a bie method for a nonlinear bvp abstract a nonlinear boundary value problem bvp governed by laplace s equation with a nonlinear boundary condition is considered this paper is concerned with the numerical solvability of a nonlinear boundary integral equation bie obtained by reformulating the nonlinear bvp we give a simple alternative to the standard collocation method for the nonlinear bie which requires less computational effort the method is a bie analogue of the product approximation method used for solving some nonlinear problems by the finite element method we obtain superconvergence of approximate solutions introduced in this paper

on weak type inequalities for dyadic maximal functions we obtain sharp estimates for the localized distribution function of the dyadic maximal function m d given the local l1 norms of and of g where g is a convex increasing function such that g x x as x using this we obtain sharp refined weak type estimates for the dyadic maximal operator

dyadic like maximal operators on integrable functions and bellman functions related to kolmogorov s inequality for each q 1 we precisely evaluate the main bellman functions associated with the behavior of dyadic maximal operators on n on integrable functions actually we do that in the more general setting of tree like maximal operators these are related to and refine the corresponding kolmogorov inequality which we show is actually sharp for this we use the effective linearization introduced by the first author in 2005 for such maximal operators on an adequate set of functions

the distribution function inequality and products of toeplitz operators and hankel operators abstract in this paper we study hankel operators and toeplitz operators through a distribution function inequality on the lusin area integral function and the littlewood paley theory a sufficient condition and a necessary condition are obtained for the boundedness of the product of two hankel operators they lead to a way to approach sarason s conjecture on products of toeplitz operators and shed light on the compactness of the product of hankel operators an elementary necessary and sufficient condition for the product of two toeplitz operators to be a compact perturbation of a toeplitz operator is obtained moreover a necessary condition is given for the product of hankel operators to be in the commutator ideal of the algebra generated by the toeplitz operators with symbols in a sarason algebra

the sharp bound for the hilbert transform on weighted lebesgue spaces in terms of the classical ap characteristic we show that the norm of the hilbert transform as an operator in the weighted space l p r for 2 p oo is bounded by a constant multiple of the first power of the classical ap characteristic of this result is sharp we also prove a bilinear imbedding theorem with simple conditions

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

bilinear embedding for real elliptic differential operators in divergence form with potentials we present a simple bellman function proof of a bilinear estimate for elliptic operators in divergence form with real coefficients and with nonnegative potentials the constants are dimension free the p range of applicability of this estimate is 1 for any real accretive nonsymmetric matrix a of coefficients

convex body domination and weighted estimates with matrix weights abstract we introduce the so called convex body valued sparse operators which generalize the notion of sparse operators to the case of spaces of vector valued functions we prove that calderon zygmund operators as well as haar shifts and paraproducts can be dominated by such operators by estimating sparse operators we obtain weighted estimates with matrix weights we get two weight a 2 a estimates that in the one weight case give us the estimate t l 2 w l 2 w c w a 2 1 2 w a c w a 2 3 2 where t is either calderon zygmund operator with modulus of continuity satisfying the dini condition or a haar shift or a paraproduct

sharp general local estimates for dyadic like maximal operators and related bellman functions for each 1q p we precisely evaluate the main bellman functions associated with the local lp lq estimates of the dyadic maximal operator on actually we do that in the more general setting of tree like maximal operators and with respect to general convex and increasing growth functions we prove that these bellman functions equal to analogous extremal problems for the hardy operator which can be viewed as a symmetrization principle for such operators under certain mild conditions on the growth functions we show that for the latter extremals exist although for the original bellman functions do not and analyzing them we give a determination of the corresponding bellman function

two weight l p estimates for paraproducts in non homogeneous settings we give a necessary and sufficient condition for the two weight l p l p estimates for paraproducts in non homogeneous settings 1 1 p we are mainly interested in the case p 2 p 2 since the case p 2 p 2 is a well known and easy corollary of the carleson embedding theorem the necessary and sufficient condition is given in terms of testing conditions of sawyer type for p 2 p 2 only one direct testing condition is required but for p 2 p 2 both direct and adjoint testing conditions are needed an interesting feature is that the adjoint testing condition is that it is a testing condition not for the adjoint of the paraproduct but for some auxiliary operator

