cyclic self dual codes it is shown that if the automorphism group of a binary self dual code satisfies a certain condition then the code contains words of weight congruent to 2 modulo 4 in particular no cyclic binary self dual code can have all its weights divisible by four the number of cyclic binary self dual codes of length n is determined and the shortest nontrivial code in this class is shown to have length 14

constacyclic codes cocycles and a u v u v construction a connection between cohomology cocycles and constacyclic codes is explored it suggests an isomorphism between cyclic codes of length mn and a direct sum of m constacyclic codes of length n the isomorphism is used i to study the discrete fourier transforms and the decomposition of group ring codes ii to give a u v u v construction over gf q when q is odd the u v u v construction gives some of the best ternary cyclic codes and classes of nearly mds cyclic codes of length 2q 2 the symmetry of the construction is used to give a new proof that there are no cyclic self dual codes over gf q when q is odd

the cauchy integral analytic capacity and uniform rectifiability several explanations concerning notation terminology and background are in order first notation by 7hi we have denoted the one dimensional hausdorff measure i e length and a z r stands for the closed disc with center z and radius r a curve f is called ad regular that is ahlfors david regular if it satisfies 1 2 with e f since the lower bound is automatic for curves this means that hl r n a z r 0 general sets satisfying 1 2 are called ad regular it is simplest to define the l2 boundedness of the cauchy singular integral operator via the truncated integrals we say that ce is bounded in l2 e without really defining the operator ce itself if there is m ox such that

on the algebraic structure of quasi cyclic codes i finite fields a new algebraic approach to quasi cyclic codes is introduced the key idea is to regard a quasi cyclic code over a field as a linear code over an auxiliary ring by the use of the chinese remainder theorem crt or of the discrete fourier transform dft that ring can be decomposed into a direct product of fields that ring decomposition in turn yields a code construction from codes of lower lengths which turns out to be in some cases the celebrated squaring and cubing constructions and in other cases the u spl upsi u spl upsi and vandermonde constructions all binary extended quadratic residue codes of length a multiple of three are shown to be attainable by the cubing construction quinting and septing constructions are introduced other results made possible by the ring decomposition are a characterization of self dual quasi cyclic codes and a trace representation that generalizes that of cyclic codes

ap weights for nondoubling measures in rn and applications we study an analogue of the classical theory of ap weights in r n without assuming that the underlying measure is doubling then we obtain weighted norm inequalities for the centered hardy littlewood maximal function and corresponding weighted estimates for nonclassical calderon zygmund operators in the sense of ntv1 we also consider commutators of those calderon zygmund operators with bounded mean oscillation functions bmo extending the main result from crw finally we study self improving properties of poincare b m o type inequalities within this context more precisely we show that if f is a locally integrable function satisfying 1 q r q f fq d a q for all cubes q then it is possible to deduce higher l p integrability result of f assuming certain simple geometric condition on the functional a

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

goodness of fit problem and scanning innovation martingales this paper is mainly devoted to the following statistical problem in the case of random variables of any finite dimension and both simple or parametric hypotheses how to construct convenient empirical processes which could provide the basis for goodness of fit tests more or less in the same way as the uniform empirical process does in the case of simple hypothesis and scalar random variables the solution of this problem is connected here with the theory of multiparameter martingales and the theory of function parametric processes namely for the limiting gaussian processes some kind of filtration is introduced and so called scanning innovation processes are constructed the adapted standard wiener processes in one to one correspondence with initial gaussian processes this is done for the function parametric versions of the processes

bergman type singular integral operators and the characterization of carleson measures for besov sobolev spaces on the complex ball the purposes of this paper are twofold first we extend the method of non homogeneous harmonic analysis of nazarov treil and volberg to handle bergman type singular integral operators the canonical example of such an operator is the beurling transform on the unit disc second we use the methods developed in this paper to settle the important open question about characterizing the carleson measures for the besov sobolev space of analytic functions b sigma 2 on the complex ball of bbb c d in particular we demonstrate that for any sigma 0 the carleson measures for the space are characterized by a t1 condition the method of proof of these results is an extension and another application of the work originated by nazarov treil and the first author

