counting dimensions of l harmonic functions in this article we will consider second order uniformly elliptic operators of divergence form defined on lri with measurable coefficients mainly we will give estimates on the dimension of space of solutions that grow at most polynomially of degree d more precisely in terms of a rectangular coordinate system xi n a second order uniformly elliptic operator of divergence form l acting on a function f e h c rin is given by

new abstract hardy spaces the aim of this paper is to propose an abstract construction of spaces which keep the main properties of the already known hardy spaces h1 we construct spaces through an atomic or molecular decomposition we prove some results about continuity from these spaces into l1 and some results about interpolation between these spaces and the lebesgue spaces we also obtain some results on weighted norm inequalities finally we present partial results in order to understand a characterization of the duals of hardy spaces

semigroup kernels poisson bounds and holomorphic functional calculus letlbe the generator of a continuous holomorphic semigroupswhose action is determined by an integral kernelkon a scale of spaceslp x under mild geometric assumptions on x we prove that iflhas a boundedh functional calculus onl2 x andksatisfies bounds typical for the poisson kernel thenlhas a boundedh functional calculus onlp x for eachp 1 moreover if x is of polynomial type andksatisfies second order gaussian bounds we establish criteria forlto have a bounded hormander functional calculus or a bounded davies helffer sjostrand functional calculus

duality of hardy and bmo spaces associated with operators with heat kernel bounds the introduction and development of hardy and bmo spaces on euclidean spaces r in the 1960s and 1970s played an important role in modern harmonic analysis and applications in partial differential equations these spaces were studied extensively in 32 22 18 19 31 and many others an l function f on r is in the hardy spaceh r if the area integral function of the poisson integral e t f satisfies

on fundamental solutions of generalized schrodinger operators abstract we consider the generalized schrodinger operator where is a nonnegative radon measure in r n n 3 assuming that satisfies certain scale invariant kato conditions and doubling conditions we establish the following bounds for the fundamental solution of in r n ce e 2 d x y x y n 2 x y ce e 1 d x y x y n 2 where d x y is the distance function for the modified agmon metric m x dx2 associated with we also study the boundedness of the corresponding riesz transforms 1 2 on lp r n dx

two species coagulation approach to consensus by group level interactions we explore the self organization dynamics of a set of entities by considering the interactions that affect the different subgroups conforming the whole to this end we employ the widespread example of coagulation kinetics and characterize which interaction types lead to consensus formation and which do not as well as the corresponding different macroscopic patterns the crucial technical point is extending the usual one species coagulation dynamics to the two species one this is achieved by means of introducing explicitly solvable kernels which have a clear physical meaning the corresponding solutions are calculated in the long time limit in which consensus may or may not be reached the lack of consensus is characterized by means of scaling limits of the solutions the possible applications of our results to some topics in which consensus reaching is fundamental such as collective animal motion and opinion spreading dynamics are also outlined

abstract hardy sobolev spaces and interpolation the purpose of this work is to describe an abstract theory of hardy sobolev spaces on doubling riemannian manifolds via an atomic decomposition we study the real interpolation of these spaces with sobolev spaces and finally give applications to riesz inequalities

use of abstract hardy spaces real interpolation and applications to bilinear operators this paper can be considered as the sequel of bernicot and zhao j func anal 255 1761 1796 2008 where the authors have proposed an abstract construction of hardy spaces h1 they shew an interpolation result for these hardy spaces with the lebesgue spaces here we describe a more precise result using the real interpolation theory and we clarify the use of hardy spaces then with the help of the bilinear interpolation theory we then give applications to study bilinear operators on lebesgue spaces these ideas permit us to study singular operators with singularities similar to those of bilinear calderon zygmund operators in a far more abstract framework as in the euclidean case

a master equation approach to option pricing abstract a master equation approach to the numerical solution of option pricing models is developed the basic idea of the approach is to consider the black scholes equation as the macroscopic equation of an underlying mesoscopic stochastic option price variable the dynamics of the latter is constructed and formulated in terms of a master equation the numerical efficiency of the approach is demonstrated by means of stochastic simulation of the mesoscopic process for both european and american options

second order riesz transforms associated to the schrodinger operator for p 1 abstract let l v be the schrodinger operator on r n where v belongs to the class of reverse holder weights r h q for some q max 2 n 2 we show that the second order riesz transforms 2 l 1 and v l 1 are bounded from the hardy spaces h l p r n associated to l into l p r n for 0 p 1 we show also that the operators 2 l 1 map the classical hardy spaces h p r n into h p r n for a restricted range of p

