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

a boundedness criterion for generalized calderon zygmund operators in 1965 a p calderon showed the l2 boundedness of the so called first calderon commutator this is one of the first examples of a non convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the hilbert transform these properties together with the l2 boundedness imply the lp boundedness for all p s in 1 oo many operators in analysis such as certain classes of pseudo differential operators and the cauchy integral operator on a curve are associated with kernels having these properties for these operators one of the major questions is if they are bounded on l2 we are going to give necessary and sufficient conditions for such an operator to be bounded on l2 they are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in bmo in the case of the aforementioned first commutator this can be checked by an integration by parts in the first section we present some basic notions and state the theorem which is proved in sections 2 and 3 in section 4 we show how to recover some classical results in sections 5 and 6 we construct a functional calculus for small perturbations of a and in section 7 we show a connection between the theory of calderon zygmund operators and kato s conjecture it is a pleasure to express our thanks to r r coifman and y meyer for suggesting many elegant simplifications in our proofs and most of the applications we also wish to thank stephen semmes for several pertinent remarks

on the constructions of constant weight codes two methods of constructing binary constant weight codes from 1 codes over gf q and 2 constant weight codes over gf q are presented several classes of binary optimum constant weight codes are derived from these methods in general we show that binary optimum constant weight codes which achieve the johnson bound can be constructed from optimum codes over gf q which achieve the plotkin bound finally several classes of optimum constant weight codes over gf q are constructed

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

linear size optimal q ary constant weight codes and constant composition codes an optimal constant composition or constant weight code of weight w has linear size if and only if its distance d is at least 2w 1 when d 2w the determination of the exact size of such a constant composition or constant weight code is trivial but the case of d 2w 1 has been solved previously only for binary and ternary constant composition and constant weight codes and for some sporadic instances this paper provides a construction for quasicyclic optimal constant composition and constant weight codes of weight w and distance 2w 1 based on a new generalization of difference triangle sets as a result the sizes of optimal constant composition codes and optimal constant weight codes of weight w and distance 2w 1 are determined for all such codes of sufficiently large lengths this solves an open problem of etzion the sizes of optimal constant composition codes of weight w and distance 2w 1 are also determined for all w 6 except in two cases

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 two weight inequalities for singular integrals with applications to the hilbert transform and the sarason conjecture abstract we prove two weight norm inequalities for calderon zygmund singular integrals that are sharp for the hilbert transform and for the riesz transforms in addition we give results for the dyadic square function and for commutators of singular integrals as an application we give new results for the sarason conjecture on the product of unbounded toeplitz operators on hardy spaces

weighted and vector valued inequalities for potential operators on etablit des inegalites ponderees et a valeurs vectorielles pour quelques operateurs integraux de type potentiel qui appliquent les fonctions definies sur r n dans des fonctions sur r n 1 r n 0

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

logarithmic bump conditions and the two weight boundedness of calderon zygmund operators abstract we prove that if a pair of weights u v satisfies a sharp a p bump condition in the scale of all log bumps or certain loglog bumps then haar shifts map l p v into l p u with a constant quadratic in the complexity of the shift this in turn implies the two weight boundedness for all calderon zygmund operators this gives a partial answer to a long standing conjecture we also give a partial result for a related conjecture for weak type inequalities to prove our main results we combine several different approaches to these problems in particular we use many of the ideas developed to prove the a 2 conjecture as a byproduct of our work we also disprove a conjecture by muckenhoupt and wheeden on weak type inequalities for the hilbert transform this is closely related to the recent counterexamples of reguera scurry and thiele

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

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

two weight inequality for the hilbert transform a real variable characterization i let and w be locally finite positive borel measures on r which do not share a common point mass assume that the pair of weights satisfy a poisson a2 condition and satisfy the testing conditions below for the hilbert transform h ih 1i 2dw i ih w1i 2d w i with constants independent of the choice of interval i then h maps l2 to l2 w verifying a conjecture of nazarov treil and volberg the proof uses basic tools of nonhomogeneous analysis with two components particular to the hilbert transform the first component is a global to local reduction which is a consequence of prior work by lacey sawyer shen and uriarte tuero the second component an analysis of the local part is the particular contribution of this article

