conditions for some semi linear parabolic equations consider the cauchy problem of the semi linear parabolic equation u tu vup w 0 p 1 in rn 0 t t 0 u x 0 u0 x 0 and u0 v w lloc1 rn we establish a necessary condition on the nonlinear potential v v x so that the above has a positive solution for some w w x 0 u0 u0 x 0 an application to the case when v is the important inverse square potential is also given

on the beauville form of the known irreducible symplectic varieties we study the global geometry of o grady s ten dimensional irreducible symplectic variety we determine its second betti number its beauville form and its fujiki constant

green function for schrodinger operator and conditioned feynman kac gauge consider the schrodinger operator h 2 q in a bounded c1 1 domain d with q klocd d q satisfies the condition sup spec 2 q d v x y g x y e y x exp 0 t q x t d t x y d where xt is the conditioned brownian motion starting at x and ending at y with the life time t and v g can be extended to a continuous and positive function on d d specifically we give the comparison theorem there exists a constant c c d q 0 such that 1 c g x y v x y c g x y x y d another result is that v nz x z x d z d exists and represents the poisson kernel for the dirichlet problem corresponding to schrodinger operator h

the ptolemy and zbăganu constants of normed spaces abstract in every inner product space h the ptolemy inequality holds the product of the diagonals of a quadrilateral is less than or equal to the sum of the products of the opposite sides in other words x y z w x z y w z y x w for any points w x y z in h it is known that for each normed space x there exists a constant c such that for any w x y z x we have x y z w c x z y w z y x w the smallest such c is called the ptolemy constant of x and is denoted by c p x we study the relationships between this constant and the geometry of the space x and hence with metric fixed point theory in particular we relate the ptolemy constant c p to the zb ganu constant c z and prove that if x is a banach space with c z x 1 3 2 then x has uniform normal structure and therefore the fixed point property for nonexpansive mappings we derive general lower and upper bounds for both c p and c z and calculate the precise values of these two constants for several normed spaces we also present a number of conjectures and open problems

sobolev multipliers maximal functions and parabolic equations with a quadratic nonlinearity abstract we develop a general framework to describe global mild solutions to a cauchy problem with small initial values concerning a general class of semilinear parabolic equations with a quadratic nonlinearity this class includes the navier stokes equations the subcritical dissipative quasi geostrophic equation and the parabolic elliptic keller segel system

numerical calculation of the density of prime numbers with a given least primitive root in this paper the densities d i of prime numbers p having the least primitive root g p i where i is equal to one of the initial positive integers less than 32 have been numerically calculated the computations were carried out under the assumption of the generalised riemann hypothesis the results of these computations were compared with the results of numerical frequency estimations

singular quasilinear and hessian equations and inequalities we solve the existence problem in the renormalized or viscosity sense and obtain global pointwise estimates of solutions for quasilinear and hessian equations with measure coefficients and data including the following model problems pu uq fk u uq u 0 on rn or on a bounded domain rn here p is the p laplacian defined by pu div u u p 2 and fk u is the k hessian i e the sum of the k k principal minors of the hessian matrix d2u k 1 2 n and are general nonnegative measurable functions or measures on

a_p weights and quantitative estimates in the schrodinger setting suppose l delta v is a schrodinger operator on mathbb r n with a potential v belonging to certain reverse holder class rh sigma with sigma ge n 2 the aim of this paper is to study the a p weights associated to l denoted by a p l which is a larger class than the classical muckenhoupt a p weights we first prove the quantitative a p l bound for the maximal function and the maximal heat semigroup associated to l then we further provide the quantitative a p q l bound for the fractional integral operator associated to l we point out that all these quantitative bounds are known before in terms of the classical a p q constant however since a p q subset a p q l the a p q l constants are smaller than a p q constant hence our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to l next we prove two weight inequalities for the fractional integral operator these have been unknown up to this point finally we also have a study on the exp log link between a p l and bmo l the bmo space associated with l and show that for w in a p l log w is in bmo l and that the reverse is not true in general

