integral representation with weights ii division and interpolation let f be a r m matrix of holomorphic functions that is generically surjective we provide explicit integral representation of holomorphic such that f provided that is holomorphic and annihilates a certain residue current with support on the set where f is not surjective we also consider formulas for interpolation as applications we obtain generalizations of various results previously known for the case r 1

estimates of solutions of the h p and bmoa corona problem we prove new sharper estimates of solutions to the h p corona problem in strictly pseudoconvex domains in particular we show that the constant is independent of the number of generators we also obtain sharper estimates for solutions to the bmoa corona problem the proofs also lead to new results about the taylor spectrum of analytic toeplitz operators on h p and bmoa

integral formulas and an explicit version of the fundamental principle soit p c n c m une application polynomiale et on considere le systeme d equations differentielles p j d f t 0 j 1 m ou d i t et f est une fonction lisse dans un domaine convexe borne r n on presente une version explicite qui emploie des formules integrales et des courants residus du principe fondamental de ehrenpreis et palamodov

decomposition of residue currents given a submodule j o r 0 and a free resolution of j one can de ne a certain vector valued residue current whose annihilator is j we make a decomposition of the current with respect to assj that corresponds to a primary decomposition of j as a tool we introduce a class of currents that includes usual residue and principal value currents in particular these currents admit a certain type of restriction to analytic varieties and more generally to constructible sets

explicit serre duality on complex spaces abstract in this paper we use recently developed calculus of residue currents together with integral formulas to give a new explicit analytic realization as well as a new analytic proof of serre duality on any reduced pure n dimensional paracompact complex space x at the core of the paper is the introduction of certain fine sheaves b x n q of currents on x of bidegree n q such that the dolbeault complex b x n becomes in a certain sense a dualizing complex in particular if x is cohen macaulay then b x n is an explicit fine resolution of the grothendieck dualizing sheaf

effective bezout identities in q z1 zn abstract if p sub 1 p sub m are n variate polynomials with integral coefficients and no common zeros in c exp n brownawell has shown in 1986 that there exist q sub 1 q sub m polynomials with integral coefficients and nu is an element of z such that p sub 1 q sub 1 p sub m q sub m nu and max deg q sub j max deg p sub j exp n on the other hand if h logarithm of the largest coefficient of all the p sub j and h sub 1 is the corresponding quantity for the q sub j then there is no sharp estimate of h sub 1 in terms of h and max deg p sub j in this paper we show that when the variety of common zeros at infinity of the p sub j is discrete then essentially we have h sub 1 d exp cn h for an absolute constant c if there were an algorithm to compute the q sub j in d exp cn time one would obtain exactly the above estimate current algorithms require about d exp n squared operations

residue current approach to ehrenpreis malgrange type theorem for linear differential equations with constant coefficients and commensurate time lags abstract we introduce a ring h of partial difference differential operators with constant coefficients initially defined by h glusing lursen for ordinary difference differential operators and investigate its cohomological properties combining this ring theoretic observation with the integral representation technique developed by m andersson we solve a certain type of division with bounds in the last section we deduce from this injectivity properties of various function modules over h as well as the density results of exponential polynomial solutions for partial difference differential equations

integral representation with weights i we describe a new approach to representation formulas for holomorphic functions including the cauchy fantappie leray formula and provide a general method to generate weighted formulas

explicit representation of membership in polynomial ideals we introduce a new division formula on projective space which provides explicit solutions to various polynomial division problems with sharp degree estimates we consider simple examples as the classical macaulay theorem as well as a quite recent result by hickel related to the effective nullstellensatz we also obtain a related result that generalizes max noether s classical af bg theorem

division formulas on projective varieties we introduce a division formula on a possibly singular projective subvariety x of complex projective space mathbb p n which e g provides explicit representations of solutions to various polynomial division problems on the affine part of x especially we consider a global effective version of the briancon skoda huneke theorem

courants du type residuel attaches a une intersection complete abstract we construct in complete intersection s case elementary currents which describe the local ideal and give a decomposition in it for holomorphic function

a dolbeault grothendieck lemma on complex spaces via koppelman formulas let x be a complex space of pure dimension we introduce fine sheaves a x q of 0 q currents which coincides with the sheaves of smooth forms on the regular part of x so that the associated dolbeault complex yields a resolution of the structure sheaf hol x our construction is based on intrinsic and quite explicit semi global koppelman formulas

solvable model for spatiotemporal chaos we show that the dynamical behavior of a coupled map lattice where the individual maps are bernoulli shift maps can be solved analytically for integer couplings we calculate the invariant density of the system and show that it displays a nontrivial spatial behavior we also introduce and calculate a generalized spatiotemporal correlation function

gradient estimates below the duality exponent we show sharp local a priori estimates and regularity results for possibly degenerate non linear elliptic problems with data not lying in the natural dual space we provide a precise non linear potential theoretic analog of classical potential theory results due to adams duke math j 42 765 778 1975 and adams and lewis studia math 74 169 182 1982 concerning morrey spaces imbedding regularity properties for this we introduce a technique allowing for a non local representation of solutions via riesz potentials in turn yielding optimal local estimates simultaneously in both rearrangement and non rearrangement invariant function spaces in fact we also derive sharp estimates in lorentz spaces covering borderline cases which remained open for some while

ideals of smooth functions and residue currents abstract let f f 1 f m be a holomorphic mapping in a neighborhood of the origin in c n we find sufficient condition in terms of residue currents for a smooth function to belong to the ideal in c or ck generated by f if f is a complete intersection the condition is necessary more generally we give a sufficient condition for an element of class c or ck in the koszul complex induced by f to be exact for the proofs we introduce explicit homotopy formulas for the koszul complex induced by f