random martingales and localization of maximal inequalities abstract let x d be a metric measure space for r 0 consider the hardy littlewood maximal operator m r f x def sup r r 1 b x r b x r f d we show that if there is an n 1 such that one has the microdoubling condition b x 1 1 n r b x r for all x x and r 0 then the weak 1 1 norm of m r has the following localization property m r l 1 x l 1 x sup r 0 m r r n r l 1 x l 1 x an immediate consequence is that if x d is ahlfors david n regular then the weak 1 1 norm of m r is n log n generalizing a result of stein and stromberg 1983 47 we show that this bound is sharp by constructing a metric measure space x d that is ahlfors david n regular for which the weak 1 1 norm of m 0 is n log n the localization property of m r is proved by assigning to each f l 1 x a distribution over random martingales for which the associated random doob maximal inequality controls the weak 1 1 inequality for m r

a class of constacyclic codes a class of q sup m ary constacyclic codes is defined which has the property that the q ary image is equivalent to a class of shortened cyclic codes this description leads to the construction of nonlinear codes as an example one can construct nonlinear codes of length n q sup m 1 over the field gf q sup m m spl les m with a minimum distance greater than n 1 r m m where r is the rate for higher rates these codes are the best known the codes are easily encoded and decoded

construction and encoding of qc ldpc codes using group rings quasi cyclic qc low density parity check ldpc codes which are known as qc ldpc codes have many applications due to their simple encoding implementation by the means of cyclic shift registers in this paper we construct qc ldpc codes from group rings a group ring is a free module at the same time a ring constructed in a natural way from any given ring and any given group we present a structure based on the elements of a group ring for constructing qc ldpc codes some of the previously addressed methods for constructing qc ldpc codes based on finite fields are special cases of the proposed construction method the constructed qc ldpc codes perform very well over the additive white gaussian noise channel with iterative decoding in terms of bit error probability and block error probability simulation results demonstrate that the proposed codes have competitive performance in comparison with the similar existing ldpc codes finally we propose a new encoding method for the proposed group ring based qc ldpc codes that can be implemented faster than the current encoding methods the encoding complexity of the proposed method is analyzed mathematically and indicates a significate reduction in the required number of operations even when compared to the available efficient encoding methods that have linear time and space complexities

non homogeneous t1 theorem for bi parameter singular integrals abstract we prove a non homogeneous t 1 theorem for certain bi parameter singular integral operators moreover we discuss the related non homogeneous journe s lemma and product bmo theory

weighted kolmogorov smirnov test accounting for the tails accurate goodness of fit tests for the extreme tails of empirical distributions is a very important issue relevant in many contexts including geophysics insurance and finance we have derived exact asymptotic results for a generalization of the large sample kolmogorov smirnov test well suited to testing these extreme tails in passing we have rederived and made more precise the approximate limit solutions found originally in unrelated fields first in l turban j phys a 25 127 1992 and later in p l krapivsky and s redner am j phys 64 546 1996

classification of pseudo cyclic mds codes linear codes are considered a code is characterized by the length n the dimension k and the minimum distance d an n k d code over the finite field gf q is said to be maximum distance separable mds if d n k 1 a krishna and d v sarwate 1990 investigated the existence of pseudo cyclic mds codes over gf q of length n where n divides q 1 or q 1 it is shown that the pseudo cyclic mds codes constructed by krishna and sarwate are generalised reed solomon codes pseudo cyclic codes are studied over gf q where n and q p sup h are not relatively prime it is proven that pseudo cyclic n k mds codes modulo x sup n a over gf q exist if and only if n p furthermore any pseudo cyclic p k code modulo x sup p a over gf q turns out to be mds and generalized reed solomon it is explicitly proven that some classes of pseudo cyclic n k mds codes over gf q are generalized reed solomon codes furthermore pseudo cyclic q 1 4 mds codes over gf q q even are completely classified

riesz transforms of non integer homogeneity on uniformly disconnected sets in this paper we obtain precise estimates for the l norm of the s dimensional riesz transforms on very general measures supported on cantor sets in r with d 1 s d from these estimates we infer that for the so called uniformly disconnected compact sets the capacity s associated with the riesz kernel x x is comparable to the capacity 2 3 d s 3 2 from non linear potential theory

