toeplitz and hankel operators on bergman spaces in this paper we consider toeplitz and hankel operators on the bergman spaces of the unit ball and the polydisk in c n whose symbols are bounded measurable functions we giv e necessary and sufficient conditions on the symbols for these operators to be compact we study the fredholm theory of topelitz operators for which the corresponding hankel operator is compact for these toeplitz operators the essential spectrum is computed and shown to be connected we also consider symbols that extend to continuous functions on the maximal ideal space of h for these symbols we describe when the toeplitz or hankel operators are compact

high order methods for low reynolds number flows around moving obstacles based on universal meshes summary we propose a family of methods for simulating two dimensional incompressible low reynolds number flow around a moving obstacle whose motion is prescribed the methods make use of a universal mesh a fixed background mesh that adapts to the geometry of the immersed obstacle at all times by adjusting a few elements in the neighborhood of the obstacle s boundary the resulting mesh provides a conforming triangulation of the fluid domain over which discretizations of any desired order of accuracy in space and time can be constructed using standard finite element spaces together with off the shelf time integrators we demonstrate the approach by using taylor hood elements to approximate the fluid velocity and pressure to integrate in time we consider implicit runge kutta schemes as well as a fractional step scheme we illustrate the methods and study their convergence numerically via examples that involve flow around obstacles that undergo prescribed deformations copyright 2015 john wiley sons ltd

the connection matrix theory for morse decompositions the connection matrix theory for morse decompositions is introduced the connection matrices are matrices of maps between the homology indices of the sets in the morse decomposition the connection matrices cover in a natural way the homology index braid of the morse decomposition and provide information about the structure of the morse decomposition the existence of connection matrices of morse decompositions is established and examples illustrating applications of the connection matrix are provided

the bergman space the bloch space and commutators of multiplication operators soit d le disque unite ouvert dans le plan complexe c soit h d l ensemble des fonctions analytiques bornees sur d on caracterise les fonctions feh d telles que t f t f tft f est un operateur compact ou t f est l operateur multiplication t f l a 2 l a 2 defini par t f g fg

holomorphic martingales and interpolation between hardy spaces the complex method a probabilistic proof is given to identify the complex interpolation space of h1 t and h t as hp t introduction in this note a soft propabilistic proof of p w jones s theorem on the complex interpolation space between h1 and hoc is given we shall work with n th varopoulos s space of holomorhpic martingales the observation presented in this article is that the use of a stopping time decomposition simplifies constructions of serguei v kislyakov and quanhua xu to obtain the following theorem the complex interpolation space th t h t 0 o 0 e 1 coincides with hp t provided that 1 1 03 p as is well known this result has been obtained by p w jones using l estimates to the a problem see j his work also contains the description of the real interpolation spaces for the couple hl h at about the same time jean bourgain obtained a marcinkiewicz type decomposition using completely different techniques see b1 recently in a series of papers jean bourgain b2 serguei kislyakov kl k2 k x gilles pisier p and quanhua xu xl x2 obtained deep results concerning real and complex interpolation methods between vector valued weighted hardy spaces of analytic functions s kislyakov s paper kl contains the following idea to approximate the characteristic function 1 1f1 2 by analytic functions on the circle t he starts a max l itl and then considers 1 a iha where h denotes the hilbert transform on t received by the editors august 11 1992 and in revised form february 1 1994 originally communicated to the proceedings of the ams by dale alspach 1991 mathematics subject classification primary 60g46 42b30

on algebras generated by toeplitz operators and their representations abstract we study banach and c algebras generated by toeplitz operators acting on weighted bergman spaces a 2 b 2 over the complex unit ball b 2 c 2 our key point is an orthogonal decomposition of a 2 b 2 into a countable sum of infinite dimensional spaces each one of which can be identified with a differently weighted bergman space a 2 d over the complex unit disk d moreover all elements of the above algebras leave each of the summands in the above decomposition invariant and their restriction to each level acts as a compact perturbation of a toeplitz operator on a 2 d the symbols of the generating toeplitz operators are chosen to be suitable extensions to b 2 of families s of bounded functions on d symbol classes s that generate important classical commutative and non commutative toeplitz algebras in l a 2 d are of particular interest in this paper we discuss various examples in the case of s c d and s c d l 0 1 we characterize all irreducible representations of the resulting toeplitz operator c algebras their calkin algebras are described and index formulas are provided