orlicz hardy spaces associated to operators satisfying bounded h functional calculus and davies gaffney estimates abstract let x be a metric space with doubling measure and l be an operator which has a bounded h functional calculus and satisfies davies gaffney estimates in this paper we develop a theory of orlicz hardy spaces associated to l including a molecule decomposition square function characterization and duality of orlicz hardy spaces h l x finally we show that l has a bounded holomorphic functional calculus in h l x and the riesz transform is bounded from h l x to l

bmo spaces related to schrodinger operators with potentials satisfying a reverse holder inequality we identify the dual space of the hardy type space related to the time independent schrodinger operator v with v a potential satisfying a reverse holder inequality as a bmo type space we prove the boundedness in this space of the versions of some classical operators associated to hardy littlewood semigroup and poisson maximal functions square function fractional integral operator we also get a characterization of in terms of carlesson measures

a maximal function characterization for hardy spaces associated to nonnegative self adjoint operators satisfying gaussian estimates abstract let l be a nonnegative self adjoint operator satisfying gaussian estimates on l 2 r n in this article we give an atomic decomposition for the hardy spaces h l max p r n in terms of the nontangential maximal functions associated with the heat semigroup of l and this leads eventually to characterizations of hardy spaces associated to l via atomic decomposition or the nontangential maximal functions the proof is based on a modification of a technique due to a calderon 6

on characterization of poisson integrals of schrodinger operators with bmo traces abstract let l be a schrodinger operator of the form l v acting on l 2 r n where the nonnegative potential v belongs to the reverse holder class b q for some q n let bmo l r n denote the bmo space on r n associated to the schrodinger operator l in this article we will show that a function f bmo l r n is the trace of the solution of l u u t t l u 0 u x 0 f x where u satisfies a carleson condition sup x b r b r b n 0 r b b x b r b t u x t 2 d x d t c conversely this carleson condition characterizes all the l harmonic functions whose traces belong to the space bmo l r n this result extends the analogous characterization founded by fabes johnson and neri in 13 for the classical bmo space of john and nirenberg

new orlicz hardy spaces associated with divergence form elliptic operators abstract let l be the divergence form elliptic operator with complex bounded measurable coefficients the positive concave function on 0 of strictly critical lower type p 0 1 and t t 1 1 t 1 for t 0 in this paper the authors study the orlicz hardy space h l r n and its dual space bmo l r n where l denotes the adjoint operator of l in l 2 r n several characterizations of h l r n including the molecular characterization the lusin area function characterization and the maximal function characterization are established the carleson measure characterization and the john nirenberg inequality for the space bmo l r n are also given as applications the authors show that the riesz transform l 1 2 and the littlewood paley g function g l map h l r n continuously into l the authors further show that the riesz transform l 1 2 maps h l r n into the classical orlicz hardy space h r n for p n n 1 1 and the corresponding fractional integral l for certain 0 maps h l r n continuously into h l r n where is determined by and and satisfies the same property as all these results are new even when t t p for all t 0 and p 0 1

riesz transforms associated to schrodinger operators on weighted hardy spaces let w be some ap weight and enjoy reverse holder inequality and let l v be a schrodinger operator on rn where v lloc1 rn is a non negative function on rn in this article we introduce weighted hardy spaces hl w1 rn associated to l in terms of the area function characterization and prove their atomic characters we show that the riesz transform l 1 2 associated to l is bounded on lwp rn for 1 p 2 and bounded from hl w1 rn to the classical weighted hardy space hw1 rn

regularity properties of the heat kernel and area integral characterization of hardy space hl1 related to degenerate schrodinger operators abstract in this paper we introduce the area integral associated with the heat semigroup t t t 0 generated by a degenerate schrodinger operator l we obtain a regularity estimate of the kernels of the semigroup t t t 0 this regularity together with a reproducing formula enables us to establish the area integral characterization of the hardy space associated with l

classes of hardy spaces associated with operators duality theorem and applications let l be the infinitesimal generator of an analytic semigroup on l 2 r n with suitable upper bounds on its heat kernels in auscher duong and mcintosh 2005 and duong and yan 2005 a hardy space h1 l r n and a bmo l r n space associated with the operator l were introduced and studied in this paper we define a class of h p l r n spaces associated with the operator l for a range of p 1 acting on certain spaces of morrey campanato functions defined in new morrey campanato spaces associated with operators and applications by duong and yan 2005 and they generalize the classical h p r n spaces we then establish a duality theorem between the h p l r n spaces and the morrey campanato spaces in that same paper as applications we obtain the boundedness of fractional integrals on h p l r n and give the inclusion between the classical h p r n spaces and the h p l r n spaces associated with operators

abstract hardy spaces with variable exponents abstract in this paper we introduce the definition of abstract hardy spaces with variable exponents via the atomic decomposition and molecular decomposition we prove the continuity from our variable hardy space h a t o p into l p where p r n 0 1 0 p p 1 moreover we investigate the bilinear theory finally we give an application of the abstract hardy spaces with variable exponents