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

constructions for q ary constant weight codes this paper introduces a new combinatorial construction for q ary constant weight codes which yields several families of optimal codes and asymptotically optimal codes the construction reveals intimate connection between q ary constant weight codes and sets of pairwise disjoint combinatorial designs of various types

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

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

optimal ternary constant weight codes of weight four and distance six recently chee and ling constructions for q ary constant weight codes ieee trans inf theory vol 53 no 1 135 146 jan 2007 introduced a new combinatorial construction for q ary constant weight codes which reveals a close connection between q ary constant weight codes and sets of pairwise disjoint combinatorial designs in this paper we study the problem of constructing optimal ternary constant weight codes with hamming weight four and minimum distance six using this approach the construction here exploits completely reducible super simple designs and group divisible codes the problem is solved leaving only two cases undetermined previously the sizes of constant weight codes of weight four and distance six were known only for those of length no greater than 10

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

dyadic like maximal operators on llogl functions abstract we study the following well known property of the dyadic maximal operator m d on r n see e m stein note on the class l log l studia math 32 1969 305 310 for the case of the hardy littlewood maximal function if is integrable and supported in a dyadic cube q then m d is integrable over sets of finite measure if and only if log 1 is integrable and the integral of m d can be estimated both from above and from below in terms of the integral of log 1 over q here we define and explicitly evaluate bellman functions related to this property and the corresponding estimates both upper and lower for the integrals thus producing sharp improved versions of the behavior of m d on the local l log l spaces

structurally variable control of time varying lurie systems abstractthis poses and solves the problem of structurally variable control synthesis for time varying lurie systems which has both theoretical and engineering significance a theorem on absolute stability of a set related to lurie systems solves the problem the theorem is basic for proving control algorithms that ensure absolute stability of a sliding subspace or its stability with finite reachability time moreover they ensure the attraction in the whole of the zero equilibrium state of the whole control system of the plant on n l the selection of the sliding subspace can be such that the system nonlinearity does not influence the sliding motion

bellman function and linear dimension free estimates in a theorem of bakry by using an explicit bellman function we prove a bilinear embedding theorem for the laplacian associated with a weighted riemannian manifold m having the bakry emery curvature bounded from below the embedding acting on the cartesian product of lp m and lq t m 1 p 1 q 1 involves estimates which are independent of the dimension of the manifold and linear in p as a consequence we obtain linear dimension free estimates of the lp norms of the corresponding shifted riesz transform all our proofs are analytic

the sizes of optimal q ary codes of weight three and distance four a complete solution this correspondence introduces two new constructive techniques to complete the determination of the sizes of optimal q ary codes of constant weight three and distance four

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

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

paper on absolute stability and the aizerman conjecture necessary and sufficient liapunov like conditions for both absolute stability and the aizerman conjecture are established in the framework of stationary continuous time lurie postnikov systems with multiple nonlinearities thus providing theoretical solutions for these problems in the liapunov sense the solutions are based on the use of functional families instead of a single liapunov function directly applicable results of the paper are new algebraic criteria for absolute stability one of them is proved by using a new form of a tentative system liapunov function also an estimate of the asymptotic stability domain is determined for all admissible nonlinearities the results do not impose any requirement on the linear part of the system such as its stability or controllability or observability three illustrative examples are worked out