conformal welding and koebe s theorem it is well known that not every orientation preserving homeomorphism of the circle to itself is a conformal welding but in this paper we prove several results which state that every homeomorphism is almost a welding in a precise way the proofs are based on koebe s circle domain theorem we also give a new proof of the well known fact that quasisymmetric maps are conformal weldings

characterizations of certain classes of hankel operators on the bergman spaces of the unit disk let f be an integrable function on the unit disk the hankel operator hf is densely defined on the bergman space ap by hfg fg p fg where g is a bounded analytic function and p is the bergman projection orthogonal projection from l2 to a2 extended to l1 via its integral formula in this paper the functions f for which hf extends to a bounded operator from ap to lp are characterized 1 p also characterized are the functions f for which hf extends to a compact or schatten class operator on a2 the proofs can be extended to handle any smoothly bounded domain in c in place of the unit disk

mellin transform monomial symbols and commuting toeplitz operators abstract in this paper we give necessary and sufficient conditions for a symbol that produces a toeplitz operator on the bergman space that commutes with another such operator whose symbol is a monomial the result is stated in terms of the mellin transform of the symbol as a corollary we show that a toeplitz operator with a radial symbol may only commute with another such operator with a radial symbol

commuting toeplitz operators with pluriharmonic symbols let da z denote the lebesgue volume measure on the open unit ball bn f cn normalized so that the measure of bn equals 1 the 13ergman space l2 bn is the eiilbert space consisting of holomorphic functions on bn that are also in l2 bn da for f in l90 bn the toeplitz operator tf with symbol f is desned by t h p fh where p is the orthogonal projection from l2 bnnda onto ol2tbesda n called the bergman projection let dsr be the surface area measure on the unit sphere sn for 1 p 007 h s4 is the banach space of holomorphic functions on bn with the norm defined by

solid hulls of quasi banach spaces of analytic functions and interpolation abstract we study solid hulls of abstract hardy spaces of analytic functions on the unit disk interpolation of sublinear operators is used to description of solid hulls of quasi banach spaces generated by interpolation methods associated to classical hardy spaces h p for 0 p 1 we specialize our results to solid hulls of hardy orlicz and hardy lorentz spaces and we are able to characterize multipliers from hardy lorentz spaces to any solid sequence space we give for example a complete description of multipliers with the domain in hardy lorentz zygmund spaces

multipliers for entire functions and an interpolation problem of beurling abstract we characterize the interpolating sequences for the bernstein space of entire functions of exponential type in terms of a beurling type density condition and a carleson type separation condition our work extends a description previously given by beurling in the case that the interpolating sequences are restricted to the real line an essential role is played by a multiplier lemma which permits us to link techniques from hardy spaces with entire functions of exponential type we finally present a characterization of the sampling sequences for the bernstein space also extending a density theorem of beurling

universal meshes a method for triangulating planar curved domains immersed in nonconforming meshes summary we introduce a new method to triangulate planar curved domains that transforms a specific collection of triangles in a background mesh to conform to the boundary in the process no new vertices are introduced and connectivities of triangles are left unaltered the method relies on a novel way of parameterizing an immersed boundary over a collection of nearby edges with its closest point projection to guarantee its robustness we require that the domain be c2 regular the background mesh be sufficiently refined near the boundary and that specific angles in triangles near the boundary be strictly acute the method can render both straight edged and curvilinear triangulations for the immersed domain the latter includes curved triangles that conform exactly to the immersed boundary and ones constructed with isoparametric mappings to interpolate the boundary at select points high order finite elements constructed over these curved triangles achieve optimal accuracy which has customarily proven difficult in numerical schemes that adopt nonconforming meshes aside from serving as a quick and simple tool for meshing planar curved domains with complex shapes the method provides significant advantages for simulating problems with moving boundaries and in numerical schemes that require iterating over the geometry of domains with no conformity requirements the same background mesh can be adopted to triangulate a large family of domains immersed in it including ones realized over several updates during the coarse of simulating problems with moving boundaries we term such a background mesh as a universal mesh for the family of domains it can be used to triangulate universal meshes hence facilitate a framework for finite element calculations over evolving domains while using only fixed background meshes furthermore because the evolving geometry can be approximated with any desired order numerical solutions can be computed with high order accuracy we present demonstrative examples using universal meshes to simulate the interaction of rigid bodies with stokesian fluids copyright 2014 john wiley sons ltd

