t1 theorem for besov and triebel lizorkin spaces using the discrete calderon type reproducing formula and the plancherel polya characterization for the besov and triebel lizorkin spaces thet1 theorem for the besov and triebel lizorkin spaces was proved

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

the lp neumann problem for the heat equation in non cylindrical domains we consider the lp neumann problem for the heat equation when 1 p 2 in a time varying domain using a perturbation technique and previous l2 neumann results the authors obtain a generalization of earlier work in caloric cylinders

on an operator arising in the calderon zygmund method of rotations and the bramble hilbert lemma in this article sharp estimates are obtained for an operator that is a quadratic average with respect to v of one dimensional hardy littlewood maximal functions in the directions v this operator arises naturally in fourier analysis and numerical analysis

necessary and sufficient conditions for the bounds of the calderon type commutator for the littlewood paley operator abstract for b l loc 1 r n we define the calderon type commutator for the littlewood paley operator g function by g 1 b f x 0 1 t 2 x y t x y x y n 1 b x b y f y d y 2 d t t 1 2 which is a new kind of commutator of littlewood paley operator g function in this paper the authors obtain the necessary and sufficient conditions on the function b to guarantee that g 1 b is a bounded operator on l 2 r n namely if l log l 1 2 s n 1 and b l i p r n then g 1 b is bounded on l 2 r n and the converse is if g 1 b is bounded on l 2 r n then b l i p r n

universal coalgebra a theory of systems in the semantics of programming finite data types such as finite lists have traditionally been modelled by initial algebras later final coalgebras were used in order to deal with infinite data types coalgebras which are the dual of algebras turned out to be suited moreover as models for certain types of automata and more generally for transition and dynamical systems an important property of initial algebras is that they satisfy the familiar principle of induction such a principle was missing for coalgebras until the work of aczel 1988 on a theory of non wellfounded sets in which he introduced a proof principle nowadays called coinduction it was formulated in terms of bisimulation a notion originally stemming from the world of concurrent programming languages milner 1980 park 1981 using the notion of coalgebra homomorphism the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras aczel and mendler 1989 thus the three basic notions of universal algebra algebra homomorphism of algebras and congruence turn out to correspond to coalgebra homomorphism of coalgebras and bisimulation respectively in this paper the latter are taken as the basic ingredients of a theory called universal coalgebra some standard results from universal algebra are reformulated using the afore mentioned correspondence and proved for a large class of coalgebras leading to a series of results on e g the lattices of subcoalgebras and bisimulations simple coalgebras and coinduction and a covariety theorem for coalgebras similar to birkhoff s variety theorem

sharp bounds for the commutators with variable kernels of fractional differentiations and bmo sobolev spaces abstract for 0 1 and b i bmo we introduce a new class of commutators with fractional differentiations and variable kernels which is defined by b t f x r n x x y x y n b x b y f y d y in this paper we give the sharp l 2 norm inequalities for the rough operators b t with x z l r n l q s n 1 q 2 n 1 n satisfying the mean zero value condition in its second variable in the sense that the exponent q 2 n 1 n is optimal if strengthen the smoothness of x z in its second variable we prove weight norm inequalities for these operators our results recover a previous result of murray and extend a previous result of calderon

bisimulation can t be traced in the concurrent language ccs two programs are considered the same if they are bisimilar several years and many researchers have demonstrated that the theory of bisimulation is mathematically appealing and useful in practice however bisimulation makes too many distinctions between programs we consider the problem of adding operations to ccs to make bisimulation fully abstract we define the class of gsos operations generalizing the style and technical advantages of ccs operations we characterize gsos congruence in as a bisimulation like relation called ready simulation bisimulation is strictly finer than ready simulation and hence not a congruence for any gsos language

bisimulation for probabilistic transition systems a coalgebraic approach the notion of bisimulation as proposed by larsen and skou for discrete probabilistic transition systems is shown to coincide with a coalgebraic definition in the sense of aczel and mendier in terms of a set functor this coalgebraic formulation makes it possible to generalize the concepts to a continuous setting involving borel probability measures under reasonable conditions generalized probabilistic bisimilarity can be characterized categorically application of the final coalgebra paradigm then yields an internally fully abstract semantical domain with respect to probabilistic bisimulation