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

local muckenhoupt weights on gaussian measure spaces abstract we study an analogue of a p weights which are called local weights and denoted by a p a in gauss measure spaces some classical properties of it are obtained then we characterize the weights by the boundedness of local hardy littlewood maximal operators on weighted lebesgue spaces we also get the jones s decomposition of a p a which is different from the type of orobitg and perez 24 at last we consider the class weights of a p q a and obtain the weighted boundedness of fractional maximal operator m a

the structure of reverse holder classes on metric measure spaces abstract this work extends some euclidean results on the structure of the reverse holder classes to metric measure spaces with a doubling measure satisfying an annular decay condition the equivalence between the variants used to define the muckenhoupt class a and some standard correspondences between reverse holder classes and muckenhoupt classes are utilized it is established that the logarithm of the reverse holder classes is an open and convex subset of the space b m o and that the logarithm of the intersection of the reverse holder classes is a closed and convex subset of the space b m o as an intermediary result we also establish that the intersection of the reverse holder classes consists precisely of the pointwise multipliers of the muckenhoupt class a

a central limit theorem under metric entropy with l_2 bracketing let mathbf s rho be a metric space mathbf v mathscr v mu be a probability space and f mathbf s times mathbf v rightarrow mathbb r be a real valued function on mathbf s times mathbf v which has mean zero and is lipschitz in l 2 mu with respect to rho let v be a random variable defined on mathbf v mathscr v mu and let v i i geq 1 be a sequence of independent copies of v the limiting behavior of the process s n s n 1 2 sum n i 1 f s v i is studied under an integrability condition on the metric entropy with bracketing in l 2 mu this metric entropy condition is analogous to one which implies the continuity of the limiting gaussian process a tightness result is derived which in conjunction with the results of andersen and dobric 1987 shows that a central limit theorem holds for the s n process this result generalizes those of dudley 1978 dudley 1981 and jain and marcus 1975

boundedness of calderon zygmund operators on non homogeneous metric measure spaces equivalent characterizations national natural science foundation of china 10871025 changjiang scholars and innovative research team in university of china fundamental research funds for the central universities of china 2011121040

multilinear calderon zygmund operators on the product of lebesgue spaces with non doubling measures abstract under the assumption that is a non doubling measure on r d the author proves that for the multilinear calderon zygmund operator its boundedness from the product of hardy space h 1 h 1 into l 1 2 implies its boundedness from the product of lebesgue spaces l p 1 l p 2 into l p with 1 p 1 p 2 and p satisfying 1 p 1 p 1 1 p 2

products of functions in bmo and h1 spaces on spaces of homogeneous type abstract we give an extension to certain rd space x i e space of homogeneous type in the sense of coifman and weiss which has the reverse doubling property of the definition and various properties of the product of functions in bmo x and h 1 x and functions in lipschitz space 1 p 1 x and h p x for p n n 1 where n and denote respectively the dimension and the order of x

maximal properties of the normalized cauchy transform where m is the normalized lebesgue measure on tf by the fatou theorem kf can also be defined on tf by its non tangential boundary values after such an extension one can view the cauchy integral as a transform i e an operator in lp m which sends f into the boundary values of kf it is well known and not difficult to see that the cauchy transform is unbounded for p 1 or p oc however the classical theorem by m riesz says that the cauchy transform is bounded in lp when 1 p oc further progress is due to r hunt b muckenhoupt and r wheeden who studied spaces with absolutely continuous weights lp w it was shown that the cauchy transform is bounded in lp w 1 p 00 if and only if the weight w satisfies the celebrated ap condition another classical object of complex function theory is the non tangential maximal function for any function g in d we can define its non tangential maximal function mg as

consta abelian codes over galois rings we study n length consta abelian codes a generalization of the well known abelian codes and constacyclic codes over galois rings of characteristic p sup a where n and p are coprime a twisted discrete fourier transform dft is used to generalize transform domain results of abelian and constacyclic codes to consta abelian codes further we characterize consta abelian codes invariant under two kinds of monomials whose underlying permutations are effected by i multiplying the coordinates with a unit in the appropriate mixed radix representation of the coordinate positions and ii shifting the coordinates by t positions all the codes studied here belong to the class of quasi twisted codes which are known to contain some good codes we show that the dual of a consta abelian code invariant under the two monomials is also a consta abelian code closed under both monomials

