on realizations of solutions of the kdv equation by the c 0 semigroup method the c0 semigroup method is used to find exact solutions of the korteweg de vries kdv equation ut uxxx 6uux various new kinds of solutions such as multisoliton solutions with unbounded phase numbers null solutions and solutions of wavelet type are found the realization program consists of two steps firstly the scalar kdv equation will be translated into an operator equation ut uxxx 3 uu x on the banach algebra of a given banach space the solution formula for the operator equation has been constructed proposition 2 1 secondly once all of the operators u x t are of rank one solutions of the scalar kdv equation will be extracted by using the trace functional the use of fractional powers of a closed unbounded linear operator in a banach space also plays an essential role in the present work

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

riemann stieltjes operators and multipliers on q p spaces in the unit ball of c n this paper is devoted to characterizing the riemann stieltjes operators and pointwise multipliers acting on mobius invariant spaces q p which unify bmoa and bloch space in the scale of p the boundedness and compactness of these operators on q p spaces are determined by means of an embedding theorem i e q p spaces boundedly embedded in the non isotropic tent type spaces t q infinity c 2010 elsevier inc all rights reserved

error analysis of a specialized numerical method for mathematical models from neuroscience abstract the exponential euler method is a nonstandard approximation scheme that was developed specifically for the hodgkin huxley differential equation models that arise in neuroscience and was one of the discretization schemes used in the neural systems package called genesis in this article we show the scheme is first order accurate develop a second order accurate extension and suggest ways the method can be used to compute approximations to certain time dependent partial differential equations we also apply the scheme to an integro differential equation model of neuronal activity and furnish sample computations that illustrate our theoretical results

growth of sobolev norms in the cubic nonlinear schrodinger equation with a convolution potential fix s 1 colliander et al proved in invent math 181 39 113 2010 the existence of solutions of the cubic defocusing nonlinear schrodinger equation in the two torus whose s sobolev norm undergoes arbitrarily large growth as time evolves in this paper we generalize their result to the cubic defocusing nonlinear schrodinger equation with a convolution potential moreover we show that the speed of growth is the same as the one obtained for the cubic defocusing nonlinear schrodinger equation in guardia and kaloshin growth of sobolev norms in the cubic defocusing nonlinear schrodinger equation to appear in the journal of the european mathematical society 2012

transfer of energy to high frequencies in the cubic defocusing nonlinear schrodinger equation we consider the cubic defocusing nonlinear schrodinger equa tion on the two dimensional torus we exhibit smooth solutions for which the support of the conserved energy moves to higher fourier modes this behavior is quantified by the growth of higher sobolev norms given any 1 k 1 s 1 we construct smooth initial data u0 withu0 h s k at some time t this growth occurs despite the hamiltonian s bound onu t h 1 and despite the conservation of the quantityu t l2

strong type estimates for homogeneous besov capacities capacitary strong type estimates for homogeneous besov spaces are established moreover the result is used to study embeddings and dualities of function spaces

the boundedness of riesz transforms for hermite expansions on the hardy spaces abstract let l x 2 be a hermite operator where is the laplacian on r d in this paper we first characterize the hardy spaces h l 1 r d associated with l by a new version of area integral then we use it to prove the boundedness of riesz transforms r j l j 1 2 for l on h l 1 r d moreover we characterize the hardy space h l 1 r d by r j l this gives a negative answer to a question asked by thangavelu

generalized coorbit space theory and inhomogeneous function spaces of besov lizorkin triebel type coorbit space theory is an abstract approach to function spaces and their atomic decom positions the original theory developed by feichtinger and grochenig in the late 1980ies heavily uses integrable representations of locally compact groups their theory covers in particular homogeneous besov lizorkin triebel spaces modulation spaces bergman spaces and the recent shearlet spaces however inhomogeneous besov lizorkin triebel spaces cannot be covered by their group theoretical approach later it was recognized by fornasier and the first named author 24 that one may replace coherent states related to the group representation by more general abstract continuous frames in the first part of the present paper we significantly extend this abstract generalized coorbit space theory to treat a wider variety of coorbit spaces a unified approach towards atomic decompositions and banach frames with new results for general coorbit spaces is presented in the second part we apply the abstract setting to a specific framework and study coorbits of what we call peetre spaces they allow to recover inhomogeneous besov lizorkin triebel spaces of var ious types of interest as coorbits we obtain several old and new wavelet characterizations based on explicit smoothness decay and vanishing moment assumptions of the respective wavelet as main examples we obtain results for weighted spaces muckenhoupt doubling general 2 microlocal spaces besov lizorkin triebel morrey spaces spaces of dominating mixed smoothness and even mixtures of the mentioned ones due to the generality of our approach there are many more examples of interest where the abstract coorbit space theory is applicable

