boundedness of inhomogeneous journe s type operators on multi parameter local hardy spaces abstract in this paper we establish a boundedness criterion of a class of inhomogeneous journe s type of multi parameter singular integral operators on multi parameter local hardy spaces h p r n 1 r n k recently introduced in ding et al 2019 the consideration of such inhomogeneous journe s type of multi parameter singular integral operators is motivated by the study of the multi parameter pseudo differential operators on multi parameter local hardy spaces the lack of the vanishing moments of both the corresponding functions in the multi parameter local calderon reproducing formula and the multi parameter local littlewood paley stein functions as well as the lack of vanishing moments of atoms in the multi parameter local hardy spaces h p r n 1 r n k with large size all create substantial difficulties in our proofs

multilinear calderon zygmund theory abstract a systematic treatment of multilinear calderon zygmund operators is presented the theory developed includes strong type and endpoint weak type estimates interpolation a multilinear t 1 theorem and a variety of results regarding multilinear multiplier operators

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

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 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

average sampling and reconstruction of reproducing kernel signals in mixed lebesgue spaces abstract in this paper we mainly study the average sampling and reconstruction of signals in a reproducing kernel subspace of the mixed lebesgue space l p q r m n first the sampling stability for two kinds of average sampling functionals is considered then the corresponding iterative approximation projection algorithms with exponential convergence are established finally the asymptotic pointwise error estimates are given for reconstructing a signal from its average samples corrupted by additive random noise

a hormander type multiplier theorem for multilinear operators abstract in this paper we prove a hormander type multiplier theorem for multilinear operators as a corollary we can weaken the regularity assumption for multilinear fourier multipliers to assure the boundedness

on the atomic decomposition for hardy spaces on lipschitz domains of rn let be a special lipschitz domain on rn and l be a second order elliptic self adjoint operator in divergence form l div a on lipschitz domain subject to neumann boundary condition in this paper we give a simple proof of the atomic decomposition for hardy spaces hnp of for a range of p by means of nontangential maximal function associated with the poisson semigroup of l

on multilinear square function and its applications to multilinear littlewood paley operators with non convolution type kernels abstract let m 2 n 1 and x r n define the multilinear square function t by t f x 0 r n m k t x y 1 y m j 1 m f j y j d y 1 d y m 2 d t t 1 2 where the kernel k satisfies a class of integral smooth conditions which is much weaker than the standard calderon zygmund kernel conditions in this paper we first established the l p 1 w 1 l p m w m l p estimate of t when each p i 1 and weak type l p 1 w 1 l p m w m l p estimate of t when there is a p i 1 where i 1 m i p p i and each w i is a nonnegative function on r n as applications of the above results we obtained the boundedness of multilinear littlewood paley operators with non convolution type kernels including multilinear g function marcinkiewicz integral and g function

an inequality for characteristic functions and its applications to uncertainty relations and the quantum zeno effect an inequality concerning characteristic functions is established it is useful in studying zero neighbourhood behaviours of characteristic functions the physical implications for the time energy uncertainty relations and the quantum zeno effect are indicated

necessary and sufficient conditions for the quantum zeno and anti zeno effect a necessary and sufficient condition for the occurrence of the quantum zeno effect is given refining a recent conjecture of luo wang and zhang an analogous condition is derived for the quantum anti zeno effect both results rely on a formal connection between the quantum anti zeno effect and the weak law of large numbers

extension of multilinear fractional integral operators to linear operators on mixed norm lebesgue spaces in kenig and stein math res lett 6 1 1 15 1999 https doi org 10 4310 mrl 1999 v6 n1 a1 the following type of multilinear fractional integral begin aligned int mathbb r mn frac f 1 l 1 x 1 ldots x m x cdots f m 1 l m 1 x 1 ldots x m x x 1 cdots x m lambda dx 1 ldots dx m end aligned was studied where l i are linear maps from mathbb r m 1 n to mathbb r n satisfying certain conditions they proved the boundedness of such multilinear fractional integral from l p 1 times cdots times l p m 1 to l q when the indices satisfy the homogeneity condition in this paper we show that the above multilinear fractional integral extends to a linear operator for functions in the mixed norm lebesgue space l mathbf p which contains l p 1 times cdots times l p m 1 as a subset under less restrictions on the linear maps l i we give a complete characterization of the indices mathbf p q and lambda for which such an operator is bounded from l mathbf p to l q and for m 1 or n 1 we give necessary and sufficient conditions on l 1 ldots l m 1 mathbf p p 1 ldots p m 1 q and lambda such that the operator is bounded