commutators of singular integrals and fractional integral operators the commutators of integral and differential operators are very important tools in harmonic analysis and pde recently the third author of this paper gave a survey front math china 2011 6 5 821 833 on some late results and put forward a few problems on commutators of many important operators in harmonic analysis in this paper we present some results about the commutators of singular integrals and fractional integrals with rough or variable kernels which can be seen as a supplement of the aforementioned survey

spline reproducing kernels on r and error bounds for piecewise smooth lbv problems abstract reproducing kernel method for approximating solutions of linear boundary value problems is valid in hilbert spaces composed of continuous functions but its convergence is not satisfactory without additional smoothness assumptions we prove 2nd order uniform convergence for regular problems with coefficient piecewise of sobolev class h2 if the coefficients are globally of class h2 more refined phantom boundary nsc rkhs method is derived and the order of convergence rises to 3 or 4 according to whether the problem is piecewise of class h3 or h4 the algorithms can be successfully applied to various non local linear boundary conditions e g of simple integral form the paper contains also a new explicit formula for general spline reproducing kernels in hm a b if the inner product f g m i m f i g i f m g m depends on any fixed reference point a b the piecewise nsc rkhs methods are then applied to two example regular lbv problems in h3 and h5 exactness of the resulting numerical solutions the degree of convergence and their dependency of the reference point a b are presented in attached figures

language based performance prediction for distributed and mobile systems we present a framework for performance prediction of distributed and mobile systems we rely on process calculi and their structural operational semantics the dynamic behaviour is described through transition systems whose transitions are labelled by encodings of their proofs that we then map into stochastic processes we enhance related works by allowing general continuous distributions resorting to a notion of enabling between transitions we also discuss how the number of resources available affects the overall model finally we introduce a notion of bisimulation that takes stochastic information into account and prove it to be a congruence when only exponential distributions are of interest our equivalence induces a lumpable partition on the underlying markov process

mixed norm estimates for a class of nonisotropic directional maximal operators and hilbert transforms abstract for all d 2 and p 1 max 2 d 1 2 we prove sharp l p to l p l q estimates modulo an endpoint for a directional maximal operator associated to curves generated by the dilation matrices exp log t p where p has real entries and eigenvalues with positive real part for the corresponding hilbert transform we prove an analogous result for all d 2 and p 1 2 as corollaries we prove l p bounds for variable kernel singular integral operators and nikodym type maximal operators taking averages over certain families of curved sets in r d

global morrey regularity of strong solutions to the dirichlet problem for elliptic equations with discontinuous coefficients abstract well posedness is proved in the space w2 p w1 p0 for the dirichlet problem equation omitted if the principal coefficients aij x of the uniformly elliptic operator belong to vmo l

on the size of special families of linear operators abstract we continue the study started in 12 of the search for algebraic structures one can find within the sets of injective linear functions we shall focus on the cases when the operators are considered both on finite dimensional and infinite dimensional domains we also study the set of continuous surjective linear operators

a tutorial on empa a theory of concurrent processes with nondeterminism priorities probabilities and time we give an overview of empa a process algebra for modeling and analyzing concurrent systems where nondeterminism priorities probabilities and time are present aiming at a reasonable trade off between the expressive power of the calculus and the complexity of its underlying theory

compactness for the commutators of singular integral operators with rough variable kernels abstract let t be the singular integral operator with variable kernel defined by t f x p v r n x x y x y n f y d y where x y is homogeneous of degree zero in the second variable y and s n 1 x z d z 0 for any x r n in this paper the authors prove that if l r n l q s n 1 for some q 2 n 1 n then the commutator generated by a cmo r n function and t and the associated lacunary maximal operator are compact on l 2 r n the associated continuous maximal commutator is also considered

analyticity of layer potentials and l 2 solvability of boundary value problems for divergence form elliptic equations with complex l infty coefficients we consider divergence form elliptic operators of the form l dv a x nabla defined in r n 1 x t in r n times r n geq 2 where the l infty coefficient matrix a is n 1