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

new sobolev spaces via generalized poincare inequalities on metric measure spaces in this paper we introduce some new function spaces of sobolev type on metric measure spaces these new function spaces are defined by variants of poincare inequalities associated with generalized approximations of the identity and they generalize the classical sobolev spaces on euclidean spaces we then obtain two characterizations of these new sobolev spaces including the characterization in terms of a variant of local sharp maximal functions associated with generalized approximations of the identity for the well known hajlasz sobolev spaces on metric measure spaces we also establish some new characterizations related to generalized approximations of the identity finally we clarify the relations between the sobolev type spaces introduced in this paper and the hajlasz sobolev spaces on metric measure spaces

maximal function characterizations for hardy spaces on spaces of homogeneous type with finite measure and applications abstract we prove nontangential and radial maximal function characterizations for hardy spaces associated to a non negative self adjoint operator satisfying gaussian estimates on a space of homogeneous type with finite measure this not only addresses an open point in the literature but also gives a complete answer to the question posed by coifman and weiss in the case of finite measure we then apply our results to give maximal function characterizations for hardy spaces associated to second order elliptic operators with neumann and dirichlet boundary conditions schrodinger operators with dirichlet boundary conditions and fourier bessel operators

the hodge de rham laplacian and lp boundedness of riesz transforms on non compact manifolds let m be a complete non compact riemannian manifold satisfying the volume doubling property as well as a gaussian upper bound for the corresponding heat kernel we study the boundedness of the riesz transform d 12 on both hardy spaces hp and lebesgue spaces lp under two different conditions on the negative part r of the ricci curvature first we prove that if r is subcritical for some 0 1 then the riesz transform d 12 on differential 1 forms is bounded from the associated hardy space h p 1t m to lp m for all p 1 2 as a consequence d 12 is bounded on lp for all p 1 p0 where p0 2 depends on and the constant appearing in the doubling property second we prove that if 01 r 12v t 1p1 p1dtt 1 r 12v t 1p2 p2dtt for some p1 2 and p2 3 then the riesz transform d 12 is bounded on lp for all 1 2 under conditions on r and the potential v we prove both positive and negative results on the boundedness of da 12 on lp

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

a_p weights and quantitative estimates in the schrodinger setting suppose l delta v is a schrodinger operator on mathbb r n with a potential v belonging to certain reverse holder class rh sigma with sigma ge n 2 the aim of this paper is to study the a p weights associated to l denoted by a p l which is a larger class than the classical muckenhoupt a p weights we first prove the quantitative a p l bound for the maximal function and the maximal heat semigroup associated to l then we further provide the quantitative a p q l bound for the fractional integral operator associated to l we point out that all these quantitative bounds are known before in terms of the classical a p q constant however since a p q subset a p q l the a p q l constants are smaller than a p q constant hence our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to l next we prove two weight inequalities for the fractional integral operator these have been unknown up to this point finally we also have a study on the exp log link between a p l and bmo l the bmo space associated with l and show that for w in a p l log w is in bmo l and that the reverse is not true in general

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

littlewood paley functions associated to second order elliptic operators we prove l p estimates for the littlewood paley function associated with a second order divergence form operator l div a with bounded measurable complex coefficients in n

bmo estimates for non commutative vector valued lipschitz functions abstract we construct markov semi groups t and associated bmo spaces on a finite von neumann algebra m and obtain results for perturbations of commutators and non commutative lipschitz estimates in particular we prove that for any a m self adjoint and f r r lipschitz there is a markov semi group t such that for x m f a x bmo m t c a b s f a x we obtain an analogue of this result for more general von neumann valued functions f r n n by imposing hormander mikhlin type assumptions on f in establishing these result we show that markov dilations of markov semi groups have certain automatic continuity properties we also show that markov semi groups of double operator integrals admit standard and reversed markov dilations

hardy and bmo spaces associated to divergence form elliptic operators consider a second order divergence form elliptic operator l with complex bounded measurable coefficients in general operators based on l such as the riesz transform or square function may lie beyond the scope of the calderon zygmund theory they need not be bounded in the classical hardy bmo and even some l p spaces in this work we develop a theory of hardy and bmo spaces associated to l which includes in particular a molecular decomposition maximal and square function characterizations duality of hardy and bmo spaces and a john nirenberg inequality