boundedness of multi parameter fourier multiplier operators on triebel lizorkin and besov lipschitz spaces abstract the main purpose of this paper is three fold first we prove that under the limited smoothness conditions multi parameter fourier multiplier operators are bounded on multi parameter triebel lizorkin and besov lipschitz spaces by the littlewood paley decomposition and the strong maximal operator second we offer a different and more direct method to deal with the boundedness instead of transforming fourier multiplier operators into multi parameter calderon zygmund operators third we also prove the boundedness of multi parameter fourier multiplier operators on weighted multi parameter triebel lizorkin and besov lipschitz spaces when the fourier multiplier is only assumed with limited smoothness

a projection based error analysis of hdg methods we introduce a new technique for the error analysis of hybridizable discontinuous galerkin hdg methods the technique relies on the use of a new projection whose design is inspired by the form of the numerical traces of the methods this renders the analysis of the projections of the discretization errors simple and concise by showing that these projections of the errors are bounded in terms of the distance between the solution and its projection our studies of influence of the stabilization parameter are reduced to local analyses of approximation by the projection we illustrate the technique on a specific hdg method applied to a model second order elliptic problem

on the coupling of boundary integral and mixed finite element methods we present a numerical method for solving an exterior dirichlet problem in the plane the technique consists in coupling boundary integral and mixed finite element methods an artificial boundary is introduced separating an interior region from an exterior one from an integral representation of the solution in the exterior domain we deduce two integral equations which relate the solution and its normal derivative over the artificial boundary these integral equations are incorporated into the so called mixed formulation of the problem in the interior region and a finite element method is used to approximate the resulting variational problem

sufficient conditions for the anti zeno effect the ideal anti zeno effect means that a perpetual observation leads to an immediate disappearance of the unstable system we present a straightforward way to derive sufficient conditions under which such a situation occurs expressed in terms of the decaying states and spectral properties of the hamiltonian they show in particular that the gap between zeno and anti zeno effects is in fact very narrow

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

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

boundedness of singular integrals on the flag hardy spaces on heisenberg group abstract we prove that the classical one parameter convolution singular integrals on the heisenberg group are bounded on multiparameter flag hardy spaces which satisfy the intermediate dilation between the one parameter anisotropic dilation and the product dilation on c n r implicitly

littlewood paley characterization and duality of weighted anisotropic product hardy spaces the authors study anisotropic product hardy spaces hwp a associated with a pair a a1 a2 of expansive dilations and a class of product muckenhoupt weights a a on rn rm this article is a continuation of their earlier work in bownik et al 2010 8 the authors establish the littlewood paley g function characterization and the transform characterization of hwp a 0 p 1 the authors also introduce the weighted anisotropic product campanato space lp w a and establish its transform characterization as an application the authors identify the dual space of hwp a with lp w a the results of this article improve the existing results for weighted product hardy spaces on r r and are new even in the weighted isotropic setting

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

quasi banach valued inequalities via the helicoidal method abstract we extend the helicoidal method from 1 to the quasi banach context proving in this way multiple banach and quasi banach vector valued inequalities for paraproducts and for the bilinear hilbert transform bht as an immediate application we obtain mixed norm estimates for in the whole range of lebesgue exponents one of the novelties in the quasi banach framework that is when 0 r 1 which we expect to be useful in other contexts as well is the linearization of the operator k t f k g k r 1 r achieved by dualizing its weak l p quasinorms through l r see proposition 8 another important role is played by the sharp evaluation of the operatorial norm t i 0 f 1 f g 1 g 1 h r which is obtained by dualizing the weak l p quasinorms through l with r in the banach case the linearization of the operator and the sharp estimates for the localized operatorial norm can be both achieved through the classical generalized restricted type l 1 dualization

and variants of multiparameter calderon zygmund theory we consider calderon zygmund operators on product domains under certain weak conditions on the kernel a singular integral operator can be proved to be bounded on h p r r r 0 p 1 if its behaviour on l 2 and on certain scalar valued and vector valued rectangle atoms is known another result concerns an extension of the authors results on l p variants of calderon zygmund theory 1 23 to the product domain setting as an application one obtains estimates for fourier multipliers and pseudo differential operators

a direct coupling of local discontinuous galerkin and boundary element methods the coupling of local discontinuous galerkin ldg and boundary element methods bem which has been developed recently to solve linear and nonlinear exterior transmission problems employs a mortar type auxiliary unknown to deal with the weak continuity of the traces at the interface boundary as a consequence the main features of ldg and bem are maintained and hence the coupled approach benefits from the advantages of both methods in this paper we propose and analyze a simplified procedure that avoids the mortar variable by employing ldg subspaces whose functions are continuous on the coupling boundary the continuity can be implemented either directly or indirectly via the use of lagrangian multipliers in this way the normal derivative becomes the only boundary unknown and hence the total number of unknown functions is reduced by two we prove the stability of the new discrete scheme and derive an a priori error estimate in the energy norm a numerical example confirming the theoretical result is provided the analysis is also extended to the case of nonlinear problems and to the coupling with other discontinuous galerkin methods