burkholder integrals morrey s problem and quasiconformal mappings inspired by morrey s problem on rank one convex functionals and the burkholder integrals of his martingale theory we find that the burkholder functionals bp p 2 are quasiconcave when tested on deformations of identity f id c with bp df x 0 pointwise or equivalently deformations such that df 6 p p 2jf in particular this holds in explicit neighbourhoods of the identity map among the many immediate consequences this gives the strongest possible l estimates for the gradient of a principal solution to the beltrami equation fz z fz for any p in the critical interval 2 6 p 6 1 1 f examples of local maxima lacking symmetry manifest the intricate nature of the problem

fast multilevel augmentation methods for nonlinear boundary integral equations we propose a fast algorithm for the solution of the nonlinear boundary integral equation resulting from a reformulation of a boundary value problem of the laplace equation with nonlinear boundary conditions the fast algorithm is developed by using the multilevel augmentation method introduced recently by chen wu and xu for general nonlinear integral equations in conjunction with a matrix truncation strategy and an error control technique of numerical integrations for integrals appeared in the process of solving the equation we prove that the proposed algorithm has an optimal convergence order up to a logarithmic factor and a nearly linear computational complexity order measured in the number of multiplications and functional evaluations numerical experiments are presented to demonstrate its approximation accuracy and computational efficiency verifying the theoretical estimates and to compare performance of the proposed algorithm with that of the atkinson and chandler algorithm

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

riesz transforms and spectral multipliers of the hodge laguerre operator abstract on r d endowed with the laguerre probability measure we define a hodge laguerre operator l acting on differential forms here is the laguerre exterior differentiation operator defined as the classical exterior differential except that the partial derivatives x i are replaced by the laguerre derivatives x i x i and is the adjoint of with respect to inner product on forms defined by the euclidean structure and the laguerre measure we prove dimension free bounds on l p 1 p for the riesz transforms l 1 2 and l 1 2 as applications we prove the strong hodge de rahm kodaira decomposition for forms in l p and deduce existence and regularity results for the solutions of the hodge and de rham equations in l p we also prove that for suitable functions m the operator m l is bounded on l p 1 p

invertible toeplitz products we will discuss invertibility of toeplitz products tftg for analytic f and g on the bergman space and the hardy space we will furthermore describe when these toeplitz products are fredholm

annihilation of boundary singularities via suitable green s functions abstract a numerical integral equation method is presented for the accurate solution of viscous flow problems in which there are inherent boundary singularities where applicable this method has the advantage over existing singularity subtraction and singularity incorporation methods since natural boundary conditions can be employed without any special form of pre or post processing results are presented for the benchmark problem of stokes flow in a lid driven cavity

solution of nonlinear elliptic equations with boundary singularities by an integral equation method a boundary integral equation bie formulation is presented for the numerical solution of certain two dimensional nonlinear elliptic equations subject to nonlinear boundary conditions by applying the kirchoff transformation all nonlinear aspects are first transferred to the boundary of the solution domain then the accurate solution of problems in which there are boundary singularities is demonstrated by including the analytic nature of the singular solution in only those regions nearest the singularity because of this only a minor modification of the classical nonlinear bie is required and this results in a substantial improvement in the accuracy of the numerical results throughout the entire solution domain

the treatment of singularities in the calculation of stress intensity factors using the boundary integral equation method abstract a version of the boundary integral equation method applicable to elliptic boundary value problems for domains with curved boundaries and mixed boundary conditions dirichlet neumann and third type is described some problems arising when the region includes a slit crack are discussed and a number of novel ways for dealing with the resulting singularity are presented methods for computing the stress intensity factor sif are described and their numerical performance is assessed in three of these methods the sif appears directly as one of the unknowns in the computation or a simple linear combination of these unknowns which is a most welcome feature since it avoids the uncertainties associated with the commonly employed graphical and extrapolation techniques

classifications and existence of positive solutions of higher order nonlinear iterative functional differential equations classification schemes for eventually positive solutions of a class of higher order nonlinear iterative functional differential equations of the form r t x m 1 t f t x t x t x t 0 in the case when ds r s are given in terms of their magnitudes and necessary as well as sufficient conditions for the existence of these positive solutions are also provided