the conforming virtual element method for polyharmonic problems abstract in this work we exploit the capability of virtual element methods in accommodating approximation spaces featuring high order continuity to numerically approximate differential problems of the form p u f p 1 more specifically we develop and analyze the conforming virtual element method for the numerical approximation of polyharmonic boundary value problems and prove an abstract result that states the convergence of the method in suitable norms

approximating continuous functions by holomorphic and harmonic functions si est un domaine de widom dans le plan et f est une fonction harmonique bornee sur non holomorphe on demontre que l algebre h f contient toutes les fonctions uniformement continues sur

l estimates for the bar partial problem in a half planeproblem in a half plane l approche de dualite pour trouver des solutions de f est suffisante pour plusieurs problemes apparaissant en theorie h p on donne des operateurs solution qui permettent de resoudre des problemes non resolubles par la dualite

time discrete higher order ale formulations stability arbitrary lagrangian eulerian ale formulations deal with pdes on deformable domains upon extending the domain velocity from the boundary into the bulk with the purpose of keeping mesh regularity this arbitrary extension has no effect on the stability of the pde but may influence that of a discrete scheme we examine this critical issue for higher order time stepping without space discretization we propose time discrete discontinuous galerkin dg numerical schemes of any order for a time dependent advection diffusion model problem in moving domains and study their stability properties the analysis hinges on the validity of the reynolds identity for dg exploiting the variational structure and assuming exact integration we prove that our conservative and nonconservative dg schemes are equivalent and unconditionally stable the same results remain true for piecewise polynomial ale maps of any degree and suitable quadrature that guarantees the validity of the reynolds identity this approach general

torsional springs for two dimensional dynamic unstructured fluid meshes abstract dynamic fluid grids are commonly used for the solution of flow problems with moving boundaries they are often represented by a network of fictitious lineal springs that can become unreliable when the fluid mesh undergoes large displacements and or deformations in this paper we propose to control the arbitrary motion of two dimensional dynamic unstructured fluid grids with additional torsional springs we show that such springs can be designed to prohibit the interpenetration of neighboring triangles and therefore to provide the method of spring analogy with the robustness needed for enlarging its range of applications we illustrate our new dynamic mesh motion algorithm with several examples that highlight its advantages in terms of robustness quality and performance

hankel operators and toeplitz operators on the bergman space abstract we consider in this paper the question of when the semi commutator t fg t f t g on the bergman space with bounded harmonic symbols is compact several conditions equivalent to compactness of t fg t f t g are given as a consequence we prove a conjecture of axler that for bounded analytic functions f and g on the unit disk t fg t g t f is compact iff either f or g is constant on each gleason part p m except d

corona theorems for subalgebras of h qtp the main goal of this work is to give a unified proof of several corona theorems for a collection of algebras of pointwise multipliers of spaces of holomorphic functions on the unit disk d these spaces include besov spaces invariant dirichlet type spaces weighted sobolev spaces and carleson measure spaces for besov spaces

the connection map for attractor repeller pairs on montre que l application de connexion est additive les proprietes de l application connexion conduisent a une caracterisation des ensembles invariants isoles avec des points critiques hyperboliques comme une paire attracteur repulseur