boundedness of spectral multipliers of generalized laplacians on compact manifolds with boundary consider a second order strongly elliptic negative semidefinite differential operator l may be a system on a compact riemannian manifold overline m with smooth boundary where the domain of l is defined by a coercive boundary condition classically known results and also recent work in duong et al j funct anal 196 443 485 2002 and duong and mcintosh rev math iberoam 15 233 265 1999 establish sufficient conditions for l infty text bmo l continuity of varphi sqrt l where varphi in s 0 1 mathbb r and l is a suitable elliptic operator using a variant of the cheeger gromov taylor functional calculus due to mauceri et al math res lett 16 861 879 2009 and short time upper bounds on the integral kernel of e tl due to greiner arch ration mech anal 41 168 218 1971 we prove that a variant of such sufficient conditions holds for our operator l

calderon reproducing formulas and new besov spaces associated with operators abstract let l be the generator of an analytic semigroup whose heat kernel satisfies an upper bound of poisson type acting on l 2 x where x is a possibly non doubling space of polynomial upper bound on volume growth the aim of this paper is to introduce a new class of besov spaces associated with the operator l so that when l is the laplace operator or its square root acting on the euclidean space r n the new besov spaces are equivalent to the classical besov spaces depending on the choice of l the new besov spaces are natural settings for generic estimates for certain singular integral operators such as the fractional powers l since our approach does not require the doubling volume property of the underlying space it is applicable to any subset of euclidean spaces without any smoothness requirement on the boundary we will also develop a number of calderon reproducing formulas which play an important role in the theory of function spaces and are of independent interest as an application we study besov spaces associated with schrodinger operators with non negative potentials satisfying reverse holder estimates

vmo spaces associated with divergence form elliptic operators consider the second order divergence form elliptic operator l with complex bounded coefficients in this paper we introduce and develop a function space vmol of vanishing mean oscillation associated with the operator l using the theory of tent spaces and the littlewood paley theory we prove that the hardy space h l 1 of hofmann and mayboroda introduced in hofmann and mayboroda math ann 344 37 116 2009 is the dual of our rm vmo l ast in which l is the adjoint operator of l we also give an equivalent characterization of the space vmol in the context of the theory of tent spaces

some new function spaces and their applications to harmonic analysis abstract in this paper a family of spaces is introduced which seems well adapted for the study of a variety of questions related to harmonic analysis and its applications these spaces are the tent spaces they provide the natural setting for the study of such things as maximal functions the relevant space here is t p and also square functions where the space t2p is relevant as such these spaces lead to unifications and simplifications of some basic techniques in harmonic analysis thus they are closely related to lp and hardy spaces important parts of whose theory become corollaries of the description of tent spaces also as proc conf harmonic analysis cortona lect notes in math vol 992 springer verlag berlin new york 1983 already indicated where these spaces first appeared explicitly the tent spaces can be used to simplify some of the results related to the cauchy integral on lipschitz curves and multilinear analysis in retrospect one can recognize that various ideas important for tent spaces had been used if only implicitly for quite some time here one should mention carleson s inequality its simplifications and extensions the theory of hardy spaces and atomic decompositions

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

new function spaces of bmo type the john nirenberg inequality interpolation and applications in this paper we introduce and develop some new function spaces of bmo bounded mean oscillation type on spaces of homogeneous type or measurable subsets of spaces of homogeneous type the new function spaces are defined by variants of maximal functions associated with generalized approximations to the identity and they generalize the classical bmo space we show that the john nirenberg inequality holds on these spaces and they interpolate with lp spaces by the complex interpolation method we then give applications on lp boundedness of singular integrals whose kernels do not satisfy the hormander condition 2005 wiley periodicals inc

noncommutative riesz transforms a probabilistic approach for 2 p 0 in the classical setting this is certainly satisfied for the maps tt f f 1 t where t is a semigroup of measure preserving maps on a probability space in that case n l then the maps tt act on all noncommutative lp spaces lp n and in particular on the hilbert space l2 l2 n let a be the negative generator i e tt e ta for technical reasons we have to assume that there is a weakly dense algebra a n which lies in the domain of the unbounded operator a i e a a a the algebra a plays the role of smooth functions in the classical setting this assumption might be too restrictive for some classical applications but it is easily verified in the new noncommutative examples for elements x y a we may define the gradient form 2 x y a x y x a y a x y the correct definition of is the one given by sauvageot as a bilinear form with values in l1 again our arguments require stronger assumptions in fact in all our examples we will assume that the tt s are selfadjoint i e ttxy xtty and unital i e tt 1 1 then a is indeed a positive unbounded operator under these circumstances we can formulate p a meyer s problem is it true that x x 2 p c p a 1 2 x p 0 1 holds for all selfadjoint elements x a let us illustrate this question by considering the laplace operator a f f where i 2 x2 i then it is easily verified that