Showing papers in "Mathematische Nachrichten in 2006"
TL;DR: In this paper, the authors investigated polynomial decay of classical solutions of linear evolution equations for bounded strongly continuous semigroups on a Banach space, and showed that the property is closely related to the resolvent of the generator.
Abstract: We investigate polynomial decay of classical solutions of linear evolution equations. For bounded strongly continuous semigroups on a Banach space this property is closely related to polynomial growth estimates of the resolvent of the generator. For systems of commuting normal operators polynomial decay is characterized in terms of the location of the generator spectrum. The results are applied to systems of coupled wave-type equations. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
158 citations
TL;DR: In this article, the authors introduce the concept of Lipschitz continuity and prove the existence of continuous partitions of unity in a gage space, where the topology is generated by a family of semi-norms.
Abstract: Let E be a Banach space and Φ : E ℝ a 1-functional. Let be a family of semi-norms on E which separates points and generates a (possibly non-metrizable) topology on E weaker than the norm topology. This is a special case of a gage space, that is, a topological space where the topology is generated by a family of semi-metrics. We develop some critical point theory for Φ : (E, ) ℝ. In particular, we prove deformation lemmas where the deformations are continuous with respect to . In applications this yields a gain in compactness when Φ does not satisfy the Palais–Smale condition because one can work with the weak topology.
We also prove some foundational results on gage spaces. In particular, we introduce the concept of Lipschitz continuity in this setting and prove the existence of Lipschitz continuous partitions of unity. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
155 citations
TL;DR: In this article, a Lusin-area characterization for the Hardy spaces of Coifman and Weiss for p ∈ (p0, 1), where p0 = n /(n + e1) depends on the dimension n of and the regularity e 1 of the Calderon reproducing formula.
Abstract: Let (, d,μ) be a space of homogeneous type in the sense of Coifman and Weiss. Assuming that μ satisfies certain estimates from below and there exists a suitable Calderon reproducing formula in L2(), the authors establish a Lusin-area characterization for the atomic Hardy spaces Hpat() of Coifman and Weiss for p ∈ (p0, 1], where p0 = n /(n + e1) depends on the “dimension” n of and the “regularity” e1 of the Calderon reproducing formula. Using this characterization, the authors further obtain a Littlewood–Paley g*λ-function characterization for Hp () when λ > n + 2n /p and the boundedness of Calderon–Zygmund operators on Hp (). The results apply, for instance, to Ahlfors n -regular metric measure spaces, Lie groups of polynomial volume growth and boundaries of some unbounded model domains of polynomial type in ℂN. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
138 citations
TL;DR: In this paper, the authors consider the case where the potential is so singular that the associated maximally defined Schrodinger operator is self-adjoint and hence no boundary condition is required at the finite end point a.
Abstract: We examine two kinds of spectral theoretic situations: First, we recall the case of self-adjoint half-line Schrodinger operators on [a , ∞), a ∈ ℝ, with a regular finite end point a and the case of Schrodinger operators on the real line with locally integrable potentials, which naturally lead to Herglotz functions and 2 × 2 matrix-valued Herglotz functions representing the associated Weyl–Titchmarsh coefficients. Second, we contrast this with the case of self-adjoint half-line Schrodinger operators on (a , ∞) with a potential strongly singular at the end point a . We focus on situations where the potential is so singular that the associated maximally defined Schrodinger operator is self-adjoint (equivalently, the associated minimally defined Schrodinger operator is essentially selfadjoint) and hence no boundary condition is required at the finite end point a . For this case we show that the Weyl–Titchmarsh coefficient in this strongly singular context still determines the associated spectral function, but ceases to posses the Herglotz property. However, as will be shown, Herglotz function techniques continue to play a decisive role in the spectral theory for strongly singular Schrodinger operators. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
133 citations
TL;DR: In this article, a solution of the Cahn-Hilliard equation and associated Caginalp problem with dynamic boundary condition is considered and it is shown that under some conditions on the potential it converges, as t ∞, to a stationary solution.