on the minimum distance of ternary cyclic codes there are many ways to find lower bounds for the minimum distance of a cyclic code based on investigation of the defining set some new theorems are derived these and earlier techniques are applied to find lower bounds for the minimum distance of ternary cyclic codes furthermore the exact minimum distance of ternary cyclic codes of length less than 40 is computed numerically a table is given containing all ternary cyclic codes of length less than 40 and having a minimum distance exceeding the bch bound it seems that almost all lower bounds are equal to the minimum distance especially shifting which is also done by computer seems to be very powerful for length 40

finitely randomized dyadic systems and bmo on metric measure spaces abstract we study the connection between bmo and dyadic bmo in metric measure spaces using finitely randomized dyadic systems and give a garnett jones type proof for a theorem of uchiyama on a construction of certain bmo functions we obtain a relation between the bmo norm of a suitable expectation over dyadic systems and the dyadic bmo norms of the original functions in different systems the expectation is taken over only finitely randomized dyadic systems to overcome certain measurability questions applying our result we derive uchiyama s theorem from its dyadic counterpart which we also prove

besov spaces with non doubling measures suppose that is a radon measure on r d which may be non doubling the only condition on is the growth condition namely there is a constant c 0 0 such that for all x supp and r 0 b x r c 0 r n where 0 0 is a real number which depends on the non doubling measure c 0 n and d the method used to define these spaces is new even for the classical case as applications the lifting properties of these spaces by using the riesz potential operators and the dual spaces are obtained

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

transformed empirical processes and modified kolmogorov smirnov tests for multivariate distributions a general way of constructing classes of goodness of fit tests for multivariate samples is presented these tests are based on a random signed measure that plays the same role as the empirical process in the construction of the classical kolmogorov smirnov tests the resulting tests are consistent against any fixed alternative and for each sequence of contiguous alternatives a test in each class can be chosen so as to optimize the discrimination of those alternatives

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

solutions of weinstein equations representable by bessel poisson integrals of bmo functions abstract we consider the weinstein type equation l u 0 on 0 0 where l t 2 x 2 1 x 2 with 1 in this paper we characterize the solutions of l u 0 on 0 0 representable by bessel poisson integrals of bmo functions as the ones satisfying certain carleson properties

a vandermonde code construction we use a vandermonde matrix to construct a code of length mn from m codes of length n a distance estimate for the construction is given when the starting codes are suitable constacyclic codes the constructed code is cyclic and from this we obtain new cyclic codes the results generalize those of hughes see ibid vol 46 p 674 80 2000 on the u spl upsi u spl upsi construction

calderon zygmund operators with non doubling measures in this survey the authors review some basic results on calderon zygmund theory on r d with the non doubling radon measure instead of the usual doubling condition the measure is only assumed to satisfy some growth condition that is there exist positive con stants n 0 d a ndc such that b x r cr n for all x r d and r 0 in this setting the results that we review are the following the hardy space h 1 and the regularized bmo space rbmo some interpolation results related to h 1 a nd rbmo the calderon zygmund decomposition the t 1 theorem and the properties of the calderon zygmund op erator on lebesgue spaces and hardy spaces cotlar s inequality and the boundedness of the maximal calderon zygmund operator the behaviors of the multilinear calderon zygmund op erator on lebesgue spaces weighted norm inequalities for the calderon zygmund operator the boundedness of the commutator generated by the calderon zygmund operator and rbmo function moreover some new progress obtained recently is also included

estimates for maximal multilinear commutators on non homogeneous spaces abstract under the assumption that is a non doubling measure on r d the authors obtain some estimates for the maximal multilinear commutators generated by calderon zygmund operator t and rbmo functions associated to this measures