local analytic solutions of a functional differential equation αz βx z x az bx z in this paper we are concerned with the existence of analytic solution of a functional differential equation email protected z x az bx z where a b a b are four complex numbers we first discuss the existence of analytic solutions for some special cases of the above equation then by reducing the equation with the schroder transformation to the another functional equation with proportional delay an existence theorem is established for analytic solutions of the original equation for the constant l given in the schroder transformation we discuss the case 0 l 1 and l on the unit circle s 1 i e l 1 we study l is at resonance i e at a root of the unity and l is near resonance under the brjuno condition

on the boundary element method for some nonlinear boundary value problems here we analyse the boundary element galerkin method for two dimensional nonlinear boundary value problems governed by the laplacian in an interior or exterior domain and by highly nonlinear boundary conditions the underlying boundary integral operator here can be decomposed into the sum of a monotoneous hammerstein operator and a compact mapping we show stability and convergence by using leray schauder fixed point arguments due to petryshyn and necas

local analytic solutions of a functional differential equation with a deviating argument depending on the state derivative near resonance this paper is concerned with a functional differential equation x n z 1x az bx z with a deviating argument depending on the state derivative where a b are two complex numbers we first discuss the existence of analytic solutions for some special cases of the above equation then by reducing the equation with the schroder transformation to another functional differential equation with proportional delay an existence theorem is established for analytic solutions of the original equation for the constant l given in the schroder transformation besides the case 0 l 1 we focus on those l on the unit circle s 1 i e l 1 we discuss not only those l s at resonance i e at a root of the unity but also those l near resonance under the brjuno condition

sharp weighted estimates for classical operators abstract we give a general method based on dyadic calderon zygmund theory to prove sharp one and two weight norm inequalities for some of the classical operators of harmonic analysis the hilbert and riesz transforms the beurling ahlfors operator the maximal singular integrals associated to these operators the dyadic square function and the vector valued maximal operator in the one weight case we prove the sharp dependence on the a p constant by finding the best value for the exponent p such that t f l p w c n t w a p p f l p w for the hilbert transform the riesz transforms and the beurling ahlfors operator the sharp value of p was found by petermichl and volberg 2007 2008 2002 47 48 49 their proofs used approximations by the dyadic haar shift operators bellman function techniques and two weight norm inequalities our proofs again depend on dyadic approximation but avoid bellman functions and two weight norm inequalities we instead use a recent result due to a lerner 2010 34 to estimate the oscillation of dyadic operators by applying this we get a straightforward proof of the sharp dependence on the a p constant for any operator that can be approximated by haar shift operators in particular we provide a unified approach for the hilbert and riesz transforms the beurling ahlfors operator and their corresponding maximal singular integrals dyadic paraproducts and haar multipliers furthermore we completely solve the open problem of sharp dependence for the dyadic square functions and vector valued hardy littlewood maximal function in the two weight case we use the very same techniques to prove sharp results in the scale of a p bump conditions for the singular integrals considered above we show they map l p v into l p u 1 p if the pair u v satisfies sup q u 1 p a q v 1 p b q where a b p and b b p are orlicz functions this condition is sharp furthermore this condition characterizes in the scale of these a p bump conditions the corresponding two weight norm inequality for the hardy littlewood maximal operator m and its dual i e m l p v l p u and m l p u 1 p l p v 1 p muckenhoupt and wheeden conjectured that these two inequalities for m are sufficient for the hilbert transform to be bounded from l p v into l p u thus in the scale of a p bump conditions we prove their conjecture we prove similar sharp two weight results for the dyadic square function and the vector valued maximal operator

an estimate for weighted hilbert transform via square functions we show that the norm of the hilbert transform as an operator on the weighted space l 2 w is bounded by a constant multiple of the 3 2 2 power of the a 2 constant of w in other words by c sup i r i 1 i 3 2 we also give a short proof for sharp upper and lower bounds for the dyadic square function

wavelets and the angle between past and future the main result the riesz projectionp or equivalently hilbert transformt is bounded in the weighted spacel2 w wherewis a matrix valued weight if and only ifsupi 1 i iw 1 2 1 i iw 1 1 2 wheresupremumis taken over all intervalsi motivation for this problem comes from stationary processes riesz projection is bounded means the angle between past and future of a stationary process with spectral measurewis nonzero in the scalar case the result is the well known hunt muckenhoupt wheeden theorem the main step in our proof is to show that a vector haar system forms an unconditional basis inl2 w as a byproduct of our approach we get some new results about bases of wavelets in weighted spaces in both scalar and vector valued cases