polynomial approximation local polynomial convexity and degenerate cr singularities abstract we begin with the following question given a closed disc d c and a complex valued function f c d is the uniform algebra on d generated by z and f equal to c d when f c 1 d this question is complicated by the presence of points in the surface s graph d f that have complex tangents such points are called cr singularities let p s be a cr singularity at which the order of contact of the tangent plane with s is greater than 2 i e a degenerate cr singularity we provide sufficient conditions for s to be locally polynomially convex at the degenerate singularity p this is useful because it is essential to know whether s is locally polynomially convex at a cr singularity in order to answer the initial question to this end we also present a general theorem on the uniform algebra generated by z and f which we use in our investigations this result may be of independent interest because it is applicable even to non smooth complex valued f

approximation by ratios of bounded analytic functions let d be the unit disk andebe a compact subset of theh maximal ideal space m h we show that any continuous functionfon an open neighborhoodu m h ofesuch thatf u d h u d can be uniformly approximated oneby ratiosh g whereh g h andgis zero free one we also characterize those setsefor whichh gcan be replaced byhfor everyf finally we prove that any inner functionucan be written as a rational function of interpolating blaschke products it follows thatucan be uniformly approximated by ratios of interpolating blaschke products on any compact sete m h whereuis zero free

fefferman stein decomposition for q spaces and micro local quantities abstract in this paper we study the fefferman stein decomposition of q r n and give an affirmative answer to an open problem posed by essen et al 2000 one of our main methods is to characterize the structure of the predual of q r n by the micro local quantities this result indicates that the norm of the predual space of q r n depends on the micro local structure in a self correlation way

on some polynomially convex maximal real submanifolds in c2n and a related riemann hilbert problem abstract it is proved that certain maximal real graphs of maps from c n to c n are polynomially convex manifolds fibred over with fibres being such graphs are introduced and a related riemann hilbert problem is considered existence of a solution is proved and its local structure is studied

morse decompositions for delay differential equations abstract the attractor for the scalar delay differential equation x t f x t x t 1 is shown to possess a morse decomposition under the hypothesis of a negative feedback condition in the delay term using this decomposition we prove results on the asymptotic rate of oscillation of solutions of the initial value problem existence of both slowly and rapidly oscillating periodic solutions is also proved

a constructive proof of the chang marshall theorem let hw be the banach algebra of bounded analytic functions on d z z 1 equipped with the sup norm using radial limits we can identify h with a closed subalgebra of l o the algebra of bounded measurable functions on t z 1 z 1 an inner function is an h o function that is unimodular as a member of lm we note that if b is an inner function then b 6 as elements of l o in 3 lo s y a chang and d e marshall proved the following result which had been conjectured by r g douglas

time discrete higher order ale formulations a priori error analysis we derive optimal a priori error estimates for discontinuous galerkin dg time discrete schemes of any order applied to an advection diffusion model defined on moving domains and written in the arbitrary lagrangian eulerian ale framework our estimates hold without any restrictions on the time steps for dg with exact integration or reynolds quadrature they involve a mild restriction on the time steps for the practical runge kutta radau methods of any order the key ingredients are the stability results shown earlier in bonito et al time discrete higher order ale formulations stability 2013 along with a novel ale projection numerical experiments illustrate and complement our theoretical results

constructing continuous functions holomorphic off a curve abstract we give new proofs of a theorem of a browder and j wermer and a theorem of a davie which characterize the plane sets k for which ak is a dirichlet algebra the use of functional analysis in the original proofs is replaced by a construction involving bounded solutions of the t6 equation in particular this gives an explicit construction of nonconstant functions in these spaces

interpolating sequences for analytic selfmappings of the disc schwarz s lemma leads to a natural interpolation problem for analytic functions from the disc into itself the corresponding interpolating sequences are geometrically described in terms of a certain hyperbolic density

