Bilinear Singular Integral Operators,Smooth Atoms and Molecules

We present a bilinear T1 theorem in the context of Triebel–Lizorkin spaces. The proof uses atomic decomposition techniques and some a priori L ^\infty-estimates for the action of bilinear Calderón–Zygmund operators.     

Optimal error estimation for Petrov-Galerkin methods in two dimensions

Summary We examine the optimality of conforming Petrov-Galerkin approximations for the linear convection-diffusion equation in two dimensions. Our analysis is based on the Riesz representation theorem and it provides an optimal error estimate involving the smallest possible constantC. It also identifies an optimal test space, for any choice of consistent norm, as that whose image under the Riesz representation operator is the trial space. By using the Helmholtz decomposition of the Hilbert space [L ²()]², we produce a construction for the constantC in which the Riesz representation operator is not required explicitly. We apply the technique to the analysis of the Galerkin approximation and of an upwind finite element method.     

Polyhedral Voronoi Diagrams of Polyhedra in Three Dimensions

We show that the complexity of the Voronoi diagram of a collection of disjoint polyhedra in general position in 3-space that have n vertices overall, under a convex distance function induced by a polyhedron with O(1) facets, is O(n ²⁺), for any > 0. We also show that when the sites are n segments in 3-space, this complexity is O(n ² (n) log n). This generalizes previous results by Chew et al. and by Aronov and Sharir, and solves an open problem put forward by Agarwal and Sharir. Specific distance functions for which our results hold are the L ₁ and L _\infty metrics. These results imply that we can preprocess a collection of polyhedra as above into a near-quadratic data structure that can answer -approximate Euclidean nearest-neighbor queries amidst the polyhedra in time O(log (n/) ), for an arbitrarily small > 0.     

Boundedness of the commutator of Marcinkiewicz integral with rough variable kernel

In this paper the author proves that the commutator of the Marcinkiewicz integral operator with rough variable kernel is bounded from the homogeneous Sobolev space Lγ^2（R^n） to the Lebesgue space L^2（R^n）, which is a substantial improvement and extension of some known results.     

The Central Approximation Theorems for the Method of Left Gamma Quasi-Interpolants in L p spaces

The optimal degree of approximation of the method of Gammaoperators G _n in L _p spaces is O(n ^-1). In order to obtain much faster convergence, quasi-interpolants G _n ^(k) of G _n in the sense of Sablonnière are considered. We show that for fixed k the operator-norms G _n ^(k) _p are uniformly bounded in n. In addition to this, for the first time in the theory of quasi-interpolants, all central problems for approximation methods (direct theorem, inverse theorem, equivalence theorem) could be solved completely for the L _p metric. Left Gamma quasi-interpolants turn out to be as powerful as linear combinations of Gammaoperators [6].     

A note on the crossed product $$ \mathcal{R}(\mathcal{A},\alpha ) $$ associated with PSL2(ℝ)

Given the hyperbolic measure dxdy/y ² on the upper half plane ℍ, the rational actions of PSL₂(ℝ) on ℍ induces a continuous unitary representation α of this group on the Hilbert space L ²(ℍ, dxdy/y ²). Supposing that = {M _f : f ∈ L ^∞ (ℍ, dxdy/y ²)}, we show that the crossed product is of type I. In fact, the crossed product is *-isomorphic to the von Neumann algebra , where is the abelian group von Neumann algebra generated by the left regular representation of K. This work was supported by the Youth Foundation of Sichuan Education Department of China (Grant No. 2003B017)     

Parametrized Area Integrals on Hardy Spaces and Weak Hardy Spaces

In this paper, the authors prove that if Ω satisfies a class of the integral Dini condition, then the parametrized area integral μΩ,S^ρ is a bounded operator from the Hardy space H1 （R^n） to L1 （R^n） and from the weak Hardy space H^1,∞ （R^n） to L^1,∞ （R^n）, respectively. As corollaries of the above results, it is shown that μΩ,S^ρ is also an operator of weak type These conclusions are substantial improvement and （1, 1） and of type （p,p） for 1 〈 p 〈 2, respectively extension of some known results.     

The hardy and bellman operators in spaces connected with H( \mathbb{T} ) and BMO( \mathbb{T} )

Assume that 1 ≤ p < ∞ and a function f ∈ L ^p [0, π] has the Fourier series $ \sum\limits_{n = 1}^\infty {a_n } Assume that 1 ≤ p < ∞ and a function f ∈ L ^p [0, π] has the Fourier series cos nx. According to one result of G.H. Hardy, the series cos nx is the Fourier series for a certain function (f) ∈ L ^p [0, π]. But if 1 < p ≤ ∞ and f ∈ L ^p [0, π], then the series cos nx is the Fourier series for a certain function (f) ∈ L ^p [0, π]. Similar assertions are true for sine series. This allows one to define the Hardy operator on L ^p (), 1 ≤ p < ∞, and to define the Bellman operator on L ^p (), 1 < p ≤ ∞. In this paper we prove that the Bellman operator boundedly acts in VMO(), and the Hardy operator also maps a certain subspace C() onto VMO(). We also prove the invariance of certain classes of functions with given majorants of modules of continuity or best approximations in the spaces H(), L(), VMO() with respect to the Hardy and Bellman operators. Original Russian Text ? S.S. Volosivets and B.I. Golubov, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 5, pp. 4–13.     

Study of a filtration expanded to include an honest time

Summary Let (,, P) be a measurable space, and { _t} be a filtration on (,). Then, given a fixed honest timeL a new filtrationG _t} is defined, the smallest containing { _t} and for whichL is a stopping time, and the martingales, semimartingales and stopping times of this new filtration are characterised.     