two weight hilbert transform and lipschitz property of jacobi matrices associated to hyperbolic polynomials abstract we are going to prove a lipschitz property of jacobi matrices built by orthogonalizing polynomials with respect to measures in the orbit of classical perron frobenius ruelle operators associated to hyperbolic polynomial dynamics this lipschitz estimate will not depend on the dimension of the jacobi matrix it is obtained using some sufficient conditions for two weight boundedness of the hilbert transform it has been proved in f peherstorfer a volberg p yuditskii limit periodic jacobi matrices with prescribed p adic hull and a singular continuous spectrum math res lett 13 2 3 2006 215 230 for all polynomials with sufficiently big hyperbolicity and in the most symmetric case t 0 that the lipschitz estimate becomes exponentially better when the dimension of the jacobi matrix grows this allows us to get for such polynomials the solution of a problem of bellissard in other words to prove the limit periodicity of the limit jacobi matrix we suggest a scheme how to approach bellissard s problem for all hyperbolic dynamics by uniting the methods of the present paper and those of f peherstorfer a volberg p yuditskii limit periodic jacobi matrices with prescribed p adic hull and a singular continuous spectrum math res lett 13 2 3 2006 215 230 on the other hand the nearness of jacobi matrices under consideration in operator norm implies a certain nearness of their canonical spectral measures one can notice that this last claim just gives us the classical commutative perron frobenius ruelle theorem it is concerned exactly with the nearness of such measures in particular in many situations we can see that the classical perron frobenius ruelle theorem is a corollary of a certain non commutative observation concerning the quantitative nearness of pertinent jacobi matrices in operator norm

local analytic solution of a second order functional differential equation with a state derivative dependent delay abstract this paper is concerned with a second order functional differential equation x z 1 x az bx z with a state derivative dependent delay by reducing the equation to another functional differential equation with proportional delay an existence theorem is established for analytic solutions of the original equation and systematic methods for deriving explicit solutions are also given we not only prove the convergence of the formal solution under the diophantine condition i e eigenvalues is far from unit roots but also make progresses without the diophantine condition i e the convergence is equivalent to the well known small divisor problems

the functional differential equation x t x x t abstract a classification of the solutions of the functional differential equation x t x x t is given and it is proved that every solution either vanishes identically or is strictly monotonie for monotonically increasing solutions existence and uniqueness of the solution x are proved with the condition x t0 x0 where t0 x0 is any given pair of reals in some specified subset of r 2 every monotonically increasing solution is thus obtained it is analytic and depends analytically on t0 and x0 only for t0 x0 1 is the question of analyticity still open

bellman functions and dimension free lp estimates for the riesz transforms in bessel settings abstract in this article we prove dimension free l p boundedness of riesz transforms associated with a bessel differential operator we obtain explicit estimates of the l p norms for the bessel riesz transforms in terms of p establishing a linear behavior with respect to p we use the bellman function technique to prove a bilinear dimension free inequality involving poisson semigroups defined through this bessel operator

the bellman functions of dyadic like maximal operators and related inequalities abstract for each p 1 we precisely evaluate the main bellman functions associated with the dyadic maximal operator on r n and the dyadic carleson imbedding theorem actually we do that in the more general setting of tree like maximal operators these provide refinements of the sharp lp inequalities for those operators for this we introduce an effective linearization for such maximal operators on an adequate set of functions

failure of the matrix weighted bilinear carleson embedding theorem abstract we prove failure of the natural formulation of a matrix weighted bilinear carleson embedding theorem featuring a matrix valued carleson sequence as well as products of norms for the embedding we show that assuming an a 2 weight is also not sufficient indeed a uniform bound on the conditioning number of the matrix weight is necessary and sufficient to get the bilinear embedding we prove the optimal dependence of the embedding on this quantity we show that any improvement of a recent matrix weighted bilinear embedding featuring a scalar carleson sequence and inner products instead of norms must fail in particular replacing the scalar sequence by a matrix sequence results in failure even when maintaining the formulation using inner products any formulation using norms even in the presence of a scalar carleson sequence must fail as a positive result we prove the so called matrix weighted redundancy condition in full generality

analytic solutions of an iterative functional differential equation this paper is concerned with a functional differential equation x z 1 x az bx z where a b are two complex numbers by constructing a convergent power series solution y z of a auxiliary equation of the form b 2y z y m 2z ay mz my mz ay z analytic solutions of the form 1b y my 1 z az for the original differential equation are obtained