Abstract: We consider a solution of the Cahn–Hilliard equation or an associated Caginalp problem with dynamic boundary condition in the case of a general potential and prove that under some conditions on the potential it converges, as t ∞, to a stationary solution. The main tool will be the Łojasiewicz–Simon inequality for the underlying energy functional. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
122 citations
TL;DR: In this article, the authors give a characterization of d-dimensional modulation spaces with moderate weights by means of the ddimensional Wilson basis and prove that pseudodifferential operators with generalized Weyl symbols are bounded on these modulation spaces.
Abstract: We give a characterization of d-dimensional modulation spaces with moderate weights by means of the d-dimensional Wilson basis. As an application we prove that pseudodifferential operators with generalized Weyl symbols are bounded on these modulation spaces.
88 citations
TL;DR: In this article, the authors introduced a Littlewood-Paley decomposition related to any sub-Laplacian on a Lie group G of polynomial volume growth.
Abstract: We introduce a Littlewood--Paley decomposition related to any sub-Laplacian on a Lie group G of polynomial volume growth; this allows us to prove a Littlewood--Paley theorem in this general setting and to provide a dyadic characterization of Besov spaces B^{s,q}_p(G), s in R, equivalent to
the classical definition through the heat kernel.
83 citations
TL;DR: In this paper, the authors introduce a family of metrics on the configuration space Γ, which makes it a Polish space, and consider the topological and metrical properties of configuration spaces.
Abstract: We study some topological and metrical properties of configuration spaces. In particular, we introduce a family of metrics on the configuration space Γ, which makes it a Polish space. Compact functions on Γ are also considered. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
68 citations
TL;DR: In this article, the authors define an abstract setting to treat wave equations equipped with time-dependent acoustic boundary conditions on bounded domains of Rn, and develop a spectral theory which also allows to prove a conjecture proposed in [13].
Abstract: We define an abstract setting to treat wave equations equipped with time-dependent acoustic boundary conditions on bounded domains of Rn. We prove a well-posedness result and develop a spectral theory which also allows to prove a conjecture proposed in [13]. Concrete problems are also discussed. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
60 citations
TL;DR: In this paper, the same formula holds for all convex bodies with C3 boundary and everywhere positive curvature where x(x) denotes the generalized Gaus-Kronecker curvature.
Abstract: Let K be a convex body in Rd. A random polytope is the convex hull [x1, …, xn] of finitely many points chosen at random in K. IE(K, n) is the expectation of the volume of a random polytope of n randomly chosen points. I. Barany showed that we have for convex bodies with C3 boundary and everywhere positive curvature
where x(x) denotes the Gaus-Kronecker curvature. We show that the same formula holds for all convex bodies if x(x) denotes the generalized Gaus-Kronecker curvature.
56 citations
TL;DR: In this paper, the concept of Neumann functions for the Laplacian was extended to higher-order Neumann (Neumann-n) problems for Δnu = f.
Abstract: Rewriting the higher order Poisson equation Δnu = f in a plane domain as a system of Poisson equations it is immediately clear what boundary conditions may be prescribed in order to get (unique) solutions. Neumann conditions for the Poisson equation lead to higher-order Neumann (Neumann-n ) problems for Δnu = f . Extending the concept of Neumann functions for the Laplacian to Neumann functions for powers of the Laplacian leads to an explicit representation of the solution to the Neumann-n problem for Δnu = f . The representation formula provides the tool to treat more general partial differential equations with leading term Δnu in reducing them into some singular integral equations. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
TL;DR: In this paper, the authors explore the extent to which basic differential operators (such as Laplace-Beltrami, Lame, Navier-Stokes, etc.) and boundary value problems on a hypersurface in ℝn can be expressed globally, in terms of the standard spatial coordinates in √ n. The approach also provides, in some important cases, useful simplifications as well as new interpretations of classical operators and equations.
Abstract: We explore the extent to which basic differential operators (such as Laplace–Beltrami, Lame, Navier–Stokes, etc.) and boundary value problems on a hypersurface in ℝn can be expressed globally, in terms of the standard spatial coordinates in ℝn. The approach we develop also provides, in some important cases, useful simplifications as well as new interpretations of classical operators and equations. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
TL;DR: In this paper, the authors examined iterated function systems consisting of a countably infinite number of contracting mappings (IIFS) and showed that IIFS can be approximated by appropriately chosen IFS both in terms of Hausdorff distance and of Haudorff dimension, and compared the descriptive power of IFS and IIFS as mechanisms defining closed and bounded sets.