a higher order nonlocal operator method for solving partial differential equations abstract a higher order nonlocal operator method for the solution of boundary value problems is developed the proposed higher order nonlocal operator brings several advantages as compared to the original nonlocal operator method ren et al 2020 which only ensures first order convergence furthermore it can be applied to directly and efficiently obtain all partial derivatives of higher orders simultaneously without the need of using shape functions only the functionals based on the nonlocal operators termed as operator functional are needed to obtain the final discrete system of equations which significantly facilitates the implementation several numerical examples are presented to show the effectiveness and accuracy of the proposed higher order nonlocal operator method including the solution of the poisson equation in 2 5 dimensional space kirchhoff and von karman plate problems incompressible elastic materials as well as phase field modeling of fracture

bounded mobius invariant q k spaces abstract we prove the corona theorem for the banach algebra q k h under some assumptions of the weight k and a fefferman stein type decomposition of the boundary values of q k is obtained

approximation by polynomials in two diffeomorphisms soit 1 k z et soient f et g des diffeomorphismes c k de c dans c ayant des degres opposes alors c f g est dense dans l espace de frechet c k c

product domains multi cauchy transforms and the equation abstract solution operators for the equation u f are constructed on general product domains in c n when the factors are one dimensional the operator is a simple integral operator specific derivatives of f are integrated against iterated cauchy kernels for higher dimensional factors the solution is constructed by solving sub equations with modified data on the factors estimates in several norms are proved

estimates for 8706 μ 1 and calderon s theorem on the cauchy integral etude du caractere borne de l operateur integral de cauchy a l aide d une transformation quasi conforme

high order finite element methods for moving boundary problems with prescribed boundary evolution abstract we introduce a framework for the design of finite element methods for two dimensional moving boundary problems with prescribed boundary evolution that have arbitrarily high order of accuracy both in space and in time at the core of our approach is the use of a universal mesh a stationary background mesh containing the domain of interest for all times that adapts to the geometry of the immersed domain by adjusting a small number of mesh elements in the neighborhood of the moving boundary the resulting method maintains an exact representation of the prescribed moving boundary at the discrete level or an approximation of the appropriate order yet is immune to large distortions of the mesh under large deformations of the domain the framework is general making it possible to achieve any desired order of accuracy in space and time by selecting a preferred and suitable finite element space on the universal mesh for the problem at hand and a preferred and suitable time integrator for ordinary differential equations we illustrate our approach by constructing a particular class of methods and apply them to a prescribed boundary variant of the stefan problem we present numerical evidence for the order of accuracy of our schemes in one and two dimensions

quasitriangular subalgebras of semifinite von neumann algebras are closed arveson has shown in 1 that if a is a nest algebra of operators acting on a separable hilbert space h with nest of order type the extended natural numbers and if k h is the ideal of compact operators on h then the quasitriangular algebra a k h is norm closed in 6 fall arveson and muhly extended arveson s result proving that in b h general quasitriangular algebras i e compact perturbations of nest algebras are always norm closed a key step in their proof and a result of independent interest first obtained by erdos in s is the a weak density of a n k h in a from this fact they deduced that quasitriangular algebras were norm closed by using an argument depending on the duality k h b h every semifinite von neumann algebra m has an ideal of compact operators that behaves like k h namely the norm closed two sided ideal k generated by the finite projections of the algebra it is a natural question

a new discretization for mth laplace equations with arbitrary polynomial degrees this paper introduces new mixed formulations and discretizations for m th laplace equations of the form 1 m delta m u f for arbitrary m 1 2 3 dots based on novel helmholtz type decompositions for tensor valued functions the new discretizations allow for ansatz spaces of arbitrary polynomial degree and the lowest order choice coincides with the nonconforming fems of crouzeix and raviart for m 1 and of morley for m 2 since the derivatives are directly approximated the lowest order discretizations consist of piecewise affine and piecewise constant functions for any m 1 2 dots moreover a uniform implementation for arbitrary m is possible besides the a priori and a posteriori analysis this paper proves optimal convergence rates for adaptive algorithms for the new discretizations