square function non tangential maximal function estimates and the dirichlet problem for non symmetric elliptic operators a we consider divergence form elliptic operators l div a x defined in the half space rn 1 n 2 where the coefficient matrix a x is bounded measurable uniformly elliptic t independent and not necessarily symmetric we establish square function non tangential maximal function estimates for solutions of the homogeneous equation lu 0 and we then combine these estimates with the method of approximability to show that l harmonic measure is absolutely continuous with respect to surface measure i e n dimensional lebesgue measure on the boundary in a scale invariant sense more precisely that it belongs to the class a with respect to surface measure equivalently that the dirichlet problem is solvable with data in lp for some p previously these results had been known only in the case n 1

generalized area operators on hardy spaces abstract we show that if 0 0 and 1 p 0 then the operatorgfmaps the hardy sobolev spacehp intolp d if and only if the functiong d 1 z belongs to the morrey spacelp 1 p in casep 1 this condition is equivalent to the condition that t i c i 1 for all arcsicontained in the circle wheret i is the tent overicontained in the unit disk

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

dynamics of neuronal waves first of all by studying the existence and stability of traveling wave fronts of the following nonlinear nonlocal model equation u t au x u alpha mathop int mathbb r k x y h u y t theta rm d y beta mathop int mathbb r k x y h u y t theta rm d y we derive relation between speed index function and stability index function for each of the waves this model was derived when studying working memory in synaptically coupled neuronal networks which incorporates low persistent activity rate and high saturating rate we will investigate dynamics of neuronal waves for this purpose we will be concerned with the equation for several different kinds of symmetric and asymmetric kernels and will compare speeds of the waves our analysis and results on the speed index functions facilitate our investigation on stability of the waves and the estimates of speeds secondly we are concerned with standing waves of the nonlinear nonhomogeneous system of integral differential equations begin array lll u t u w alpha mathop int limits mathbb r k x y h u y t theta rm d y beta mathop int limits mathbb r k x y h u y t theta rm d y mathcal i x t w t varepsilon u gamma w end array and the scalar equation begin array lll u t u alpha mathop int limits mathbb r k x y h u y t theta rm d y beta mathop int limits mathbb r k x y h u y t theta rm d y mathcal i x t end array

l bounds for maxwell gauged equations in r1 1 and their applications abstract by making use of local energy conservation we estimate the l norm of the first derivative of the solution to the maxwell gauged o 3 sigma model in r 1 1 as an application of the bound we obtain the growth of the h 2 norm of the solution quadratic polynomial growth is obtained by improving exponential growth a similar idea can be applied to the maxwell klein gordon equations in r 1 1

on certain maximal functions and approach regions if f e l ir and u x y is the poisson integral of j a classical theorem of fatou 2 asserts that u has nontangential limits almost everywhere on r littlewood 5 and later zygmund 9 showed that this conclusion is false if nontangential approach is replaced by approach along translates of a curve which approaches the boundary tangentially in this paper we find a necessary and sufficient condition on an approach region for the associated maximal function to be appropriately bounded it then turns out that there are many approach regions which are not contained in any nontangential region but for which the conclusions of fatou s theorem remain true suppose q c ir has the property that whenever xi u e b and whenever x x y y it follows that x v e q then the maximal function associated to such a set is bounded if and only if the cross sectional measure at height y is bounded by a constant times y we also study regions where the cross sectional area is larger than y we introduce a modified maximal function which is again bounded this result leads to a simple proof of the boundedness of certain tangential maximal functions of poisson integrals of potentials studied in 161 the plan of this paper is as follows in section 1 we motivate our main ideas by discussing two particular examples the method of proof here is somewhat different from the proofs of the more general results in section 2 we obtain the necessary and sufficient conditions for fatou s theorem in section 3 we study more general approach regions and the appropriately modified maximal functions in section 4 we show how this maximal function controls the tangential boundary behavior of poisson integrals of potentials

