赋以混合范数的各向异性Besov类在不同度量下的嵌入定理

本文给出了赋以混合范数的各向异性多元Besov类BPθr(Rd)的一个表现定理, 利用此表现定理,证明了它在不同度量下的一个嵌入定理BPθr(Rd)→Bqθr'(Rd),其中 1≤P≤q≤∞,r'=(1-∑jd=1(1/pj-1/qj)1/rj)r．     

Measures under the flat norm as ordered normed vector space

The space of real Borel measures \(\mathcal {M}(S)\) on a metric space S under the flat norm (dual bounded Lipschitz norm), ordered by the cone \(\mathcal {M}_+(S)\) of nonnegative measures, is considered from an ordered normed vector space perspective in order to apply the well-developed theory of this area. The flat norm is considered in place of the variation norm because subsets of \(\mathcal {M}_+(S)\) are compact and semiflows on \(\mathcal {M}_+(S)\) are continuous under much weaker conditions. In turn, the flat norm offers new challenges because \(\mathcal {M}(S)\) is rarely complete and \(\mathcal {M}_+(S)\) is only complete if S is complete. As illustrations serve the eigenvalue problem for bounded additive and order-preserving homogeneous maps on \(\mathcal {M}_+(S)\) and continuous semiflows. Both topics prepare for a dynamical systems theory on \(\mathcal {M}_+(S)\).     

Local stability of mappings with bounded distortion on Heisenberg groups

This is the second of the author’s three papers on stability in the Liouville theorem on the Heisenberg group. The aim is to prove that each mapping with bounded distortion of a John domain on the Heisenberg group is close to a conformal mapping with order of closeness \(\sqrt {K - 1} \) in the uniform norm and order of closeness K ? 1 in the Sobolev norm L _p ¹ for all \(p < \tfrac{C}{{K - 1}}\).In this paper we prove a local variant of the desired result: each mapping on a ball with bounded distortion and distortion coefficient K near to 1 is close on a smaller ball to a conformal mapping with order of closeness \(\sqrt {K - 1} \) in the uniform norm and order of closeness K ? 1 in the Sobolev norm L _p ¹ for all \(p < \tfrac{C}{{K - 1}}\). We construct an example that demonstrates the asymptotic sharpness of the order of closeness of a mapping with bounded distortion to a conformal mapping in the Sobolev norm.     

Convergence analysis of the rectangular Morley element scheme for second order problem in arbitrary dimensions

We present the convergence analysis of the rectangular Morley element scheme utilised on the second order problem in arbitrary dimensions. Specifically, we prove that the convergence of the scheme is of O(h) order in energy norm and of O(h²) order in L² norm on general d-rectangular triangulations. Moreover, when the triangulation is uniform, the convergence rate can be of O(h²) order in energy norm, and the convergence rate in L² norm is still of O(h²) order, which cannot be improved. Numerical examples are presented to demonstrate our theoretical results.     

Hilbert投影距离与Banach空间的范数

In this paper, we establish some relations between the Hilbert＇s projective metric and the norm on a Banach space and show that the metric and the norm induce equivalent convergences at certain set. As applications, we utilize the main results to discuss the eigenvalue problems for a class of positive homogeneous operators of degree a and the positive solutions for a class of nonlinear algebraic system.     

CVaR norm and applications in optimization

This paper introduces the family of CVaR norms in \({\mathbb {R}}^{n}\) , based on the CVaR concept. The CVaR norm is defined in two variations: scaled and non-scaled. The well-known \(L_{1}\) and \(L_{\infty }\) norms are limiting cases of the new family of norms. The D-norm, used in robust optimization, is equivalent to the non-scaled CVaR norm. We present two relatively simple definitions of the CVaR norm: (i) as the average or the sum of some percentage of largest absolute values of components of vector; (ii) as an optimal solution of a CVaR minimization problem suggested by Rockafellar and Uryasev. CVaR norms are piece-wise linear functions on \({\mathbb {R}}^{n}\) and can be used in various applications where the Euclidean norm is typically used. To illustrate, in the computational experiments we consider the problem of projecting a point onto a polyhedral set. The CVaR norm allows formulating this problem as a convex or linear program for any level of conservativeness.     