unified analysis of finite element methods for problems with moving boundaries we present a unified analysis of finite element methods for problems with prescribed moving boundaries in particular we study an abstract parabolic problem posed on a moving domain with prescribed evolution discretized in space with a finite element space that is associated with a moving mesh that conforms to the domain at all times the moving mesh is assumed to evolve smoothly in time except perhaps at a finite number of remeshing times where the solution is transferred between finite element spaces via a projection a key result of our analysis is an abstract estimate for the l 2 norm of the error between the exact and semidiscrete solutions at a fixed positive time expressed in terms of the total variation in time of a quantity that measures the difference between the exact solution at time t and its elliptic projection onto the finite element space at time t specializing the abstract estimate to particular choices of the mesh motion strategy finite element space and projector leads to e

bounded functions in mobius invariant dirichlet spaces abstract for p 0 1 let q p q p 0 be the space of analytic functions f on the unit disk with sup w f w d p w 1 f w d p 0 where d p means the weighted dirichlet norm and w is the mobius map of onto itself with w 0 w in this paper we prove the corona theorem for the algebra q p h q p 0 h then we provide a fefferman stein type decomposition for q p q p 0 and finally we describe the interpolating sequences for q p h q p 0 h

a constructive proof of the fefferman stein decomposition of bmo rn let s vec r left left f r 1 f cdots r n f right f in h 1 left text r n text r right right subset h 1 left text r n text r n 1 right 22 1 begin gathered s vec r left vec g left g j right j 0 cdots n right in text bmo left text r n text r n 1 right sum limits j 0 n vec r j g j 0 text in bmo left text r n text r n 1 right left text r right end gathered 22 2 j 0

toeplitz algebras on the disk abstract let b be a douglas algebra and let b be the algebra on the disk generated by the harmonic extensions of the functions in b in this paper we show that b is generated by h d and the complex conjugates of the harmonic extensions of the interpolating blaschke products invertible in b every element s in the toeplitz algebra t b generated by toeplitz operators on the bergman space with symbols in b has a canonical decomposition s t s r for some r in the commutator ideal c t b and s is in c t b iff the berezin transform s vanishes identically on the union of the maximal ideal space of the douglas algebra b and the set m 1 of trivial gleason parts

interpolating sequences in the polydisc on donne une condition suffisante pour que des suites de points soient des suites d interpolation pour l ensemble h d n des fonctions analytiques bornees definies sur le polydisque d n de c n on discute quelques conditions d interpolation des domaines generaux

on polynomial convexity of compact subsets of totally real submanifolds in cn abstract let k be a compact subset of a totally real manifold m where m is either a c 2 smooth graph in c 2 n over c n or m u 1 0 for a c 2 smooth submersion u from c n to r 2 n k k n in this case we show that k is polynomially convex if and only if for a fixed neighbourhood u defined in terms of the defining functions of m there exists a plurisubharmonic function on c n such that k 0 u

stability and geometric conservation laws for ale formulations the aim of this paper is to investigate a model ale scheme with respect to various possible choices of time discretizations for each time scheme we investigate the relationships between stability and the so called geometric conservation laws gcl we shall see that gcl condition proves neither necessary nor sufficient for stability in doing so we review some known theoretical results and we prove some new stability results for space time ale discretizations some new error estimates are also presented several numerical experiments confirm the theory

interpolation between vector valued hardy spaces abstract let 0 p and let h p h x denote the space of x valued harmonic functions on the half space with boundary values almost everywhere and poisson maximal function in l p r n and h p x the closure of the x valued analytic polynomials on the disc under the norm given by sup 0 r f r p it is shown that if 0 p 0 p 1 1 p 1 p 0 p 1 then h p0 h x 0 h p1 h x 1 0 h p h x 0 with the restriction p 1 we prove h p0 h x 0 h p1 h x 1 0 p h h p x 0 p a counterexample for the case p 1 is given for the case of real interpolation it is also proved that h p0 x 0 h p1 x 1 0 is in general smaller than h p x 0 finally bmo x is also considered as the end point for interpolation

the pluriharmonic toeplitz operators on the polydisk in this paper we completely characterize the compact semi commutator of two toeplitz operators with bounded pluriharmonic symbols on the bergman space of the polydisk several necessary and sufficient conditions are obtained for the commutator of two toeplitz operators with bounded pluriharmonic symbols on the polydisk to be compact