semigroup kernels poisson bounds and holomorphic functional calculus letlbe the generator of a continuous holomorphic semigroupswhose action is determined by an integral kernelkon a scale of spaceslp x under mild geometric assumptions on x we prove that iflhas a boundedh functional calculus onl2 x andksatisfies bounds typical for the poisson kernel thenlhas a boundedh functional calculus onlp x for eachp 1 moreover if x is of polynomial type andksatisfies second order gaussian bounds we establish criteria forlto have a bounded hormander functional calculus or a bounded davies helffer sjostrand functional calculus

a singular sphere covering inequality uniqueness and symmetry of solutions to singular liouville type equations we derive a singular version of the sphere covering inequality which was recently introduced in gui and moradifam invent math https doi org 10 1007 s00222 018 0820 2 2018 suitable for treating singular liouville type problems with superharmonic weights as an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains as well as new self contained proofs of previously known results such as the uniqueness of spherical convex polytopes first established in luo and tian proc am math soc 116 4 1119 1129 1992 furthermore we derive new symmetry results for the spherical onsager vortex equation

the sphere covering inequality and its applications in this paper we show that the total area of two distinct surfaces with gaussian curvature equal to 1 which are also conformal to the euclidean unit disk with the same conformal factor on the boundary must be at least 4 pi in other words the areas of these surfaces must cover the whole unit sphere after a proper rearrangement we refer to this lower bound of total area as the sphere covering inequality the inequality and its generalizations are applied to a number of open problems related to moser trudinger type inequalities mean field equations and onsager vortices etc and yield optimal results in particular we prove a conjecture proposed by chang and yang acta math 159 3 4 215 259 1987 in the study of nirenberg problem in conformal geometry

mean field equation of liouville type with singular data topological degree we consider the following mean field equation gv h ev mh ev 1 4 j 1n j qj 1 on m where m is a compact riemann surface with volume 1 h is a positive c1 function on m and and j are constants satisfying j 1 in this paper we derive the topological degree counting formula for noncritical values of we also give several applications of this formula including existence of the curvature 1 metric with conic singularities doubly periodic solutions of electroweak theory and a special case of self gravitating strings 2015 wiley periodicals inc

on the mean field type bubbling solutions for chern simons higgs equation abstract this paper is the second part of our comprehensive study on the structure of the solutions for the following chern simons higgs equation 0 1 u 1 e 2 e u 1 e u 4 j 1 n p j in u is doubly periodic on where is a parallelogram in r 2 and e 0 is a small parameter in part 1 29 we proved the non coexistence of different bubbles in the bubbling solutions and obtained an existence result for the chern simons type bubbling solutions under some nearly necessary conditions mean field type bubbling solutions for 0 1 have been constructed in 27 in this paper we shall study two other important issues for the mean field type bubbling solutions the necessary conditions for the existence and the local uniqueness the results in this paper lay the foundation to find the exact number of solutions for 0 1

weighted lp estimates for powers of selfadjoint operators abstract we prove l p and weighted l p estimates for bounded functions of a selfadjoint operator satisfying a pointwise gaussian estimate for its heat kernel as an application we obtain weighted estimates for fractional powers of an electromagnetic schrodinger operator with singular coefficients the proofs are based on a modification of techniques due to hebisch 1990 17 and auscher and martell 2006 4

weighted norm inequalities off diagonal estimates and elliptic operators part i general operator theory and weights this is the first part of a series of four articles in this work we are interested in weighted norm estimates we put the emphasis on two results of different nature one is based on a good inequality with two parameters and the other uses calderon zygmund decomposition these results apply well to singular non integral operators and their commutators with bounded mean oscillation functions singular means that they are of order 0 non integral that they do not have an integral representation by a kernel with size estimates even rough so that they may not be bounded on all lp spaces for 1 p 8 pointwise estimates are then replaced by appropriate localized lp lq estimates we obtain weighted lp estimates for a range of p that is different from 1 8 and isolate the right class of weights in particular we prove an extrapolation theorem a la rubio de francia for such a class and thus vector valued estimates

second order riesz transforms associated with magnetic schrodinger operators abstract let a 1 i a 2 v be a magnetic schrodinger operator on r n where a l loc 2 r n n and 0 v l loc 1 r n we show that l p r n boundedness of l j l k a 1 and v a 1 for some interval of p automatically implies boundedness of the same operators and their commutators on l w p r n for certain muckenhoupt weights w and on the musielak orlicz hardy type spaces