nearly weakly orthonormal sequences singular value estimates and calderon zygmund operators abstract the direct way to estimate the singular values of a compact operator is to decompose it as a sum of orthogonal rank one pieces however such decompositions can generally not be found in practice we give a variation of the decomposition using nearly weakly orthonormal nwo sequences the nwo decomposition is easier to do in examples but is still strong enough to give good singular value estimates we illustrate this by giving sharp trace ideal estimates for the double layer potential and for the first and higher commutators of multiplication operators and calderon zygmund operators in particular nwo sequences seem well suited for dealing with operators which depend nonlinearly on their symbol we also show that the class of operators which admit nwo expansions is well behaved under composition under conjugation by weights and under conjugation by changes of variable

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

long time stability of small finite gap solutions of the cubic nonlinear schrodinger equation on t2 abstract in this paper we study long time stability of a class of nontrivial quasi periodic solutions depending on one spacial variable of the cubic defocusing non linear schrodinger equation on the two dimensional torus we prove that these quasi periodic solutions are orbitally stable for finite but long times provided that their fourier support and their frequency vector satisfy some complicated but explicit condition which we show holds true for most solutions the proof is based on a normal form result more precisely we expand the hamiltonian in a neighborhood of a quasi periodic solution we reduce its quadratic part to diagonal constant coefficients through a kam scheme and finally we remove its cubic terms with a step of nonlinear birkhoff normal form the main difficulty is to impose second and third order melnikov conditions this is done by combining the techniques of reduction in order of pseudo differential operators with the algebraic analysis of resonant quadratic hamiltonians

uniformly convex and smooth banach spaces and lp boundedness properties of littlewood paley and area functions associated with semigroups abstract in this paper we obtain new characterizations of the uniformly convex and smooth banach spaces these characterizations are established by using l p boundedness properties of littlewood paley functions and area integrals associated with heat semigroups and involving fractional derivatives our results apply for instance by considering the heat semigroups defined by hermite and laguerre operators that do not satisfy the markovian property

regularity of ground state solutions of dispersion managed nonlinear schrodinger equations abstract we consider the dispersion managed nonlinear schrodinger equation dmnls in the case of zero residual dispersion using dispersive properties of the equation and estimates in bourgain spaces we show that the ground state solutions of dmnls are smooth the existence of smooth solutions in this case matches the well known smoothness of the solutions in the case of nonzero residual dispersion in the case x r 2 we prove that the corresponding minimization problem with zero residual dispersion has no solution

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

tent space well posedness for parabolic cauchy problems with rough coefficients abstract we study the well posedness of cauchy problems on the upper half space r n 1 associated to higher order systems t u 1 m 1 div m a m u with bounded measurable and uniformly elliptic coefficients we address initial data lying in l p 1 p and bmo p spaces and work with weak solutions our main result is the identification of a new well posedness class given for p 1 by distributions satisfying m u t m p 2 where t m p 2 is a parabolic version of the tent space of coifman meyer stein in the range p 2 this holds without any further constraints on the operator and for p it provides a carleson measure characterization of bmo with non autonomous operators we also prove higher order l p well posedness previously only known for the case m 1 the uniform l p boundedness of propagators of energy solutions plays an important role in the well posedness theory and we discover that such bounds hold for p close to 2 this is a consequence of local weak solutions being locally holder continuous with values in spatial l l o c p for some p 2 what is also new for the case m 1

weighted tent spaces with whitney averages factorization interpolation and duality in this paper we introduce a new scale of tent spaces which covers the weighted tent spaces of coifman meyer stein and of hofmann mayboroda mcintosh and some other tent spaces considered by dahlberg kenig pipher and auscher axelsson in studying boundary value problems for elliptic systems the strong factorizations within our tent spaces with applications to quasi banach complex interpolation and to multiplier duality theory are then established this way we unify and extend the corresponding results obtained by coifman meyer stein cohn verbitsky and hytonen rosen