a general sos theory for the specification of probabilistic transition systems this article focuses on the formalization of the structured operational semantics approach for languages with primitives that introduce probabilistic and non deterministic behavior we define a general theoretic framework and present the rule format that guarantees that bisimulation equivalence in the probabilistic setting is a congruence for any operator defined in this format we show that the bisimulation is fully abstract w r t the format and possibilistic trace equivalence in the sense that bisimulation is the coarsest congruence included in trace equivalence for any operator definable within the format in other words bisimulation is the smallest congruence relation guaranteed by the format we also provide a conservative extension theorem and show that languages that include primitives for exponentially distributed time behavior such as imc and markov automata based language fit naturally within our framework

wave front sets local smoothing and bourgain s circular maximal theorem the purpose of this paper is to improve certain known regularity results for the wave equation and to give a simple proof of bourgain s circular maximal theorem 1 we use easy wave front analysis along with techniques previously used in proofs of the carleson sj6lin theorem see 3 5 7 and in the proof of sharp regularity properties of fourier integral operators 13 the circular means operators are defined by

a simple nsc rkhs algorithm for solving linear boundary value problems error bounds and implementations abstract the simple algorithm is in essence a normal spline collocation method introduced by v k gorbunov in 1989 the paper presents a reproducing kernel hilbert space approach to the method that gives rise to basic error inequalities for a natural h 2 class of mth order linear problems the approximation of any normal solution v is c m 1 uniform i e v a t v n a t c a n 2 v h m 2 for a m where n is diameter of the partition of the domain by nodes simplicity of the algorithm in sobolev spaces is justified by explicit formulas for the kernel functions discovered by gorbunov and v v petrishchev in 2003 and underestimated in the opinion of the author as an application numerical solutions in h 5 and in h 6 are compared with those computed by other authors the simple method is then applied to a fifth order nonlocal boundary value problem in h 6 and in h 7

polynomial growth estimates for multilinear singular integral operators on demontre un theoreme pour des mesures de carleson et on l applique a l operateur de kato en dimension 1 on demontre un critere de limitation general on donne une application aux commutateurs de coolderon on etudie les d commutateurs

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

on an inverse type problem for the heat equation in parabolic regular graph domains in this paper we prove some results concerning inverse free boundary type problems below the continuous threshold for the heat equation in the setting of parabolic regular graph domains

dispersion of small amplitude solutions of the generalized korteweg de vries equation abstract we study the long time behavior of small solutions of the initial value problem for the generalized korteweg de vries equation t u x 3 u x f u 0 gkdv u x 0 g x for the case where f w w s with s 1 4 23 57 3 8625 our results imply that if g l11 g l22 is sufficiently small then sup r 1 t 1 3 u t l in particular the solution tends to zero in the supremum norm the proofs make use of duhamel s formula and dispersion estimates for the linear propagator as well as chain and leibniz rules for fractional derivatives of compositions d f u lp and products d fg lp 0

a weighted inequality for the maximal bochner riesz operator on r2 for e r2 let 7 1 2 2 i it is a well known theorem of carleson and sjolin that t defines a bounded operator on z 4 if a 0 in this paper we obtain an explicit weighted inequality of the form sup t f x 2w x dxti f 2paw x dx 0 0 this strengthens the above theorem of carleson and sjolin the method gives information on the maximal operator associated to general suitably smooth radial fourier multipliers of r2

l2 solvability of boundary value problems for divergence form parabolic equations with complex coefficients abstract we consider parabolic operators of the form t l l div a x t in r n 2 x t x x n 1 t r n r r x n 1 0 n 1 we assume that a is a n 1 n 1 dimensional matrix which is bounded measurable uniformly elliptic and complex and we assume in addition that the entries of a are independent of the spatial coordinate x n 1 as well as of the time coordinate t for such operators we prove that the boundedness and invertibility of the corresponding layer potential operators are stable on l 2 r n 1 c l 2 r n 2 c under complex l perturbations of the coefficient matrix subsequently using this general result we establish solvability of the dirichlet neumann and regularity problems for t l by way of layer potentials and with data in l 2 assuming that the coefficient matrix is a small complex perturbation of either a constant matrix or of a real and symmetric matrix