the theory of weights and the dirichlet problem for elliptic equations in this work we consider questions from the area of partial differential equations and the related questions from harmonic analysis specifically we will study the dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients roughly speaking we consider the following question suppose that we are given two operators lo and li and that we have good estimates for the dirichlet problem for lo good in this context could mean several different things for instance it might mean that there exists p 1 p oo so that the dirichlet problem is solvable in a suitable sense with lp data what are the optimal conditions on the difference between the coefficients of lo and l1 which guarantee that the dirichlet problem for li also has good estimates this question has been extensively studied recently cf f j k r f and d4 our work uses in a crucial way the results in r f the main results in d4 and r f are proved using an ingenious differential inequality for a family of harmonic measures introduced in d4 another of our goals in this article is to give new direct proofs of the results in d4 and r f without the use of this differential inequality theorems 2 4 2 5 and 2 18 these direct approaches have already had important applications k p2 an interesting class of examples for the theorems quoted above arises when one pulls back the laplacian via a quasiconformal mapping of the plane into itself specializing our theorem to this case suggests a new characterization of the class of am weights introduced in connection with several problems in harmonic analysis by muckenhoupt mul and coifman c fefferman co f we shall present in theorem 2 3 a criterion in terms of carleson measures which is very close to log w e bmo and is necessary and sufficient for w e ad

analytic solutions of a second order iterative functional differential equation abstract this paper is concerned with a second order iterative functional differential equation x z x m z 2 its analytic solutions are discussed by locally reducing the equation to another functional differential equation with proportional delays 2 y z y z y z y z y z 3 y m z 2 and by constructing a convergent power series solutions for the latter equation

the numerical solution of plane potential problems by improved boundary integral equation methods abstract a modified boundary integral bie method which facilitates accurate solution of laplacian boundary value problems is presented this method is designed specifically for treatment of problems in which singularities occur on the interface between two regions with different physical properties and is illustrated by application to two physical problems analytic expressions for the integrals arising in the piecew se linear and piecewise quadratic bie approximations are also presented these analytical expressions afford an appreciable reduction in computational time when compared with previously employed quadrature formulae

smooth solutions of a nonhomogeneous iterative functional differential equation with variable coefficients using the fixed point theorems of banach and schauder in this paper we discuss the existence uniqueness and stability of smooth solutions of a nonhomogeneous iterative functional differential equation with variable coefficients

sharp logarithmic estimates for positive dyadic shifts the paper contains the study of sharp logarithmic estimates for positive dyadic shifts a a given on probability spaces x x equipped with a tree like structure for any k 0 k 0 we determine the smallest constant l l k l l k such that e a f d k r f d l k e where t t 1 log t 1 t t t 1 log t 1 t e is an arbitrary measurable subset of x and f is an integrable function on x the proof exploits bellman function method we extract the above estimate from the existence of an appropriate special function enjoying certain size and concavity type conditions as a corollary a dual exponential bound is obtained

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

sharp a2 inequality for haar shift operators as a corollary to the main result of the paper we give a new proof of the inequality rm t f l 2 w lesssim w a 2 f l 2 w where t is either the hilbert transform amer j math 129 5 1355 1375 2007 a riesz transform proc amer math soc 136 4 1237 1249 2008 or the beurling operator duke math j 112 2 281 305 2002 the weight w is non negative and the linear growth in the a2 characteristic on the right is sharp prior proofs relied strongly on haar shift operators cr acad sci paris ser i math 330 6 455 460 2000 and bellman function techniques the new proof uses haar shifts and then uses an elegant two weight t 1 theorem of nazarov treil volberg math res lett 15 3 583 597 2008 to immediately identify relevant carleson measure estimates which are in turn verified using an appropriate corona decomposition of the weight w

extremal problems related to maximal dyadic like operators abstract we obtain sharp estimates for the localized distribution function of the dyadic maximal function m d when belongs to l p using this we obtain sharp estimates for the quasi norm of m d in l p given the localized l 1 norm and certain weak l p conditions