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

bmol hn spaces and carleson measures for schrodinger operators abstract let l h n v be a schrodinger operator on the heisenberg group h n where h n is the sub laplacian and the nonnegative potential v belongs to the reverse holder class b q 2 here q is the homogeneous dimension of h n in this article we investigate the dual space of the hardy type space h l 1 h n associated with the schrodinger operator l which is a kind of bmo type space bmo l h n defined by means of a revised sharp function related to the potential v we give the fefferman stein type decomposition of bmo l functions with respect to the adjoint riesz transforms r j l for l and characterize bmo l h n in terms of the carleson measure we also establish the bmo l boundedness of some operators such as the adjoint riesz transforms r j l the littlewood paley function s q l the lusin area integral s q l the hardy littlewood maximal function and the semigroup maximal function all results hold for stratified groups as well

influence of sodium currents on speeds of traveling wave fronts in synaptically coupled neuronal networks abstract the main purpose of this paper is to apply mathematical analysis to investigate the influence of sodium currents on the speeds of traveling wave fronts the authors use speed index functions to investigate the speeds of traveling wave fronts of some scalar integral differential equations arising from synaptically coupled neuronal networks the mathematical model equation is u t f u r k x y h u y t 1 c x y d y where 0 c 0 and 0 are constants satisfying the condition 0 2 f the function f u represents sodium currents the function k denotes synaptic coupling in a neuronal network and the heaviside step function h is defined by h u 0 for all u 0 h 0 1 2 and h u 1 for all u 0

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

a new class of function spaces connecting triebel lizorkin spaces and q spaces abstract let s r 0 p 1 and q 1 in this paper we introduce a new class of function spaces f p q s r n which unify and generalize the triebel lizorkin spaces with both p 1 and p and q spaces by establishing the carleson measure characterization of q space we then determine the relationship between triebel lizorkin spaces and q spaces which answers a question posed by dafni and xiao in g dafni j xiao some new tent spaces and duality theorem for fractional carleson measures and q r n j funct anal 208 2004 377 422 moreover via the hausdorff capacity we introduce a new class of tent spaces f t p q s r n 1 and determine their dual spaces f w p q s q r n where s r p q 1 max p q 1 0 q max p q and t denotes the conjugate index of t 1 as an application of this we further introduce certain hardy hausdorff spaces f h p q s r n and prove that the dual space of f h p q s r n is just f p q s q r n when p q 1

bilinear estimates and applications to 2d nls the three bilinearities uv uv uv for functions u v r2 0 t 7 c are sharply estimated in function spaces xs b associated to the schrodinger operator i t these bilinear estimates imply local wellposedness results for schrodinger equations with quadratic nonlinearity improved bounds on the growth of spatial sobolev norms of finite energy global in time and blow up solutions of the cubic nonlinear schrodinger equation and certain generalizations are also obtained

a smoothing effect and polynomial growth of the sobolev norms for the kp ii equation abstract we prove that the eventual growth in time of the sobolev norms of the solutions of the kp ii equation is at most polynomial

pointwise multipliers and decomposition theorems in f q s in this paper we give a characterization of the pointwise multipliers of the holomorphic triebel lizorkin spaces f q s and we apply these results to prove a corona decomposition theorem for these spaces

bergman projections and operators on hardy spaces we show that toeplitz like operators of the formtsuf ps uf wherepsis a weighted bergman projection are bounded on the hardy spaceshp for 1 p for certain symbols udefined on the unit disk in particular tsuis bounded ifuis of the formu h g wherehis a bounded harmonic function andg is the green potential of a borel measure satisfying the condition that 1 z d z is a carleson measure ifuis of the formu h g where is a positive measure then these two conditions are also necessary thattsube bounded onhp we also consider the cases 0 p 1