process algebra for performance evaluation this paper surveys the theoretical developments in the field of stochastic process algebras process algebras where action occurrences may be subject to a delay that is determined by a random variable a huge class of resource sharing systems like large scale computers client server architectures networks can accurately be described using such stochastic specification formalisms the main emphasis of this paper is the treatment of operational semantics notions of equivalence and sound and complete axiomatisations of these equivalences for different types of markovian process algebras where delays are governed by exponential distributions starting from a simple actionless algebra for describing time homogeneous continuous time markov chains we consider the integration of actions and random delays both as a single entity like in known markovian process algebras like tipp pepa and empa and as separate entities like in the timed process algebras timed csp and tccs in total we consider four related calculi and investigate their relationship to existing markovian process algebras we also briefly indicate how one can profit from the separation of time and actions when incorporating more general non markovian distributions

bialgebras for structural operational semantics bialgebras and distributive laws are an abstract categorical framework to study various flavors of structural operational semantics this paper aims to introduce the reader to the basics of bialgebras for operational semantics and to sketch the state of the art in this research area

maximal operators along surfaces of revolution in lebesgue mixed norm spaces in this paper we establish the boundedness of certain maximal operators along hyperspaces in lebesgue mixed norm spaces

fast variants of the nsc rkhs algorithm for solving linear boundary value problems the fast algorithms work in the sobolev hilbert spaces on a finite interval 0 t and arise from the normal spline collocation method introduced in 4 by gorbunov and developed within the last 7years by chinese mathematicians as a reproducing kernel hilbert space method for a natural h 4 class of mth order linear problems ly f two extended variants of the method applied to an operator h m 2 h 2 yield 4th order uniform approximations of any normal solution v together with its derivatives up to order m 1 i e v b v n b c 0

restriction and kakeya phenomena for finite fields the restriction and kakeya problems in euclidean space have received much attention in the last few decades and they are related to many problems in harmonic analysis partial differential equations pdes and number theory in this paper we initiate the study of these problems on finite fields in many cases the euclidean arguments carry over easily to the finite setting and are in fact somewhat cleaner but there are some new phenomena in the finite case which deserve closer study

representation of the exact solution for infinite system of linear equations in this paper representation of the exact solution for the infinite system of linear equations is given and we give a sufficient and necessary condition for the infinite system of linear equations which has solutions

terminal coalgebras in well founded set theory abstract this paper shows that in order to obtain the theorem of aczel and mendler on the existence of terminal coalgebras for an endofunctor on the category of sets it is entirely unnecessary to delve into such exotica as non well founded set theory in addition we discuss the canonical map from the initial algebra for an endofunctor on sets to the terminal coalgebra and show that in many cases it embeds the former as a dense subset of the latter in a certain natural topology by way of example we calculate the terminal coalgebra for various simple endofunctors

caloric measure in parabolic flat domains in this paper we define parabolic chord arc domains and show that in a parabolic chord arc domain with vanishing constant the logarithm of the density of caloric measure with respect to a certain projective lebesgue measure is of vanishing mean oscillation we also obtain a partial converse to this result our work generalizes to the heat equation recent work of kenig and toro for the laplacian

regularity properties of commutators and layer potentials associated to the heat equation in recent years there has been renewed interest in the solution of parabolic boundary value problems by the method of layer potentials in this paper we consider graph domains d x t x f t in 9j2 where the boundary function f is in i112 bmo this class of domains would appear to be the minimal smoothness class for the solvability of the dirichlet problem for the heat equation by the method of layer potentials we show that for 1 p oo the boundary single layer potential operator for d maps lp into the homogeneous sobolev space i1i2 lp this regularity result is obtained by studying the regularity properties of a related family of commutators along the way we prove lp estimates for a class of singular integral operators to which the t 1 theorem of david and journe does not apply the necessary estimates are obtained by a variety of real variable methods

bisimulation through probabilistic testing abstract we propose a language for testing concurrent processes and examine its strength in terms of the processes that are distinguished by a test by using probabilistic transition systems as the underlying semantic model we show how a testing algorithm can distinguish with a probability arbitrarily close to one between processes that are not bisimulation equivalent we also show a similar result in a slightly stronger form for a new process relation called 2 3 bisimulation which lies strictly between that of simulation and bisimulation finally the ultimately strength of the testing language is shown to identify a new process relation called probabilistic bisimulation which is strictly stronger than bisimulation