on the collocation method for a nonlinear boundary integral equation abstract in this paper we study a potential problem with a nonlinear boundary condition using the green representation formula for a harmonic function we reformulate the nonlinear boundary value problem as a nonlinear boundary integral equation we shall give a brief discussion of the solvability of the integral equation the aim of this paper however is to analyse the collocation method for finding an approximate solution to this equation using the theory of a proper and a stable mappings we prove the unique solvability of the collocation equations and the asymptotic error estimates to do this we assume that the nonlinearity is strongly monotone

analytic solutions of a second order nonautonomous iterative functional differential equation abstract in this paper a second order nonautonomous iterative functional differential equation is considered by reducing the equation with the schroder transformation to another functional differential equation without iteration of the unknown function we give existence of its local analytic solutions we first discuss the case that the constant given in the schroder transformation does not lie on the unit circle in c and the case that the constant lies on the circle but fulfills the diophantine condition then we further study the case that the constant is a unit root in c but the diophantine condition is offended finally we investigate analytic solutions of the form of power functions

analytic solutions of an iterative functional differential equation abstract this paper is concerned with an iterative functional differential equation x z x m z where x m z x x x z is the m th iterate of the function x z by constructing a convergent power series solution y z of a companion equation of the form y z y z y m z analytic solutions of the form y y 1 z for the original differential equation are obtained

perfect dyadic operators weighted t 1 theorem and two weight estimates abstract perfect dyadic operators were first introduced in 1 where a local t b theorem was proved for such operators in 3 it was shown that for every singular integral operator t with locally bounded kernel on r n r n there exists a perfect dyadic operator t such that t t is bounded on l p d x for all 1 p in this paper we show a decomposition of perfect dyadic operators on real line into four well known operators two selfadjoint operators paraproduct and its adjoint based on this decomposition we prove a sharp weighted version of the t 1 theorem for such operators which implies a 2 conjecture for such operators with constant which only depends on t 1 bmo d t 1 bmo d and the constant in testing conditions for t moreover the constant depends on these parameters at most linearly in this paper we also obtain sufficient conditions for the two weight boundedness for a perfect dyadic operator and simplify these conditions under additional assumptions that weights are in the muckenhoupt class a d

linear bound for the dyadic paraproduct on weighted lebesgue space l2 w the dyadic paraproduct is bounded in weighted lebesgue spaces lp w if and only if the weight w belongs to the muckenhoupt class apd however the sharp bounds on the norm of the dyadic paraproduct are not known even in the simplest l2 w case in this paper we prove that the bound on the norm of the dyadic paraproduct in the weighted lebesgue space l2 w depends linearly on the a2d characteristic of the weight w using bellman function techniques and extrapolate this result to the lp w case

on the bellman function of nazarov treil and volberg we give an explicit formula for the bellman function associated with the dual bound related to the unconditional constant of the haar system

extremizers and sharp weak type estimates for positive dyadic shifts we find the exact bellman function for the weak l1 norm of local positive dyadic shifts we also describe a sequence of functions self similar in nature which in the limit extremize the local weak type 1 1 inequality

boundary element methods for potential problems with nonlinear boundary conditions galerkin boundary element methods for the solution of novel first kind steklov poincare and hypersingular operator boundary integral equations with nonlinear perturbations are investigated to solve potential type problems in two and three dimensional lipschitz domains with nonlinear boundary conditions for the numerical solution of the resulting newton iterate linear boundary integral equations we propose practical variants of the galerkin scheme and give corresponding error estimates we also discuss the actual implementation process with suitable preconditioners and propose an optimal hybrid solution strategy

local invertible analytic solution of a functional differential equation with deviating arguments depending on the state derivative abstract this paper is concerned with a functional differential equation of the form x z 1 x a z b x z with the distinctive feature that the argument of the unknown function depends on the state derivative where a b are two complex numbers by reducing the equation with the schroder transformation to another functional differential equation with proportional delay we give existence of its local invertible analytic solutions we discuss not only that the constant given in schroder transformation at resonance i e at a root of the unity but also those near resonance near a root of the unity under the brjuno condition

sharp results in the integral form john nirenberg inequality we consider the strong form of the john nirenberg inequality for the l 2 based bmo we construct explicit bellman functions for the inequality in the continuous and dyadic settings and obtain the sharp constant as well as the precise bound on the inequality s range of validity both previously unknown the results for the two cases are substantially different the paper not only gives another instance in the short list of such explicit calculations but also presents the bellman function method as a sequence of clear steps adaptable to a wide variety of applications