square roots of elliptic second order divergence operators on strongly lipschitz domains l p theory we study l p estimates for square roots of second order elliptic non necessarily selfadjoint operators in divergence form l mathrm div a nabla on lipschitz domains subject to dirichlet or to neumann boundary conditions pursuing our work 4 where we considered operators on mathbb r n we obtain among other things l 1 2 f p le c nabla f p for all 1 p infty if l is real symmetric and the domain bounded which is new for 1 p 2 we also obtain similar results for perturbations of constant coefficients operators our methods rely on a singular integral representation calderon zygmund theory and quadratic estimates a feature of this study is the use of a commutator between the resolvent of the laplacian dirichlet and neumann and partial derivatives which carries the geometry of the boundary

the small deborah number limit of the doi onsager equation to the ericksen leslie equation we present a rigorous derivation of the ericksen leslie equation starting from the doi onsager equation as in the fluid dynamic limit of the boltzmann equation we first make the hilbert expansion for the solution of the doi onsager equation the existence of the hilbert expansion is connected to an open question whether the energy of the ericksen leslie equation is dissipated we show that the energy is dissipated for the ericksen leslie equation derived from the doi onsager equation the most difficult step is to prove a uniform bound for the remainder in the hilbert expansion this question is connected to the spectral stability of the linearized doi onsager operator around a critical point by introducing two important auxiliary operators the detailed spectral information is obtained for the linearized operator around all critical points however these are not enough to justify the small deborah number limit for the inhomogeneous doi onsager equation since the elastic stress in the velocity equation is also strongly singular for this we need to establish a precise lower bound for a bilinear form associated with the linearized operator in the bilinear form the interactions between the part inside the kernel and the part outside the kernel of the linearized operator are very complicated we find a coordinate transform and introduce a five dimensional space called the maier saupe space such that the interactions between two parts can been seen explicitly by a delicate argument of completing the square however the lower bound is very weak for the part inside the maier saupe space in order to apply them to the error estimates we have to analyze the structure of the singular terms and introduce a suitable energy functional

product hardy spaces associated to operators with heat kernel bounds on spaces of homogeneous type the aim of this article is to develop the theory of product hardy spaces associated with operators which possess the weak assumption of davies gaffney heat kernel estimates in the setting of spaces of homogeneous type we also establish a calderon zygmund decomposition on product spaces which is of independent and use it to study the interpolation of these product hardy spaces we then show that under the assumption of generalized gaussian estimates the product hardy spaces coincide with the lebesgue spaces for an appropriate range of p

differentiation of the integral hardy spaces and calderon zygmund operators in the product setting abstract this article concerns strong differentiation and operators on product hardy spaces we first show that an example created by papoulis of a function on r 2 whose integral is strongly differentiable almost everywhere but the integral of its absolute value fails to be strongly differentiable on a set of positive measure belongs to the product hardy space h 1 r r the methods that we develop enable us to present a relaxed version of chang fefferman p atoms with a lower number of required vanishing moments and no smoothness needed on the elementary particles in analogy with the proof of this result we show a generalization of a theorem of r fefferman which concludes h p l p 0 p 1 boundedness of multiparameter operators from their behavior on rectangle atoms in addition we extend a result of pipher concerning boundedness of multiparameter calderon zygmund operators from h p to l p

multiparameter hardy space theory on carnot caratheodory spaces and product spaces of homogeneous type this paper is inspired by the work of nagel and stein in which the lp 1 p theory has been developed in the setting of the product carnot caratheodory spaces m m1 mn formed by vector fields satisfying hormander s finite rank condition the main purpose of this paper is to provide a unified approach to develop the multiparameter hardy space theory on product spaces of homogeneous type this theory includes the product hardy space its dual the product bmo space the boundedness of singular integral operators and calderon zygmund decomposition and interpolation of operators as a consequence we obtain the endpoint estimates for those singular integral operators considered by nagel and stein 2004 in fact we will develop most of our theory in the framework of product spaces of homogeneous type which only satisfy the doubling condition and some regularity assumption on the metric all of our results are established by introducing certain banach spaces of test functions and distributions developing discrete calderon identity and discrete littlewood paley stein theory our methods do not rely on the journe type covering lemma which was the main tool to prove the boundedness of singular integrals on the classical product hardy spaces