entropy conditions in two weight inequalities for singular integral operators abstract a new type of bumping of the muckenhoupt a 2 condition on weights is introduced it is based on bumping the entropy integral of the weights in particular one gets assuming mild regularity conditions on the corresponding young functions the bump conjecture proved in 20 25 as a corollary of entropy bumping but our entropy bumps cannot be reduced to the bumping with orlicz norms in the solution of bump conjecture they are effectively smaller henceforth we get somewhat stronger results than the one that solves the bump conjecture in 20 25 new results concerning one sided bumping conjecture are obtained all the results hold in the general non homogeneous situation

bellman function for extremal problems in bmo in this paper we develop the method of nding sharp estimates by using a bellman function in such a form the method appears in the proofs of the classical john nirenberg inequality and l p estimations of bmo functions in the present paper we elaborate a method of solving the boundary value problem for the homogeneous monge amp ere equation in a parabolic strip for suciently smooth boundary conditions in such a way we have obtained an algorithm for constructing an exact bellman function for a large class of integral functionals on the bmo space

muckenhoupt wheeden conjectures for fractional integral operators abstract the muckenhoupt wheeden conjecture is disproved for fractional integral operator both the strong p p and the weak p p type conjectures are proved to be false the arguments rely upon a property of the characteristic function of an approximating sequence of the cantor set

the tb theorem on non homogeneous spaces 0 introduction main objects and results 3 0 1 main results 4 0 2 an application of t1 heorem electric intensity capacity 7 0 3 how to interpret calderon zygmund operator t 9 0 3 1 bilinear form is defined on lipschitz functions 10 0 3 2 bilinear form is defined for smooth functions 11 0 3 3 apriori boundedness 12 0 4 plan of the paper 15

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

functional calculus for generators of symmetric contraction semigroups we prove that every generator of a symmetric contraction semigroup on a finite measure space admits for 1 p a hormander type holomorphic functional calculus on lp in the sector of angle p arcsin 1 2 p the obtained angle is optimal

optimal quaternary constant weight codes with weight four and distance five constant weight codes play an important role in coding theory the problem of determining the sizes for optimal quaternary constant weight codes with length n weight 4 and minimum hamming distance 5 n 5 4 4 codes has been investigated in several papers although some constructions and several infinite families for such codes with length n 0 1 mod 4 have been given the problem is still far from complete in this paper we determine the size of an optimal n 5 4 4 code for each integer n 4 leaving 55 lengths unsolved especially for length n 0 1 mod 4 the existence problem of the equivalent combinatorial object namely the generalized steiner system is solved leaving only seven values undetermined

fractional integrals on weighted and spaces we characterize the pairs of doubling weights u v on rn such that iiwiihj ch hj 0 0 is the fractional integral operator we also consider the behavior of an associated maximal function applications of the results to sobolev inequalities in weighted lp spaces are given 1 introduction and notation in this paper we will characterize the pairs of doubling weights it v on rn such that pafwm 0 is the fractional integral operator we shall also give applications of these results to sobolev inequalities in weighted lp spaces a weight function u is said to belong to ds p 1 if u tq 1 and every cube q c rn where u q denotes the u measure of q and tq is the cube with the same center as q but with t times the edgelength we write dx j idu analogously u g rdv v 0 if u tq ctnuu q for every t 1 and every cube q it is not hard to see that if u g d then u g rdv for some v 0 we write u rhr r 1 if 1 1 r 1 rprr u x t dx i x dx 0 and e s then we denote as usual f x t f 0t x where t x t n x t and n f x sup f y t vfithta x y t yern t 0 x y 0

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

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

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

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

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

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

a two weight inequality for the hilbert transform assuming an energy hypothesis let and be locally finite positive borel measures on r subject to the pair of weights satisfying a side condition we characterize boundedness of the hilbert transform h from l2 to l2 in terms of the a2 condition i i i x xi 2d x i i i x xi 2d x 12 c i and the two testing conditions for all intervals i in r ih 1i x 2d x c id x ih 1i x 2d x c id x the proof uses the beautiful corona argument of nazarov treil and volberg there is a range of side conditions termed energy conditions at one endpoint the energy conditions are also a consequence of the testing conditions above and at the other endpoint they are the pivotal conditions of nazarov treil and volberg we detail an example which shows that the pivotal conditions are not necessary for boundedness of the hilbert transform