analyticity of the global attractor for damped forced periodic korteweg de vries equation abstract we consider here a weakly damped forced periodic kdv equation we prove that if the forcing term is analytic in space then the global attractor is also contained into a space of analytic functions this result was conjectured in 23

long time instability and unbounded sobolev orbits for some periodic nonlinear schrodinger equations we study the energy cascade problematic for some nonlinear schrodinger equations on mathbb t 2 in terms of the growth of sobolev norms we define the notion of long time strong instability and establish its connection to the existence of unbounded sobolev orbits this connection is then explored for a family of cubic schrodinger nonlinearities that are equal or closely related to the standard polynomial one u 2u most notably we prove the existence of unbounded sobolev orbits for a family of hamiltonian cubic nonlinearities that includes the resonant cubic nls equation a k a the first birkhoff normal form

factorization of tent spaces and hankel operators it is shown that the factorization tpq tp t q for tent spaces proved by r r coifman et al 1985 j funct anal 62 304 335 for p q q 2 holds true for all 0 p q from this certain strong factorization theorems are derived for spaces hps of fractional derivatives of hp functions and more general triebel spaces in particular it is proved that hps hp bmoas applications considered include characterizations of symbols of bounded hankel operators h hp hqs complex interpolation of tent spaces and carleson measure theorems for derivatives of hp functions

quasiconformal mappings and chord arc curves given a quasiconformal mapping p on the plane what conditions on its dilatation guarantee that p r is rectifiable and pir is locally abso lutely continuous we show in this paper that if satisfies certain quadratic carleson measure conditions with small norm then p r is a chord arc curve with small constant and p x p 0 g0 ea t dt for x e r with a e bmo having small norm conversely given any such map from r c we show that it has an extension to c with the right kind of dilatation similar results hold with r replaced by a chord arc curve examples are given that show that it would be hard to improve these results applications are given to conformal welding and the theorem of coifman and meyer on the real analyticity of the riemann mapping on the manifold of chord arc curves let p be a quasiconformal map of the plane onto itself thus p is a homeomorphism with locally integrable distributional derivatives and ps upz where u e l c il ul100 1 here we use the notations fz af azf 2 z id f fz 2 aa ida f this function u is called the complex dilatation of p the mapping theorem for q c maps states that for each u e l c il ul100 1 there is a q c map p on c with dilatation u and p is unique up to normalization a basic problem is to understand how geometric properties of p are reflected in u for example one would like to have natural conditions on u which imply that p r is rectifiable and p ir is absolutely continuous this question arises naturally when considering problems in conformal mappings and conformal welding in this paper we obtain such estimates for the mapping theorem and we also give some applications our results involve bmo aoo weights chord arc curves and carleson measures and so we first review the appropriate definitions a locally integrable function f on r lies in bmo if llfil sup lil if x fil dx received by the editors june 1 1986 1980 mathematics subject classification 1985 revision primary 30g60 secondary 42b20 30e20

multilinear weighted convolution of l 2 functions and applications to nonlinear dispersive equations the x s b spaces as used by beals bourgain kenig ponce vega klainerman machedon and others are fundamental tools to study the low regularity behavior of nonlinear dispersive equations it is of particular interest to obtain bilinear or multilinear estimates involving these spaces by plancherel s theorem and duality these estimates reduce to estimating a weighted convolution integral in terms of the l 2 norms of the component functions in this paper we systematically study weighted convolution estimates on l 2 as a consequence we obtain sharp bilinear estimates for the kdv wave and schrodinger x s b spaces

abstract capacitary estimates and the completeness and separability of certain classes of non locally convex topological vector spaces abstract we are concerned with establishing completeness and separability criteria for large classes of topological vector spaces which are typically non locally convex including lebesgue like spaces lorentz spaces orlicz spaces mixed normed spaces tent spaces and discrete triebel lizorkin and besov spaces for vector spaces of measurable functions we also derive pointwise convergence results our approach relies on abstract capacitary estimates and works in certain cases of interest even in the absence of a background measure space and or of a vector space structure

optimal bounds for the growth of sobolev norms of solutions of a quadratic szegő equation in this paper we study a quadratic equation on the one dimensional torus involving the szeg projector onto functions with nonnegative frequencies thanks to a lax pair structure we construct a flow on bmo t im which propagates h s regularity for any s 0 whereas the energy level corresponds to s 1 2 then for each s 1 2 we exhibit solutions whose h s norm goes to exponentially fast and we show that this growth is optimal