Order Norm Completions of Cone Metric Spaces

In this article, a completion theorem for cone metric spaces and a completion theorem for cone normed spaces are proved. The completion spaces are constructed by means of an equivalence relation defined via an ordered cone norm on the Banach space E whose cone is strongly minihedral and ordered closed. This order norm has to satisfy the generalized absolute value property. In particular, if E is a Dedekind complete Banach lattice, then, together with its absolute value norm, satisfy the desired properties.     

Geometric Characterization of Weyl’s Discrepancy Norm in Terms of Its <Emphasis Type="Italic">n</Emphasis>-Dimensional Unit Balls

Weyl’s discrepancy measure induces a norm on ? ⁿ which shows a monotonicity and a Lipschitz property when applied to differences of index-shifted sequences. It turns out that its n-dimensional unit ball is a zonotope that results from a multiple sheared projection from the (n+1)-dimensional hypercube which can be interpreted as a discrete differentiation. This characterization reveals that this norm is the canonical metric between sequences of differences of values from the unit interval in the sense that the n-dimensional unit ball of the discrepancy norm equals the space of such sequences.     

Remarks on the Convergence of Pseudospectra

We establish the convergence of pseudospectra in Hausdorff distance for closed operators acting in different Hilbert spaces and converging in the generalised norm resolvent sense. As an assumption, we exclude the case that the limiting operator has constant resolvent norm on an open set. We extend the class of operators for which it is known that the latter cannot happen by showing that if the resolvent norm is constant on an open set, then this constant is the global minimum. We present a number of examples exhibiting various resolvent norm behaviours and illustrating the applicability of this characterisation compared to known results.     

Positive semidefinite indicator property for norm-numerical range

In this paper, using an elementary method, we prove that if norm-numerical range related to an absolute norm \(\Vert .\Vert \) on \(\mathbb {C}^n\) satisfies the positive semidefinite indicator property, then \(\Vert .\Vert \) will be a multiple of Euclidian norm.     

Unbounded disjointness preserving linear functionals and operators

Let E and F be Banach lattices. We show first that the disjointness preserving linear functionals separate the points of any infinite dimensional Banach lattice E, which shows that in this case the unbounded disjointness preserving operators from \({E\to F}\) separate the points of E. Then we show that every disjointness preserving operator \({T:E\to F}\) is norm bounded on an order dense ideal. In case E has order continuous norm, this implies that every unbounded disjointness preserving map \({T:E\to F}\) has a unique decomposition T = R + S, where R is a bounded disjointness preserving operator and S is an unbounded disjointness preserving operator, which is zero on a norm dense ideal. For the case that E = C(X), with X a compact Hausdorff space, we show that every disjointness preserving operator \({T:C(X)\to F}\) is norm bounded on a norm dense sublattice algebra of C(X), which leads then to a decomposition of T into a bounded disjointness preserving operator and a finite sum of unbounded disjointness preserving operators, which are zero on order dense ideals.     

Real-Linear Isometries on Spaces of Functions of Bounded Variation

Let X and Y be subsets of the real line with at least two points. We study the surjective real-linear isometries \({T:BV(X)\longrightarrow BV(Y)}\) between the spaces of functions of bounded variation on X and Y with respect to the supremum norm \({\|\cdot\|_\infty}\) and the complete norm \({\|\cdot\|:=\max(\|\cdot\|_\infty,\mathcal{V}(\cdot))}\), where \({\mathcal{V}(\cdot)}\) denotes the total variation of a function. Additively norm preserving maps between these spaces are also characterized as a corollary.     

Sur la forme de la boule unité de la norme stable unidimensionnelle

We study the stable norm on the first homology of a Riemannian polyhedron. In the one-dimensional case (metric graphs), the geometry of the unit ball of this norm is completely described by the combinatorial structure of the graph. For a smooth manifold of dimension ≥3 and using polyhedral techniques, we show that a large class of polytopes appears as unit ball of the stable norm associated to some metric conformal to a given one. Received: 18 March     

解析Sobolev型空间上的算子与算子代数

介绍Hardy-Sobolev空间和Fock空间及其算子与算子代数研究方面所做的工作,包括对这两类空间上几类特殊算子有界性、紧性、Fredholm性、指标理论、谱和本性谱、范数和本性范数、Schatten-p类的讨论,以及由它们所生成的C~*-代数的研究.     

An hp finite element method for 4th order singularly perturbed problems

We present the analysis for the hp finite element approximation of the solution to singularly perturbed fourth order problems, using a balanced norm. In Panaseti et al. (2016) it was shown that the hp version of the Finite Element Method (FEM) on the so-called Spectral Boundary Layer Mesh yields robust exponential convergence when the error is measured in the natural energy norm associated with the problem. In the present article we sharpen the result by showing that the same hp-FEM on the Spectral Boundary Layer Mesh gives robust exponential convergence in a stronger, more balanced norm. As a corollary we also get robust exponential convergence in the maximum norm. The analysis is based on the ideas in Roos and Franz (Calcolo 51, 423–440, 2014) and Roos and Schopf (ZAMM 95, 551–565, 2015) and the recent results in Melenk and Xenophontos (2016). Numerical examples illustrating the theory are also presented.     

Renormages de Quelquesℒ(K)

LetD([0, 1]) be the space of left continuous real valued functions on [0, 1] which have a right limit at each point. We show thatD([0, 1]) has no equivalent norm which is Gateau differentiable. Hence the class of spaces which can be renormed by a Gateau differentiable norm fails the three spaces property. We show that there is no norm onℒ([0, Ω]) such that its dual is strictly convex. However, there is an equivalent Fréchet differentiable norm on this space.      

The monotonicity condition and a priori estimates of the Hölder norm for a class of nondiagonal elliptic systems with quadratic nonlinearity

The Dirichlet problem for quasilinear elliptic systems of equations with nondiagonal principal matrix and additional terms with strong nonlinearity in the gradient is considered. An a priori condition on the behavior of solutions in a neighborhood of a fixed point is stated, which allows us to estimate locally the Hölder norm of a solution in terms of its energy norm. Owing to the monotonicity inequality proved for one class of such systems, the a priori assumption on the behavior of solutions is reduced to an optimal one while estimating the Hölder norm of the solution. Bibliography: 27 titles.     

Multiplication Operators on the Iterated Logarithmic Lipschitz Spaces of a Tree

We introduce a class of iterated logarithmic Lipschitz spaces ${\mathcal{L}^{(k)}, k \in \mathbb{N}}$ , on an infinite tree which arise naturally in the context of operator theory. We characterize boundedness and compactness of the multiplication operators on ${\mathcal{L}^{(k)}}$ and provide estimates on their operator norm and their essential norm. In addition, we determine the spectrum, characterize the multiplication operators that are bounded below, and prove that on such spaces there are no nontrivial isometric multiplication operators and no isometric zero divisors.     

Norm estimates of ω-circulant operator matrices and isomorphic operators for ω-circulant algebra

An n ×nω-circulant matrix which has a specific structure is a type of important matrix. Several norm equalities and inequalities are proved for ω-circulant operator matrices with ω = e^iθ (0 ≤ θ < 2π) in this paper. We give the special cases for norm equalities and inequalities, such as the usual operator norm and the Schatten p-norms. Pinching type inequality is also proposed for weakly unitarily invariant norms. Meanwhile, we present that the set of ω-circulant matrices with complex entries has an idempotent basis. Based on this basis, we introduce an automorphism on the ω-circulant algebra and then show different operators on linear vector space that are isomorphic to the ω-circulant algebra. The function properties, other idempotent bases and a linear involution are discussed for ω-circulant algebra. These results are closely related to the special structure of ω-circulant matrices.