qualitative properties of solutions to some nonlinear elliptic equations in r 2 o introduction elliptic equations in this paper we investigate properties of the solutions to the au r x e u x x r 2 for functions r x which are positive near infinity equations of this kind arise from a variety of situations such as from prescribing gaussian curvature in geometry i 2 and from combustion theory in physics 3 recently a series ofworks have been sought to understand the existence and the qualitative properties of the solutions of ni i 5 and ni cheng 4 1 considered the case where r x is nonpositive mcowen 6 1 and aviles 7 investigated the situation where r x 0 in some order as ixl in our previous paper 1 we consider a special case where r is a constant we proved that the solutions are radially symmetric and hence classified all the solutions in this paper we consider more general functions r x first we obtain the asymptotic behavior of the solution near infinity consequently we prove that all the solutions satisfy an identity which is somewhat of a generalization of the well known kazdan warner condition finally using the asymptotic behavior together with the further development of the method employed in our previous paper i 1 we show that all the solutions are radially symmetric provided r is radially symmetric and nonincreasing this part can be viewed as the completion of 1 throughout this paper we assume that the function r x is positive near infinity in 1 we study the asymptotic behavior of the solution u x of let fl 1 2n r2 r x e dx under some appropriate conditions we show that the solutions approach

polynomial time reductions from multivariate to bi and univariate integral polynomial factorization consider a polynomial f with an arbitrary but fixed number of variables and with integral coefficients we present an algorithm which reduces the problem of finding the irreducible factors of f in polynomial time in the total degree of f and the coefficient lengths of f to factoring a univariate integral polynomial together with a lenstra s h lenstra s and l lovasz polynomial time factorization algorithm for univariate integral polynomials math ann 261 1982 pp 515 534 this algorithm implies the following theorem factoring an integral polynomial with a fixed number of variables into irreducibles except for the constant factors can be accomplished in deterministic polynomial time in the total degree and the size of its coefficients our algorithm can be generalized to factoring multivariate polynomials with coefficients in algebraic number fields and finite fields in polynomial time we also present a different algorithm based on an effective version of a hilbert irreducibility theorem w

maximal inequalities and riesz transform estimates on lp spaces for magnetic schrodinger operators i the paper concerns the magnetic schrodinger operator h a v j 1n 1i xj aj 2 v on rn under certain conditions given in terms of the reverse holder inequality on the magnetic field and the electric potential we prove some lp estimates on the riesz transforms of h and we establish some related maximal inequalities

asymptotic blow up analysis for singular liouville type equations with applications abstract we generalize the pointwise estimates obtained in 2 19 and 34 concerning blow up solutions of the liouville type equation u x 2 w x e u in with r 2 open and bounded 1 and w any lipschitz continuous function which satisfies 0 a w b we focus to the case left open in 2 and 34 where the parameter n whose analysis is much more involved as we need to resolve the difficulty of a genuinely non radial behaviour of blow up solutions in the worst situation there is no chance in general to resolve the profile in the form of a solution of a liouville equation in r 2 instead we need to adopt iterated blow up arguments next we refine our blow up analysis to cover a class of planar liouville type problems see 1 27 1 28 below arising from the study of cosmic strings cfr 28 35 in this context we are able to distinguish between a single blow up radial profile and the case of multiple blow up profiles typical of non radial solutions as a consequence we obtain a radial symmetry result which is interesting in itself but also contributes towards the sharp solvability issue for the planar problem 1 27 1 28

classification of solutions of elliptic equations arising from a gravitational o 3 gauge field model abstract in this paper we study an elliptic equation arising from the self dual maxwell gauged o 3 sigma model coupled with gravity when the parameter equals 1 and there is only one singular source we consider radially symmetric solutions there appear three important constants a positive parameter a representing a scaled gravitational constant a nonnegative integer n 1 representing the total string number and a nonnegative integer n 2 representing the total anti string number the values of the products a n 1 a n 2 0 play a crucial role in classifying radial solutions by using the decay rates of solutions at infinity we provide a complete classification of solutions for all possible values of a n 1 and a n 2 this improves previously known results