weighted rearrangement inequalities for local sharp maximal functions several weighted rearrangement inequalities for uncentered and centered local sharp functions are proved these results are applied to obtain new weighted weak type and strong type estimates for singular integrals a self improving property of sharp function inequalities is established

the local tb theorem with rough test functions abstract we prove a local tb theorem under close to minimal up to certain buffering integrability assumptions conjectured by s hofmann el escorial 2008 every cube is assumed to support two non degenerate functions b q 1 l p and b q 2 l q such that 1 2 q t b q 1 l q and 1 2 q t b q 2 l p with appropriate uniformity and scaling of the norms this is sufficient for the l 2 boundedness of the calderon zygmund operator t for any p q 1 a result previously unknown for simultaneously small values of p and q we obtain this as a corollary of a local tb theorem for the maximal truncations t and t for the l 2 boundedness of t it suffices that 1 q t b q 1 and 1 q t b q 2 be uniformly in l 0 the proof builds on the technique of suppressed operators from the quantitative vitushkin conjecture due to nazarov treil volberg

a decomposition theorem for singular integral operators on spaces of homogeneous type let x d be a space of homogeneous type under the assumption x 0 for all x x we prove a decomposition theorem for singular integral operators on x d isotropic haar expansion gives a representation of the integral operator as a series of simple shifts and rearrangements plus two paraproducts this yields a umd valued t 1 theorem on spaces of homogeneous type

some ternary and quaternary codes and associated sphere packings tables of good ternary and quaternary codes are presented and they are used in the construction of dense sphere packings results include 1 tables of the best ternary and quaternary constacyclic codes including cyclic codes up to block length 50 2 a class of optimal n 2 codes over gf q 3 the u v w mod 2u v mod u construction a new ternary code construction technique that can be used to construct the ternary reed muller codes and others and 4 tables of linear ternary and quaternary codes obtained by modifying and combining various codes here and in the literature packings are generated in even dimensions up to 100 using these codes and a sphere packing construction in dimensions 36 and 60 new record densities appear to have been achieved

characterization and semiadditivity of the c 1 harmonic capacity the c 1 harmonic capacity c plays a central role in problems of approximation by harmonic functions in the c 1 norm in n 1 in this paper we prove the comparability between the capacity c and its positive version c as a corollary we deduce the semiadditivity of c this capacity can be considered as a generalization in n 1 of the continuous analytic capacity in c moreover we also show that the so called inner boundary conjecture fails for dimensions n 1 unlike in the case n 1

multilinear commutators of calderon zygmund operators on hardy type spaces with non doubling measures abstract under the assumption that is a non negative radon measure on r d which only satisfies some growth condition the authors obtain the boundedness in some hardy type spaces of multilinear commutators generated by calderon zygmund operators or fractional integrals with rbmo functions where the hardy type spaces are some appropriate subspaces associated to the considered rbmo functions of the hardy space h 1 of tolsa

improved cotlar s inequality in the context of local tb theorems abstract in the context of local tb theorems with l p testing conditions we prove an enhanced cotlar s inequality this is related to the problem of removing the so called buffer assumption of hytonen nazarov which is the final barrier for the full solution of s hofmann s problem we also investigate the problem of extending the hytonen nazarov result to non homogeneous measures we work not just with the lebesgue measure but with measures in r d satisfying b x r c r n n 0 d the range of exponents in the cotlar type inequality depend on n without assuming buffer we get the full range of exponents p q 1 2 for measures with n 1 and in general we get p q 2 n 2 n 0 consequences for non homogeneous local tb theorems are discussed

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

consta abelian polyadic codes in this correspondence the class of polyadic codes is generalized to the class of consta abelian polyadic codes which in particular includes the class of constacyclic polyadic codes properties such as the equivalence of polyadic codes and the mth root lower bound for the minimum weight of a subcode of certain types of polyadic codes are preserved in the consta abelian case sufficient conditions for the existence of this class of codes are established for the special case of constacyclic codes the characterization of negacyclic self dual codes of length coprime to the characteristic of the field in terms of negacyclic duadic codes is also given