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

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

optimal ternary constant weight codes with weight 4 and distance 5 constant weight codes cwcs play an important role in coding theory the problem of determining the sizes for optimal ternary cwcs with length n weight 4 and minimum hamming distance 5 n 5 4 3 code has been settled for all positive integers n 10 or n 10 and n 1 mod 3 with n 13 52 58 undetermined in this paper we investigate the problem of constructing optimal n 5 4 3 codes for all lengths n with the tool of group divisible codes we determine the size of an optimal n 5 4 3 code for each integer n 4 leaving the lengths n 12 13 21 27 33 39 45 52 unsolved

absolute stability of lurie direct control systems with time varying coefficients and multiple nonlinearities absolute stability of lurie direct control systems with time varying coefficients and multiple nonlinearities is studied in this paper depending on the related methods relative magnitude of the norm unbounded coefficients was estimated by knowledge of nonsingular m matrix the lyapunov function was constructed and some absolute stability criteria of this kind of systems were obtained in addition some simple and practical corollaries were derived from the theorem of taussky the main contribution of this paper is that the criteria which we introduced allow for the situation that the norms of coefficient matrices are unbounded at last the example shows the availability of the criteria

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

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

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

carleson measures and multipliers of dirichlet type spaces a function p from 0 1 onto itself is a dirichlet weight if it is increasing p 0 and limr o x p x 0 the corresponding dirichlettype space dp consists of those bounded holomorphic functions on u z e c izl 1 such that if z 12p 1 izi is integrable with respect to lebesgue measure on u we characterize in terms of a carleson type maximal operator the functions in the set of pointwise multipliers of dp m dp g u c gf e dp vf e dp

on burkholder function for orthogonal martingales and zeros of legendre polynomials burkholder obtained a sharp estimate of bf e w p via bf e z p for martingales w differentially subordinated to martingales z his result is that bf e w p le p 1 p bf e z p where p max big p p over p 1 big what happens if the martingales have an extra property of being orthogonal martingales this property is an analog for martingales of the cauchy riemann equation for functions and it naturally appears in a problem on singular integrals see the references at the end of section 1 we establish here that in this case the constant is quite different actually bf e w p le big 1 z p over 1 z p big p bf e z p p ge 2 where z p is a specific zero of a certain solution of the legendre ode we also prove the sharpness of this estimate asymptotically 1 z p 1 z p 4j 2 0 o 1 p p to infty where j 0 is the first positive zero of the bessel function of zero order this connection with zeros of special functions and orthogonal polynomials for p n n 1 is rather unexpected

best constants for a family of carleson sequences abstract we consider a general family of carleson sequences associated with dyadic a 2 weights and find sharp or in one case simply best known upper and lower bounds for their carleson norms in terms of the a 2 characteristic of the weight the results obtained make precise and significantly generalize earlier estimates by wittwer vasyunin beznosova and others we also record several corollaries one of which is a range of new characterizations of dyadic a 2 particular emphasis is placed on the relationship between sharp constants and optimizing sequences of weights in most cases explicit optimizers are constructed our main estimates arise as consequences of the exact expressions or explicit bounds for the bellman functions for the problem and the paper contains a measure of bellman function innovation

structurally variable control of lurie systems a solution to the problem of structurally variable control synthesis for time invariant lurie systems is of both theoretical and engineering significance a theorem on absolute stability of a set a

a lower bound for ternary constant weight codes we derive a lower bound for ternary constant weight codes with minimum hamming distance three the bound is similar to a bound for binary constant weight codes with minimum distance four described by graham and sloane 1980 it improves upon the gilbert 1952 bound and coincides asymptotically with the johnson 1962 bound for fixed weight as the codeword length tends to infinity

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