riesz transform gaussian bounds and the method of wave equation for an abstract self adjoint operator l and a local operator a we study the boundedness of the riesz transform al on lp for some 0 a very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation we also discuss the relation between the gaussian bounds and the finite speed propagation property using the wave equation methods we obtain a new natural form of the gaussian bounds for the heat kernels for a large class of the generating operators we describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature

on a class of elliptic problems in r symmetry and uniqueness results in the plane r 2 we classify all solutions for an elliptic problem of liouville type involving a radial weight function as a consequence we clarify the origin of the non radially symmetric solutions for the given problem as established by chanillo and kiessling for a more general class of liouville type problems we show that rather than radial symmetry the solutions always inherit the invariance of the problem under inversion with respect to suitable circles this symmetry result is derived with the help of a shrinking sphere method

resolution of a semilinear equation in l1 let n or be a bounded regular open set of n and let x s be a continuous nondecreasing function in s measurable in x such that x 0 0 almost everywhere we solve for f l 1 the problem p u u f in u 0 on in fact for this result instead of assuming that is nondecreasing in s we need only that x s s 0 we deduce an almost necessary and sufficient condition on in order that p has a solution roughly speaking this condition is f v g with g l 1 and v l 1

uniqueness and symmetry results for solutions of a mean field equation on 2 via a new bubbling phenomenon motivated by the study of gauge field vortices we consider a mean field equation on the standard two sphere involving a dirac distribution supported at a point p of s 2 as needed for the applications we show that solutions concentrate exactly at p for some limiting value of a given parameter we use this fact to obtain symmetry about the axis op and uniqueness property for the solution the presence of the dirac measure makes such a task particularly delicate indeed we need to rule out the possibility that after blow up in a suitable scale the solution sequence may admit a double peak profile described in terms of appropriate limiting problems for instance we need to account for the presence of non axially symmetric solutions in such limiting problems in this process we establish a symmetry result about a maximal circle through p and its antipodal point p that applies to more general situations where the full axial symmetry cannot be expected

eigenvalue asymptotics and exponential decay of eigenfunctions for schrodinger operators with magnetic fields we consider the schr6dinger operator with magnetic field h va x 2 v x in rj assuming that v 0 and icurl a i v 1 is locally in certain reverse holder class we study the eigenvalue asymptotics and exponential decay of eigenfunctions introduction consider the schrodinger operator with magnetic field 0 1 h h a v jv a x v x inlr n 3 where i 1 v rn r is the electric scalar potential and a rn rn is the magnetic vector potential under some natural conditions on a and v h admits a self adjoint realization which is still denoted by h in l2 ir1n in this paper we shall study its eigenvalue asymptotics and exponential decay of its eigenfunctions let n a h denote the number of eigenvalues counting multiplicity of h smaller than a or the dimension of the spectral projection for h corresponding to x a in the case a x 0 the following phase space volume estimate due to cwickel lieb rosenbljum sil theorem 9 3 p 95 is classical 0 2 n a a v cnl i x e r n x r n 1h12 v x l where cn is a constant depending on n on the other hand there exist some simple degenerate potentials v x i e potentials which do not tend to infinity in all direction as lxi oo e g v x v xi x2 xn x2x n xn for which the right hand side of 0 2 is infinite and nevertheless a v x has a discrete spectrum furthermore in the case where a x 0 and v x 0 the phase space volume of the set where the symbol of h is less than a n i x e nr x r n e ijaj x l2 l j 1 received by the editors may 19 1995 1991 mathematics subject classification primary 35p20 35j10

parallel algorithms for algebraic problems fast parallel algorithms are presented for the following problems in symbolic manipulation of univariate polynomials computing all entries of the extended euclidean scheme of two polynomials over an arbitrary field gcd and 1cm of many polynomials factoring polynomials over finite fields and the squarefree decomposition of polynomials over fields of characteristic zero and over finite fields for the following estimates assume that the input polynomials have degree at most n and the finite field has p d elements the euclidean algorithm is deterministic and runs in parallel time o log 2 n all the other algorithms are probabilistic las vegas in the general case but when applicable to bf q or bf r they can be implemented deterministically over these fields the algorithms for gcd and lcm use parallel time o log 2 n the factoring algorithm runs in parallel time o log 2 n log 2 d 1 log p the algorithm for squarefree decomposition runs in parallel time o log 2 n