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

embedding bergman spaces into tent spaces let a p omega denote the bergman space in the unit disc mathbb d of the complex plane induced by a radial weight omega with the doubling property int r 1 omega s ds le c int frac 1 r 2 1 omega s ds the tent space t q s nu omega consists of functions such that begin aligned begin aligned vert f vert t q s nu omega q int mathbb d left int vargamma zeta f z s d nu z right frac q s omega zeta da zeta infty quad 0 q s infty end aligned end aligned here vargamma zeta is a non tangential approach region with vertex zeta in the punctured unit disc mathbb d setminus 0 we characterize the positive borel measures nu such that a p omega is embedded into the tent space t q s nu omega where 1 frac s p frac s q 0 by considering a generalized area operator the results are provided in terms of carleson measures for a p omega

the qp carleson measure problem let be a nonnegative borel measure on the open unit disk this note shows how to decide that the mobius invariant space covering and is boundedly resp compactly embedded in the quadratic tent type space interestingly the embedding result can be used to determine the boundedness resp compactness of the volterra type and multiplication operators on

the evolution of synaptically generated waves in one and two dimensional domains we study the onset and evolution of waves in an integrate and fire network of synaptically coupled neurons in one and two dimensional domains which are restricted to firing one spike we determine the critical size of initial excitation of the network necessary for the onset of propagation we also determine the ignition time we derive an integro differential equation for the evolution of the firing time as a function of spatial position we use the evolution equation to understand propagation failure we compare the behavior of the integrate and fire model with a biophysically based model finally we show that weak heterogeneity has similarly weak effects on the firing times

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

regular traveling waves in a one dimensional network of theta neurons this paper focuses on regular traveling waves or waves that propagate smoothly in a one dimensional network of theta neurons we show that when coupling strength is sufficiently large there exist two traveling wave solutions for this network moreover the cells in the network spike more than one time after joining the wave that is some form of synaptic depression or other adaptive mechanism must exist to attain single spike traveling waves we also present results about how the behavior of solutions to this network depends on the maximal synaptic coupling strength and the wave speed this is relevant to consideration of uniqueness and to bounding the range of coupling strengths for which waves exist

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

layer potentials and boundary value problems for elliptic equations with complex l coefficients satisfying the small carleson measure norm condition abstract we consider divergence form elliptic equations l u a u 0 in the half space r n 1 x t r n 0 whose coefficient matrix a is complex elliptic bounded and measurable in addition we suppose that a satisfies some additional regularity in the direction transverse to the boundary namely that the discrepancy a x t a x 0 satisfies a carleson measure condition of fefferman kenig pipher type with small carleson norm under these conditions we establish a full range of boundedness results for double and single layer potentials in l p hardy sobolev bmo and holder spaces furthermore we prove solvability of the dirichlet problem for l with data in l p r n bmo r n and c r n and solvability of the neumann and regularity problems with data in the spaces l p r n h p r n and l 1 p r n h 1 p r n respectively with the appropriate restrictions on indices assuming invertibility of layer potentials for the t independent operator l 0 a 0

standing ring blow up solutions to the n dimensional quintic nonlinear schrodinger equation we consider the quintic nonlinear schr odinger equation in dimension n 3 this problem is energy critical in dimension n 3 and energy super critical for n 4 we prove the existence of a radially symmetric blow up mechanism with l2 concentration along the unit sphere of rn this singularity formation is moreover stable by smooth and radially symmetric perturbation of the initial data this result extends the result obtained for n 2 in 29 and is the first result of description of a singularity formation in the energy supercritical class for nls type problems our main tool is the proof of the propagation of regularity outside the blow up sphere in the presence a so called log log type singularity