the euler equations in planar nonsmooth convex domains abstract as a model problem for the barotropic mode of the primitive equations of the oceans and atmosphere we consider the euler system on a bounded convex planar domain endowed with non penetrating boundary conditions for 4 3 p 2 and initial and forcing data with l p vorticity we show the existence of a weak solution enriching and extending the results of taylor 2000 32 in the physical case of a rectangular domain 0 l 1 0 l 2 a similar result holds for all 2 p as well moreover by means of a new bmo type regularity estimate for the dirichlet problem on a planar domain with corners we prove uniqueness of solutions with bounded initial vorticity

a note onbmoand bloch space of generalized α β harmonic functions abstract in this paper we study functions in the unit ball in c n annihilated by the differential operators c given by formula we obtain two equivalent carleson measure characterizations of those of such functions having boundary value in bmo we also give analogous carleson bergman measure characterizations of harmonic bloch functions

well posedness of the water waves equations 1 1 presentation of the problem the water waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect incompressible irrotational fluid under the influence of gravity in this paper we restrict our attention to the case when the surface is a graph parameterized by a function x where t denotes the time variable and x xi xd rd the horizontal spatial variables the method developed here works equally well for any integer d 1 but the only physically relevant cases are of course d 1 and d 2 the layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time independent function b x we denote by q t the fluid domain at time t the incompressibility of the fluids is expressed by 1 1 div v 0 in qu t 0

the hormander multiplier theorem for multilinear operators in this paper we provide a version of the mikhlin hormander multiplier theorem for multilinear operators in the case where the target space is l p for pe 1 this extends a recent result of tomita 15 who proved an analogous result for p 1

sobolev bmo and fractional integrals on super critical ranges of lebesgue spaces abstract in this article we explore the mapping and boundedness properties of linear and bilinear fractional integral operators acting on lebesgue spaces with large indices the prototype order fractional integral operator is the riesz potential i and the standard estimates for i are from l p into l q when 1 p n and 1 p 1 q n we show that a order linear fractional integral operator can be continuously extended to a bounded operator from l p into the sobolev bmo space i s b m o when n p and 0 s satisfy 1 p s n likewise we prove estimates for order bilinear fractional integral operators from l p 1 l p 2 into i s b m o for various ranges of the indices p 1 p 2 and s satisfying 1 p 1 1 p 2 s n

the atomic decomposition of weighted hardy spaces associated to self adjoint operators on product spaces abstract let l be a non negative self adjoint operator acting on l 2 r n satisfying a pointwise gaussian estimate for its heat kernel let w be an a p weight on r n r n 1 p in this article we obtain a weighted q atomic decomposition with q p for the weighted hardy space h l w 1 r n r n associated to l

coupling of raviart thomas and hybridizable discontinuous galerkin methods with bem we present new a priori error analyses of the coupling of the raviart thomas rt method and bem as well as of the coupling of the hybridizable discontinuous galerkin hdg method with bem the novel features of the analysis of the rt bem coupling are the superconvergence estimates for the scalar approximation of the interior problem and new bounds in weak and strong norms for the boundary variables the analysis of the hdg bem coupling is the first analysis of this coupling and shows that the coupling provides approximations with the same convergence properties as those of the rt bem coupling

analysis of a non symmetric coupling of interior penalty dg and bem we analyze a non symmetric coupling of interior penalty discontinuous galerkin and boundary element methods in two and three dimensions main results are discrete coercivity of the method and thus unique solvability and quasi optimal convergence the proof of coercivity is based on a localized variant of the varia

existence and boundedness of multilinear littlewood paley operators on campanato spaces abstract in this paper we established the boundedness of multilinear littlewood paley operators on campanato spaces including the multilinear g function multilinear lusin s area integral and multilinear littlewood paley g function we proved that if the above operators are finite for one point then they are finite almost everywhere moreover norm inequalities do hold for the above operators on the corresponding campanato spaces

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

multi parameter local hardy spaces abstract though multi parameter hardy space theory has been well developed in the past half century not much has been studied for a local hardy space theory in the multi parameter settings such multi parameter local hardy spaces can play an important role in studying the boundedness of multi parameter pseudo differential operators multi parameter singular integrals of non convolution type and applications to partial differential equations etc by establishing a bi parameter local reproducing formula bi parameter local hardy space h p r n 1 r n 2 is introduced in this paper this space coincides with l p r n 1 n 2 when p 1 while p 1 this space is substantially different from the classical bi parameter hardy spaces h p r n 1 r n 2 we will establish its atomic decomposition of the bi parameter local hardy spaces and as an application we prove the boundedness from h p r n 1 r n 2 to l p r n 1 n 2 of the bi parameter singular integral operators in inhomogeneous journe class for p near 1 for the simplicity we have chosen to present all the results in the bi parameter setting nevertheless all of them hold for arbitrary number of parameters the multi parameter local theory developed in this paper can serve as a model case for similar theory in other multi parameter settings