A family of higher order mixed finite element methods for plane elasticity

Summary The Dirichler problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the Raviart-Thomas mixed finite elements for a scalar, second-order elliptic equation. Estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor inL ²(), and optimal order negative norm estimates are obtained inH ^s () for a range ofs depending on the index of the finite element space. An optimal order estimate inL () for the displacement error is given. Also, a quasioptimal estimate is derived in an appropriate space. All estimates are valid uniformly with respect to the compressibility and apply in the incompressible case. The formulation of the elements is presented in detail.This work was performed while Professor Arnold was a NATO Postdoctoral Fellow     

A penalty function proof of a Lagrange multiplier theorem with application to linear delay systems

Using a penalty function method, a Lagrange multiplier theorem in dual Banach spaces is proved. This theorem is applied to the optimal control of linear, autonomous time-delay systems with function space equality end condition and pointwise control restrictions. Under an additional regularity condition, the resulting Lagrange multiplier can be identified with an element ofL .     

An <Emphasis Type="Italic">L</Emphasis><Subscript>\infty</Subscript>/<Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-Constrained Quadratic Optimization Problem with Applications to Neural Networks

Pattern formation in associative neural networks is related to a quadratic optimization problem. Biological considerations imply that the functional is constrained in the L _\infty norm and in the L ₁ norm. We consider such optimization problems. We derive the Euler–Lagrange equations, and construct basic properties of the maximizers. We study in some detail the case where the kernel of the quadratic functional is finite-dimensional. In this case the optimization problem can be fully characterized by the geometry of a certain convex and compact finite-dimensional set.     

An L \infty/ L 1-Constrained Quadratic Optimization Problem with Applications to Neural Networks

Pattern formation in associative neural networks is related to a quadratic optimization problem. Biological considerations imply that the functional is constrained in the L_\infty norm and in the L₁ norm. We consider such optimization problems. We derive the Euler–Lagrange equations, and construct basic properties of the maximizers. We study in some detail the case where the kernel of the quadratic functional is finite-dimensional. In this case the optimization problem can be fully characterized by the geometry of a certain convex and compact finite-dimensional set.     

The minimal sublinear expectations and their related properties

In this paper, we prove that for a sublinear expectation ɛ[·] defined on L ²(Ω,), the following statements are equivalent:

Local <Emphasis Type="Italic">Tb</Emphasis> theorem with <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> testing conditions and general measures: square functions

Local Tb theorems with L ^p type testing conditions, which are not scale invariant, have been studied widely in the case of the Lebesgue measure. In the non-homogeneous world local Tb theorems have only been proved assuming scale invariant (L ^∞ or BMO) testing conditions. In this paper, for the first time, we overcome these obstacles in the non-homogeneous world, and prove a nonhomogeneous local Tb theorem with L ² type testing conditions. This paper is in the setting of the vertical and conical square functions defined using general measures and kernels. On the technique side, we demonstrate a trick of inserting Calderón–Zygmund stopping data of a fixed function into the construction of the twisted martingale difference operators. This built-in control of averages is an alternative to Carleson embedding.     

On convergence of information in spaces with infinite invariant measure

Summary The paper extends the ergodic theorems of information theory (Shannon-MacMillan-Breiman theorems) to spaces with an infinite invariant measure. An L ₁ difference theorem and a pointwisa ratio theorem are proved, for the information of spreading partitions. For the validity of the theorems it is assumed that the supremum f ^* of the conditional information given the increasing past is integrable. Simple necessary and sufficient conditions for the integrability of f ^* are obtained in special cases: If the initial partition is composed of one state of a null-recurrent Markov chain, then f ^* is integrable if and only if the partition of this state according to the first return times has finite entropy.Research of both authors was supported by the National Science Foundation (U. S. A.), under Grant GP 7693.     

L 2 error estimates for finite elements with interpolated boundary conditions

Summary TheL ² error in the nearly zero quadratic approximate solution to Poisson's equation is shown to be of optimal order. The same method of proof can be used to show that theL ² error with a space of cubics developed by Ridgway Scott is optimal.This work was supported by the Office of Naval Research under Contract Number ONR NR-044-453 and by the Naval Ordnance Laboratory Independent Research Fund.     

Inequalities for Rational Functions

A new hyperbolic area estimate for the level sets of finite Blaschke products is presented. The following inversion of the usual Sobolev embedding theorem is proved:

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Hereris a rational function of degreenwith poles outside . This estimate implies a new inverse theorem for rational approximation of analytic functions with respect to areaL^pnorm.     

Sobolev embedding in ℂn and the\overline \partial - equation

We prove a Sobolev embedding theorem for functions that are in a Sobolev space while their is in L^t, fort large enough. This allows us to deduceL ^p or Lipschitz estimates with loss from classical Sobolev estimates for the solution of in weakly pseudo-convex domains.     

Bounds for algebraic gossip on graphs

We study the stopping times of gossip algorithms for network coding. We analyze algebraic gossip (i.e., random linear coding) and consider three gossip algorithms for information spreading: Pull, Push, and Exchange. The stopping time of algebraic gossip is known to be linear for the complete graph, but the question of determining a tight upper bound or lower bounds for general graphs is still open. We take a major step in solving this question, and prove that algebraic gossip on any graph of size n is O(Δn) where Δ is the maximum degree of the graph. This leads to a tight bound of for bounded degree graphs and an upper bound of O(n²) for general graphs. We show that the latter bound is tight by providing an example of a graph with a stopping time of . Our proofs use a novel method that relies on Jackson's queuing theorem to analyze the stopping time of network coding; this technique is likely to become useful for future research. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 185–217, 2014