some inequalities for singular convolution operators in spaces suppose that a bounded function m satisfies a localized multiplier condition supt o ii0m tp imp 0 the purpose of this paper is to find easily verified conditions that 1 implies me mp the condition llmll 0 then we ask under which conditions supt 0 qm tp 1mp oc implies m c mp throughout this paper x will always be chosen as in the following definition x c co rn satisfies a tauber condition with respect to the dilations tp if for every x 0 there is a tx such that q tpx 0 sometimes we need special bump functions of the following kind let p e c rn be a p homogeneous distance function this means that p tpx tp x received by the editors august 21 1986 1980 mathermatics subject lassijication 1985 revision primary 42b15 42b20 42b25

inequalities for some maximal functions ii let s be a smooth compact hypersurface in rn and let u be a measure on s absolutely continuous with respect to surface measure for t in r tt denotes the dilate of u by t normalised to have the same total variation as u for f in s rn t f denotes the maximal function supt o l t f we seek conditions on u which guarantee that the a priori estimate ll0 flip clhflip f g s rn holds this estimate entails that the sublinear operator extends to a bounded operator on the lebesgue space lp rn our methods generalise e m stein s treatment of the spherical maximal function 5 a study of riesz operators g functions and analytic families of measures reduces the problem to that of obtaining decay estimates for the fourier transform of u these depend on the geometry of s and the relation between u and surface measure on s in particular we find that there are natural geometric maximal operators limited on lp rn if and only if p e q oo q is some number in 1 oo and may be greater than 2 this answers a question of s wainger posed by stein 61 0 notation and contents the following notation will be used the fourier transform f of a function f on rn is given by f f dxf x e 2 x o e rn rn by b x r we mean the open ball of radius r and centre x the unit sphere in rn is sn 1 we often use polar coordinates p and o with p in 0 x and o in sn by expressions like c cp ck r etc we denote various constants which may vary from line to line and which depend on p k and r and so on these constants will normally also depend on other parameters like n the dimension of the ambient space but we do not mention this dependence explicitly given f in s rn we denote by t secondary 42b10

weighted norm inequality for the singular integral with variable kernel and fractional differentiation abstract for 0 1 let d be the fractional differentiation operator and t be the singular integral operator with variable kernel which is defined by t f x r n x x y x y n f y d y let t and t denote the adjoint of t and the pseudo adjoint of t respectively let t 1 t 2 denote the product of t 1 and t 2 t 1 t 2 denote the pseudo product of t 1 and t 2 in this paper we give the weighted norm inequalities for commutators of these singular integral operators and the fractional differentiation operator d finally we extend all results to the morrey spaces

interior estimates in morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients abstract let us consider the nondivergence form elliptic equation a ij u x i x j in this paper we show that if the known term belongs to the morrey space l p then the second derivatives of the w 2 p solution u belong to the same space next we derive a c 1 loc regularity result that is related to a recent work by caffarelli

structural operational semantics for stochastic and weighted transition systems we introduce weighted gsos a general syntactic framework to specify well behaved transition systems where transitions are equipped with weights coming from a commutative monoid we prove that weighted bisimilarity is a congruence on systems defined by weighted gsos specifications we illustrate the flexibility of the framework by instantiating it to handle some special cases most notably that of stochastic transition systems through examples we provide weighted gsos definitions for common stochastic operators in the literature

solving fredholm integro differential equations using reproducing kernel hilbert space method in this study the numerical solution of fredholm integro differential equation is discussed in a reproducing kernel hilbert space a reproducing kernel hilbert space is constructed in which the initial condition of the problem is satisfied the exact solution ux is represented in the form of series in the space w 2 2 a b in the mean time the n term approximate solution u n x is obtained and is proved to converge to the exact solution u x furthermore we present an iterative method for obtaining the solution in the space w 2 2 a b some examples are displayed to demonstrate the validity and applicability of the proposed method the numerical result indicates that the proposed method is straightforward to implement efficient and accurate for solving linear and nonlinear fredholm integro differential equations

singular integrals related to the radon transform and boundary value problems abstract two classes of integral operators are introduced that combine features of calderon zygmund singular integrals the hilbert integral and the radon transform examples and applications arising from integral geometry and several complex variables are discussed

the boundedness for commutators of a class of maximal hypersingular integrals with variable kernels abstract in this paper the authors proved that a class of maximal hypersingular integrals with variable kernels are bounded from the sobolev space l 2 r n to the lebesgue space l 2 r n which is a substantial improvement and extension of some known results