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

positive operators and maximal operators in a filtered measure space abstract in a filtered measure space a characterization of weights for which the trace inequality of a positive operator holds is given by the use of discrete wolff s potential a refinement of the carleson embedding theorem is also introduced sawyer type characterization of weights for which a two weight norm inequality for a generalized doob s maximal operator holds is established by an application of our carleson embedding theorem moreover hytonen perez type one weight norm estimate for doob s maximal operator is obtained by the use of our two weight characterization

on the existence of weak solutions of semilinear elliptic equations and systems with hardy potentials abstract let r n n 3 be a bounded c 2 domain and x dist x put l 2 with 0 in this paper we provide various necessary and sufficient conditions for the existence of weak solutions to l u u p in u on where 0 p 0 and are measures on and respectively we then establish existence results for the system l u v p in l v u p in u v on where 1 p 0 p 0 and are measures on and are measures on we also deal with elliptic systems where the nonlinearities are more general

birational geometry of o grady s six dimensional example over the donaldson uhlenbeck compactification we determine the birational geometry of o grady s six dimensional example over the donaldson uhlenbeck compactification by looking at the locus of non locally free sheaves on the relevant moduli space

regularity and capacity for the fractional dissipative operator abstract this paper is devoted to exploring some analytic geometric properties of the regularity and capacity associated with the so called fractional dissipative operator t naturally establishing a diagonally sharp hausdorff dimension estimate for the blow up set of a weak solution to the fractional dissipative equation t u t x f t x subject to u 0 x 0 the methods used in this paper rely on effectively controlling the time dependent non local kernels and potentials with fractional order 0 1 dual representation of the capacity and frostman type theorem from geometric measure theory

families of lagrangian fibrations on hyperkahler manifolds abstract a holomorphic lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with lagrangian fibers it is known due to huybrechts that a given compact manifold admits only finitely many holomorphic symplectic structures up to deformation we prove that a given compact simple hyperkahler manifold with b 2 7 admits only finitely many deformation types of holomorphic lagrangian fibrations we also prove that all known hyperkahler manifolds are never kobayashi hyperbolic

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

maximum and anti maximum principlesand eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations in this paper we discuss some new results concerning perturbation theory for second order elliptic partial differential equations related to positivity properties of such equations we continue the study of some different notions of small perturbations and discuss their relations to comparisons of green s functions refined maximum and anti maximum principles ground state and the decay of eigenfunctions in particular we show that if v is a positive function which is a semismall perturbation of a subcritical schrodinger operator h defined on a domain omega subset mathbb r d and phi k k geq 0 are the dirichlet eigenfunctions of the equation hu lambda vu then for any k geq 0 the function phi k phi 0 is bounded and has a continuous extension up to the martin boundary of the pair omega h where phi 0 is the ground state of h with a principal eigenvalue lambda 0

an elliptic semilinear equation with source term and boundary measure data the supercritical case we give new criteria for the existence of weak solutions to equation with source term begin align delta u u q text in omega u sigma text on partial omega end align where q 1 omega is a either a bounded smooth domain or mathbb r n and sigma in mathfrak m partial omega is a nonnegative radon measure on partial omega in particular one of the criteria is expressed in terms of some bessel capacities on partial omega we also give a sufficient condition for the existence of weak solutions to equation with source mixed term begin align delta u u q 1 1 u nabla u q 2 text in omega u sigma text on partial omega end align where q 1 q 2 geq 0 q 1 q 2 1 q 2 2 sigma in mathfrak m partial omega is a radon measure on partial omega

a priori estimates and the representation formula for all positive solutions to a semilinear parabolic problem we establish some a priori estimates and a representation formula for all classical positive solutions to a semilinear parabolic problem up to now some uncertainties about the formula have been circulating