blow up analysis for a cosmic strings equation abstract in this paper we develop a blow up analysis for solutions of an elliptic pde of liouville type over the plane such solutions describe the behavior of cosmic strings parallel in a given direction for a w boson model coupled with einstein s equation we show how the blow up behavior of the solutions is characterized according to the physical parameters involved by new and surprising phenomena for example in some cases after a suitable scaling the blow up profile of the solution is described in terms of an equations that bares a geometrical meaning in the context of the uniformization of the riemann sphere with conical singularities

multiplicity results for the assigned gauss curvature problem in r2 to study the problem of the assigned gauss curvature with conical singularities on riemanian manifolds we consider the liouville equation with a single dirac measure on the two dimensional sphere by a stereographic projection we reduce the problem to a liouville equation on the euclidean plane we prove new multiplicity results for bounded radial solutions which improve on earlier results of c s lin and his collaborators based on numerical computations we also present various conjectures on the number of unbounded solutions using symmetries some multiplicity results for non radial solutions are also stated

existence of topological multi string solutions in abelian gauge field theories in this paper we consider a general form of self dual equations arising from abelian gauge field theories coupled with the einstein equations by applying the super subsolution method we prove that topological multi string solutions exist for any coupling constant which improves previously known results we provide two examples for application the self dual einstein maxwell higgs model and the gravitational maxwell gauged o 3 sigma model

on the best constant in the moser onofri aubin inequality let s2 be the 2 dimensional unit sphere and let j denote the nonlinear functional on the sobolev space h1 s2 defined by j alpha u frac alpha 16 pi int s 2 nabla u 2 d mu 0 frac 1 4 pi int s 2 u d mu 0 rm ln int s 2 e u frac d mu 0 4 pi where d mu 0 sin theta d theta wedge d phi onofri had established that j is non negative on h1 s2 provided 1 in this note we show that if j is restricted to those u in h 1 s 2 that satisfies the aubin condition int s 2 e u x j d mu 0 0 quad rm for all 1 leq j leq 3 then the same inequality continues to hold i e j u 0 whenever alpha geq frac 2 3 epsilon 0 for some epsilon 0 0 the question of chang yang on whether this remains true for all alpha geq frac 1 2 remains open

the existence of non topological multivortex solutions in the relativistic self dual chern simons theory we construct a general type of multivortex solutions of the self duality equations the bogomol nyi equations of 2 1 dimensional relativistic chern simons model with the non topological boundary condition near infinity for such construction we use a perturbation argument around the explicit solutions of the liouville equation

hardy spaces associated with magnetic schrodinger operators on strongly lipschitz domains abstract let p 0 1 be a strongly lipschitz domain in r n and a i a i a v a magnetic schrodinger operator on l 2 satisfying the dirichlet boundary condition where a a 1 a n l loc 2 r n and 0 v l loc 1 in this paper the authors introduce the hardy space h a p by the lusin area function associated with a and establish its equivalent characterization via the non tangential maximal function associated with e t a t 0 as applications the authors obtain the boundedness of the riesz transforms l k a 1 2 k 1 n from h a p to l p for p 0 1 and the fractional integral a from h a p to h a q for 0 p q 1 and n 2 1 p 1 q where l k is the closure of x k i a k in l 2

on a planar liouville type problem in the study of selfgravitating strings abstract motivated by the construction of selfgravitating strings cf yang 2001 1994 22 23 we analyze a liouville type equation on the plane derived in yang 1994 23 we establish sharp existence and uniqueness properties for the corresponding radial solutions we investigate also when the problem allows for non radial solutions

factorization of multivariate polynomials over finite fields we present a probabilistic algorithm that finds the irreducible factors of a bivariate polynomial with coefficients from a finite field in time polynomial in the input size i e in the degree of the polynomial and log cardinality of field the algorithm generalizes to multivariate polynomials and has polynomial running time for densely encoded inputs a deterministic version of the algorithm is also discussed whose running time is polynomial in the degree of the input polynomial and the size of the field

semiclassical analysis for the ground state energy of a schrodinger operator with magnetic wells abstract motivated by a recent paper by montgomery mon we give the asymptotic behavior in the semi classical sense of the ground state energy for the schrodinger operator with a magnetic field we consider the case when the locus of the minima of the intensity of the magnetic field is compact and our study is sharper when this locus is an hypersurface or a finite union of points