asymptotical stability of muckenhoupt weights through gurov reshetnyak classes we show that weights in the gurov reshetnyak class gr satisfy a weak reverse holder inequality with an explicit and asymptotically sharp bound for the exponent thus extending classical results from the euclidean setting to doubling metric mea sure spaces as an application we study asymptotical behaviour of embeddings between muckenhoupt classes and reverse holder classes

bmo and h1 for the ornstein uhlenbeck operator in this paper we develop a theory of singular integral operators acting on function spaces over the measured metric space rd where denotes the euclidean distance and the gauss measure our theory plays for the ornstein uhlenbeck operator the same role that the classical calderon zygmund theory plays for the laplacian on rd where is the lebesgue measure our method requires the introduction of two new function spaces the space bmo of functions with bounded mean oscillation and its predual the atomic hardy space h1 we show that if p is in 2 then lp is an intermediate space between l2 and bmo and that an inequality of john nirenberg type holds for functions in bmo then we show that if m is a bounded operator on l2 and the schwartz kernels of m and of its adjoint satisfy a local integral condition of hormander type then m extends to a bounded operator from h1 to l1 from l to bmo and on lp for all p in 1 as an application we show that certain singular integral operators related to the ornstein uhlenbeck operator which are unbounded on l1 and on l turn out to be bounded from h1 to l1 and from l to bmo

littlewood paley theory and the t 1 theorem with non doubling measures let be a radon measure on rd which may be non doubling the only condition that must satisfy is b x r crn for all x rd r 0 and for some fixed 0 n d in this paper littlewood paley theory for functions in lp is developed one of the main difficulties to be solved is the construction of reasonable approximations of the identity in order to obtain a calderon type reproducing formula moreover it is shown that the t 1 theorem for n dimensional calderon zygmund operators without doubling assumptions can be proved using the littlewood paley type decomposition that is obtained for functions in l2 as in the classical case of homogeneous spaces

the space for nondoubling measures in terms of a grand maximal operator let be a radon measure on r d which may be nondoubling the only condition that must satisfy is the size condition b x r c r n for some fixed 0 n d recently some spaces of type bmo and h 1 were introduced by the author these new spaces have properties similar to those of the classical spaces bmo and h 1 defined for doubling measures and they have proved to be useful for studying the l p boundedness of calderon zygmund operators without assuming doubling conditions in this paper a characterization of the new atomic hardy space h 1 p in terms of a maximal operator m is given it is shown that f belongs to h 1 if and only if f e l 1 f d 0 and m f f l 1 as in the usual doubling situation

bmo h 1 and calderon zygmund operators for non doubling measures given a radon measure mu on mathbb r d which may be non doubling we introduce a space of type bmo with respect to this measure it is shown that many properties which hold for the classical space bmo mu when mu is a doubling measure remain valid for the space of type bmo introduced in this paper without assuming mu doubling for instance calderon zygmund operators which are bounded on l 2 mu are also bounded from l infty mu into the new bmo space moreover this space also satisfies a john nirenberg inequality and its predual is an atomic space h 1 using a sharp maximal operator it is shown that operators which are bounded from l infty mu into the new bmo space and from its predual h 1 into l 1 mu must be bounded on l p mu 1 p infty from this result one can obtain a new proof of the t 1 theorem for the cauchy transform for non doubling measures finally it is proved that commutators of calderon zygmund operators bounded on l 2 mu with functions of the new bmo are bounded on l p mu 1 p infty

the molecular characterization of the hardy space h 1 on non homogeneous metric measure spaces and its application national natural science foundation of china 11171027 11101339 specialized research fund for the doctoral program of higher education of china 20120003110003 natural science foundation of fujian province of china 2013j01020 fundamental research funds for central universities of china 2013121004

trudinger inequalities without derivatives we prove that the trudinger inequality holds on connected homogeneous spaces for functions satisfying a very weak type of poincare inequality we also illustrate the connection between this result and the john nirenberg theorem for bmo