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

picard number of the generic fiber of an abelian fibered hyperkahler manifold we shall show that the picard number of the generic fiber of an abelian fibered hyperkahler manifold over the projective space is always one we then give a few applications for the mordell weil group in particular by deforming o grady s 10 dimensional manifold we construct an abelian fibered hyperkahler manifold of mordell weil rank 20 which is the maximum possible among all known ones

weighted nonlinear potential theory the potential theoretic idea of the thinness of a set at a given point is extended to the weighted nonlinear potential theoretic setting the weights representing in general singularities degeneracies and conditions on these weights are given that guarantee when two such notions are equivalent at the given point when applied to questions of boundary regularity for solutions to degenerate elliptic second order partial differential equations in bounded domains this result relates the boundary wiener criterion for one operator to that of another and in the linear case gives conditions for boundary regular points to be the same for various operators the methods also yield two weight norm inequalities for riesz potentials j llq p 1 p q oo which at least in the first order case a 1 have found some use in a number of places in analysis

moduli spaces of stable sheaves on abelian surfaces in this paper we consider basic problems on moduli spaces of stable sheaves on abelian surfaces our main assumption is the primitivity of the associated mukai vector we determine the deformation types albanese maps bogomolov factors and their weight 2 hodge structures we also discuss the deformation types of moduli spaces of stable sheaves on k3 surfaces

on the subordinate killed b m in bounded domains and existence results for nonlinear fractional dirichlet problems we take up in this paper the existence of positive continuous solutions for some nonlinear boundary value problems with fractional differential equation based on the fractional laplacian delta d frac alpha 2 associated to the subordinate killed brownian motion process z alpha d in a bounded c 1 1 domain d our arguments are based on potential theory tools on z alpha d and properties of an appropriate kato class of functions k d

weighted lp inequalities for multi parameter riesz type potentials and strong fractional maximal operators we prove weighted lp inequalities for multi parameter riesz type potentials strong fractional maximal operators and their dyadic counterparts our proofs avoid the good inequalities used earlier in the rm case and are based on our integrated multi parameter summation by parts lemma that might be of independent interest

gradient estimates via linear and nonlinear potentials we prove new potential and nonlinear potential pointwise gradient estimates for solutions to measure data problems involving possibly degenerate quasilinear operators whose prototype is given by pu in particular no matter the nonlinearity of the equations considered we show that in the case p 2 a pointwise gradient estimate is possible using standard linear riesz potentials the proof is based on the identification of a natural quantity that on one hand respects the natural scaling of the problem and on the other allows to encode the weaker coercivity properties of the operators considered in the case p 2 in the case p 2 we prove a new gradient estimate employing nonlinear potentials of wolff type

singular symplectic moduli spaces moduli spaces of semistable sheaves on a k3 or abelian surface with respect to a general ample divisor are shown to be locally factorial with the exception of symmetric products of a k3 or abelian surface and the class of moduli spaces found by o grady consequently since singular moduli space that do not belong to these exceptional cases have singularities in codimension 4 they do no admit projective symplectic resolutions

counting rational curves on text k3 surfaces the aim of these notes is to explain the remarkable formula found by yau and zaslow y z to express the number of rational curves on a k3 surface projective k3 surfaces fall into countably many families fg g 1 a surface in fg admits a gdimensional linear system of curves of genus g a naive count of constants suggests that such a system will contain a positive number say n g of rational highly singular curves the formula is

a new prime p for which the least primitive root mod p and the least primitive root mod p 2 are not equal with the aid of a computer network we have performed a search for primes p 10 12 and revealed a new prime p 6692367337 for which its least primitive root mod p and its least primitive root mod p 2 are not equal

comments on search procedures for primitive roots let p be an odd prime assuming the extended riemann hypothesis we show how to construct o logp 4 loglogp 3 residues modulo p one of which must be a primitive root in deterministic polynomial time granting some well known character sum bounds the proof is elementary leading to an explicit algorithm

a new critical phenomenon for semilinear parabolic problems abstract we study the cauchy problem of the inhomogeneous semilinear parabolic equations u up ut w 0 onmn 0 with initial valueu0 0 wheremnis a riemannian manifold with possibly nonnegative ricci curvature there is an exponentp which is critical in the following sense when 1 p the problem has a global positive solution for somew w x 0 and someu0 0