hardy spaces associated with magnetic schrodinger operators on strongly lipschitz domains abstract let p 0 1 be a strongly lipschitz domain in r n and a i a i a v a magnetic schrodinger operator on l 2 satisfying the dirichlet boundary condition where a a 1 a n l loc 2 r n and 0 v l loc 1 in this paper the authors introduce the hardy space h a p by the lusin area function associated with a and establish its equivalent characterization via the non tangential maximal function associated with e t a t 0 as applications the authors obtain the boundedness of the riesz transforms l k a 1 2 k 1 n from h a p to l p for p 0 1 and the fractional integral a from h a p to h a q for 0 p q 1 and n 2 1 p 1 q where l k is the closure of x k i a k in l 2

hardy spaces and divergence operators on strongly lipschitz domains of rn abstract let be a strongly lipschitz domain of r n consider an elliptic second order divergence operator l including a boundary condition on and define a hardy space by imposing the non tangential maximal function of the extension of a function f via the poisson semigroup for l to be in l1 under suitable assumptions on l we identify this maximal hardy space with h 1 r n if r n with h r 1 under the dirichlet boundary condition and with h z 1 under the neumann boundary condition

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

propagating activity patterns in thalamic neuronal networks singular perturbation methods are used to construct traveling waves for models of thalamic networks we first study a single layer of mutually inhibitory neurons each of which has the ability to rebound after hyperpolarization two types of waves are constructed smoothly propagating waves and lurching waves which propagate in a saltatory fashion we reduce the existence of these waves to simple boundary value problems or algebraic systems which are solved using the computational package auto the resulting calculations are compared to numerical simulations of the network finally some comments are made concerning two layer networks

imbedding potentials in tent spaces abstract we obtain characterizations of positive borel measures on b n so that the nonisotropic potential space k lp d is imbedded in the tent space t2q d where 1 q we deduce characterizations for pointwise multipliers of the space h p

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

tent space boundedness via extrapolation we study the action of operators on tent spaces such as maximal operators calderon zygmund operators riesz potentials we also consider singular non integral operators we obtain boundedness as an application of extrapolation methods in the banach range in the non banach range boundedness results for calderon zygmund operators follow by using an appropriate atomic theory we end with some consequences on amalgam 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

slow waves in mutually inhibitory neuronal networks abstract a variety of experimental and modeling studies have been performed to investigate wave propagation in networks of thalamic neurons and their relationship to spindle sleep rhythms it is believed that spindle oscillations result from the reciprocal interaction between thalamocortical tc and thalamic reticular re neurons we consider a network of tc and re cells reduced to a one layer network model and represented by a system of singularly perturbed integral differential equations geometric singular perturbation methods are used to prove the existence of a locally unique slow wave pulse that propagates along the network by seeking a slow pulse solution we reformulate the problem to finding a heteroclinic orbit in a 3d system of odes with two additional constraints on the location of the orbit at two distinct points in time in proving the persistence of the singular heteroclinic orbit difficulties arising from the solution passing near points where normal hyperbolicity is lost on a 2d critical manifold are overcome by employing results by wechselberger singularly perturbed folds and canards in r 3 thesis tu wien 1998

the regularity problem for second order elliptic operators with complex valued bounded measurable coefficients the present paper establishes a certain duality between the dirichlet and regularity problems for elliptic operators with t independent complex bounded measurable coefficients t being the transversal direction to the boundary to be precise we show that the dirichlet boundary value problem is solvable in l p subject to the square function and non tangential maximal function estimates if and only if the corresponding regularity problem is solvable in l p moreover the solutions admit layer potential representations in particular we prove that for any elliptic operator with t independent real possibly non symmetric coefficients there exists a p 1 such that the regularity problem is well posed in l p

quadratic forms for the 1 d semilinear schrodinger equation this paper is concerned with 1 d quadratic semilinear schr6dinger equations we study local well posedness in classical sobolev space hs of the associated initial value problem and periodic boundary value problem our main interest is to obtain the lowest value of s which guarantees the desired local well posedness result we prove that at least for the quadratic cases these values are negative and depend on the structure of the nonlinearity considered

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation