On Minimal Subspaces in Tensor Representations

In this paper we introduce and develop the notion of minimal subspaces in the framework of algebraic and topological tensor product spaces. This mathematical structure arises in a natural way in the study of tensor representations. We use minimal subspaces to prove the existence of a best approximation, for any element in a Banach tensor space, by means of a tensor given in a typical representation format (Tucker, hierarchical, or tensor train). We show that this result holds in a tensor Banach space with a norm stronger than the injective norm and in an intersection of finitely many Banach tensor spaces satisfying some additional conditions. Examples using topological tensor products of standard Sobolev spaces are given.     

Points de namioka espaces normants applications à la théorie isométrique de la dualité

We study in this work various aspects of the isometric theory of duality. We show that in wide classes of Banach spaces, dual spaces are characterized by the existence of a retraction fromE″ ontoE. The predual of such spaces is then unique. We study the imbedding of regularly normed spaces into dual spaces. We better the known results on loss of regularity of the norm of dual spaces. We characterize the dual norms on an Asplund space in terms of “bad differentiability”.      

On dual locally uniformly rotund norms

We show that the existence of an equivalent dual LUR norm on a dual Banach space can be characterized by a topological property similar to the fragmentability. The compact spaces homeomorphic to weak* compact subsets of a dual LUR Banach space have the same properties as the class of Radon-Nikodym compact spaces. Research supported by the DGICYT PB 95-1025.     

Norm behaviour of solutions to a parabolic–elliptic system modelling chemotaxis in a domain of ℝ3

In this paper, we study a parabolic–elliptic system defined on a bounded domain of ?³, which comes from a chemotactic model. We first prove the existence and uniqueness of local in time solution to this problem in the Sobolev spaces framework, then we study the norm behaviour of solution, which may help us to determine the blow‐up norm of the maximal solution. Copyright © 2004 John Wiley & Sons, Ltd.     

Besov-Morrey spaces: Function space theory and applications to non-linear PDE

This paper is devoted to the analysis of function spaces modeled on Besov spaces and their applications to non-linear partial differential equations, with emphasis on the incompressible, isotropic Navier-Stokes system and semi-linear heat equations. Specifically, we consider the class, introduced by Hideo Kozono and Masao Yamazaki, of Besov spaces based on Morrey spaces, which we call Besov-Morrey or BM spaces. We obtain equivalent representations in terms of the Weierstrass semigroup and wavelets, and various embeddings in classical spaces. We then establish pseudo-differential and para-differential estimates. Our results cover non-regular and exotic symbols. Although the heat semigroup is not strongly continuous on Morrey spaces, we show that its action defines an equivalent norm. In particular, homogeneous BM spaces belong to a larger class constructed by Grzegorz Karch to analyze scaling in parabolic equations. We compare Karch's results with those of Kozono and Yamazaki and generalize them by obtaining short-time existence and uniqueness of solutions for arbitrary data with subcritical regularity. We exploit pseudo-differential calculus to extend the analysis to compact, smooth, boundaryless, Riemannian manifolds. BM spaces are defined by means of partitions of unity and coordinate patches, and intrinsically in terms of functions of the Laplace operator.

     

Non-coercive Lyapunov functions for infinite-dimensional systems

We show that the existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided the speed of decay is measured in terms of the norm of the state and an additional mild assumption is satisfied. For evolution equations in Banach spaces with Lipschitz continuous nonlinearities these additional assumptions become especially simple. The results encompass some recent results on linear switched systems on Banach spaces. Finally, we derive new non-coercive converse Lyapunov theorems and give some examples showing the necessity of our assumptions.     

An interpolation theorem related to the a.e. convergence of integral operators

We show that for integral operators of general form the norm bounds in Lorentz spaces imply certain norm bounds for the maximal function. As a consequence, the a.e. convergence for the integral operators on Lorentz spaces follows from the appropriate norm estimates.

     

The cauchy problem for a shallow water type equation

We prove existence and uniqueness of local and global solutions of the periodic Cauchy problem for a higher order shallow water type equation under low regularity initial data. Using Fourier analysis we first prove local estimates in appropriate spaces and then use a contraction mapping argument and a conserved norm to get global existence.     

Adjoint for operators in Banach spaces

In this paper we show that a result of Gross and Kuelbs, used to study Gaussian measures on Banach spaces, makes it possible to construct an adjoint for operators on separable Banach spaces. This result is used to extend well-known theorems of von Neumann and Lax. We also partially solve an open problem on the existence of a Markushevich basis with unit norm and prove that all closed densely defined linear operators on a separable Banach space can be approximated by bounded operators. This last result extends a theorem of Kaufman for Hilbert spaces and allows us to define a new metric for closed densely defined linear operators on Banach spaces. As an application, we obtain a generalization of the Yosida approximator for semigroups of operators.

     

Complementation in spaces of continuous functions on compact lines

We characterize order preserving continuous surjections between compact linearly ordered spaces which admit an averaging operator, together with estimates of the norm of such an operator. This result is used to the study of strengthenings of the separable complementation property in spaces of continuous functions on compact lines. These properties include in particular continuous separable complementation property and existence of a projectional skeleton.     

Norm-one complemented hyperplanes in spaces with a Schauder basis

Some results on existence of norm one projections onto hyperplanes in spaces with a Schauder basis are given. Possible characterizations of Hilbert spaces using this property are also discussed.     

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.      

Quasihyperbolic geodesics in convex domains

We show that quasihyperbolic geodesics exist in convex domains in reflexive Banach spaces and that quasihyperbolic geodesies are quasiconvex in the norm metric in convex domains in all normed spaces.     

Convex hull properties in location theory

We present new characterizations of inner product spaces which bring into play a property of a family of optimization problems related to the norm of the space. This property concerns the existence of a solution .to some optimization problems which belongs to the convex hull of some set. We thus obtain a generalization of results of V. Klee and A. Garkavi about the Chebychev centers and also of more recent results of the author about Fermat points. Intermediate propositions concerning unicity in some optimization problems, a geometric characterization of finite dimensional inner product spaces and monotone norms seem to have their own interest.     

Counting functions of zeros of analytic functions in spaces with mixed norm

We study the behavior of counting functions of zeros of analytic in a disk functions in spaces with mixed norm, in particular, the Bergman-Dzhrbashyan spaces with standard weights. We obtain corollaries that strengthen the known results on zero sets of spaces with mixed norm.     

An iterative method for split inclusion problems without prior knowledge of operator norms

In this paper, we study the approximation of solution (assuming existence) for the split inclusion problem in uniformly convex Banach spaces which are also uniformly smooth. We introduce an iterative algorithm in which the stepsizes are selected without the need for any prior information about the bounded linear operator norm and strong convergence obtained. The novelty of our algorithm is that the bounded linear operator norm is not given a priori and stepsizes are constructed step by step in a natural way. Our results extend and improve many recent and important results obtained in the literature on the split inclusion problem and its variations.     

赋Amemiya-Orlicz范数的Musielak-Orlicz-Sobolev空间单调性在最佳逼近中的应用

利用Musielak-Orlicz-Sobolev空间的构成特点,借鉴Orlicz-Sobolev空间的单调性在最佳逼近中的一些应用,以Orlicz空间中Jensen'S不等式的推广为主要工具,讨论了赋Amemiya-Orlicz范数的Musielak-Orlicz-Sobolev空间中的最佳逼近问题,主要是唯一性、存在性、稳定性.     

Products of uniformly noncreasy spaces

We show that finite products of uniformly noncreasy spaces with a strictly monotone norm have the fixed point property for nonexpansive mappings. It gives new and natural examples of superreflexive Banach spaces without normal structure but with the fixed point property.

     

赋Orlicz范数的Orlicz空间中最佳逼近算子

在赋Ｏｒｌｉｃｚ范数的Ｏｒｌｉｃｚ空间中,给出最佳逼近算子单调性的一个充分条件和最佳逼近元存在定理．     

Nonlinear Riemann-Hilbert Problems with Unbounded Restriction Curves

Using a norm inequality for singular integral operators in pairs of weighted Lebesgue spaces we prove new existence and uniqueness results for solutions of nonlinear Riemann-Hilbert problems with noncompact restriction curves.