pseudocyclic maximum distance separable codes the n k pseudocyclic maximum distance separable mds codes modulo x sup n a over gf q are considered suppose that n is a divisor of q 1 if n is odd pseudocyclic mds codes exist for all k however if n is even nontrivial pseudocyclic mds codes exist for odd k but not for even k if a is a quadratic residue in gf q and they exist for even k but not for odd k if a is not a quadratic residue in gf q also considered is the case when n is a divisor of q 1 and it is shown that pseudocyclic mds codes exist if and only if the multiplicative order of a divides q 1 n and that when this condition is satisfied such codes exist for all k if the condition is not satisfied every pseudocyclic code of length n is the result of interleaving a shorter pseudocyclic code

multilinear operator valued calderon zygmund theory abstract we develop a general theory of multilinear singular integrals with operator valued kernels acting on tuples of umd banach spaces this in particular involves investigating multilinear variants of the r boundedness condition naturally arising in operator valued theory we proceed by establishing a suitable representation of multilinear operator valued singular integrals in terms of operator valued dyadic shifts and paraproducts and studying the boundedness of these model operators via dyadic probabilistic banach space valued analysis in the bilinear case we obtain a t 1 type theorem without any additional assumptions on the banach spaces other than the necessary umd higher degrees of multilinearity are tackled via a new formulation of the rademacher maximal function rmf condition in addition to the natural umd lattice cases our rmf condition covers suitable tuples of non commutative l p spaces we employ our operator valued theory to obtain new multilinear multi parameter operator valued theorems in the natural setting of umd spaces with property

representation of bi parameter singular integrals by dyadic operators abstract we prove a dyadic representation theorem for bi parameter singular integrals that is we represent certain bi parameter operators as averages of rapidly decaying sums of what we call bi parameter shifts a new version of the product space t 1 theorem is established as a consequence

rectifiable sets and the traveling salesman problem let k c c be a bounded set in this paper we shall give a simple necessary and sufficient condit ion for k to lie in a rectifiable curve we say that a set is a rectifiable curve if it is the image of a finite interval under a lipschitz mapping recall that for a connected set f c c f is a rectifiable curve not necessarily simple if and only if l f where l denotes one dimensional hausdorff measure this classical result follows from the fact that on any finite graph there is a tour which covers the entire graph and which crosses each edge but not necessarily each vertex at most twice if k is a finite set then we are essentially reduced to the classical traveling salesman problem tsp compute the length of the shortest hami l ton ian cycle which hits all points of k this is the same up to a constant multiple as asking for the inf imum of l f where f is a curve k c f such a f is called a spanning tree in tsp theory for infinite sets k we cannot hope in general to have k be a subset of a jordan curve what we should therefore look at is connected sets which conta in k let fmi n be the shortest minimal spanning tree then we cannot possibly solve our problem for sets k of infinite cardinality if we cannot find f i f c o fmin for any finite set k here and throughout the paper c co c1 c o etc denote various universal constants while there are several algorithms for computing l fml these algorithms work for finite graphs satisfying the triangle inequality and do not use the euclidean properties of k see 13 for an excellent discussion of some of these algorithms therefore these methods cannot solve our problem for general infinite k we present a method which is a minor modification of a well known algorithm farthest insert ion see 13 which yields a f with i f c o l fmi the farthest insert ion algorithm has been extensively studied with large numerical calculations on computers and is experimentally good in the sense that the f produced satisfy i f c o l f i for all examples which have

sobolev embedding into bmo and weak l for 1 dimensional probability measure abstract we characterize rearrangement invariant spaces x with respect to a suitable 1 dimensional probability e g log concave measure such that the sobolev embedding u bmo r c u x u l 1 r holds for any function u l 1 r whose real valued weakly derivative u belongs to x here bmo r is the space of functions with bounded mean oscillation with respect to we investigate the embedding in weak l r too

testing semiparametric conditional moment restrictions using conditional martingale transforms this paper studies conditional moment restrictions that contain unknown nonparametric functions and proposes a general method of obtaining asymptotically distribution free tests via martingale transforms examples of such conditional moment restrictions are single index restrictions partially parametric regressions and partially parametric quantile regressions this paper introduces a conditional martingale transform that is conditioned on the variable in the nonparametric function and shows that we can generate distribution free tests of various semiparametric conditional moment restrictions using this martingale transform the paper proposes feasible martingale transforms using series estimation and establishes their asymptotic validity some results from a monte carlo simulation study are presented and discussed