Abstract: We examine iterated function systems consisting of a countably infinite number of contracting mappings (IIFS). We state results analogous to the well-known case of finitely many mappings (IFS). Moreover, we show that IIFS can be approximated by appropriately chosen IFS both in terms of Hausdorff distance and of Hausdorff dimension. Comparing the descriptive power of IFS and IIFS as mechanisms defining closed and bounded sets, we show that IIFS are strictly more powerful than IFS. On the other hand, there are closed and bounded non-empty sets not describable by IIFS.
TL;DR: In this paper, the Fourier-Bros-Iagolnitzer transformation was used to study the complex interactions between electronic levels that do not intersect in the real domain, and an exponential weight was added to the L2-type Hilbert spaces associated to a complex Lagrangian manifold.
Abstract: We propose a method to study the complex interactions (or microlocal tunneling) between electronic levels that do not intersect in the real domain. The method consists in using a special kind of Fourier-Bros-Iagolnitzer transformation, and in adjoining an exponential weight to the L2-type Hilbert spaces which are associated to a complex Lagrangian manifold and have been introduced by Helffer and Sjostrand for the study of resonances.
TL;DR: In this paper, an intrinsic characterization of the restrictions of Sobolev, Triebel-Lizorkin and Besov spaces to regular subsets of ℝn via sharp maximal functions and local approximations is given.
Abstract: We give an intrinsic characterization of the restrictions of Sobolev (ℝn), Triebel–Lizorkin (ℝn) and Besov (ℝn) spaces to regular subsets of ℝn via sharp maximal functions and local approximations. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
TL;DR: In this paper, a Szego theory for orthogonal rational matrix-valued functions on the unit circle has been proposed for matrix polynomials on the other side.
Abstract: This paper contains further steps towards a Szego theory for orthogonal rational matrix-valued functions on the unit circle . It continues the investigations started in [18]–[20]. Hereby we are guided by former work of Bultheel, Gonzalez–Vera, Hendriksen, and Njastad on scalar orthogonal rational functions on the one side and by research of Delsarte, Genin, and Kamp on matrix polynomials on the other side. In this paper we derive recursion formulas for Christoffel–Darboux pairs of rational matrix functions which lead us to jqq-recursively connected pairs of rational matrix functions. Moreover, we prove a Favard-type theorem for rational matrix functions. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
TL;DR: In this article, Sobolev embeddings for Riesz potential spaces of variable exponent were used to deal with the problem of embedding Sobolevs in Riestz potential spaces.
Abstract: Our aim in this paper is to deal with Sobolev embeddings for Riesz potential spaces of variable exponent (© 2006 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim)
TL;DR: In this paper, the authors give a method to compute the space of infinitesimal deformations of a double cover of a smooth algebraic variety, which is a special case of Calabi-Yau threefolds, which are nonsingular models of double covers of the projective 3-space branched along an octic surface.
Abstract: The goal of this paper is to give a method to compute the space of infinitesimal deformations of a double cover of a smooth algebraic variety. The space of all infinitesimal deformations has a representation as a direct sum of two subspaces. One is isomorphic to the space of simultaneous deformations of the branch locus and the base of the double covering. The second summand is the subspace of deformations of the double covering which induce trivial deformations of the branch divisor. The main result of the paper is a description of the effect of imposing singularities in the branch locus.
As a special case we study deformations of Calabi–Yau threefolds which are non-singular models of double cover of the projective 3-space branched along an octic surface. We show that in that case the number of deformations can be computed explicitly using computer algebra systems. This gives a method to compute the Hodge numbers of these Calabi–Yau manifolds. In this case the transverse deformations are resolutions of deformations of double covers of projective space but not double covers of a blow-up of projective space. In the paper we gave many explicit examples. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
TL;DR: In this article, brushlet bases are constructed to form unconditional and even greedy bases for the α-modulation spaces, a family of spaces that contain the Besov and modulation spaces as special cases.
Abstract: The a -modulation spaces are a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that brushlet bases can be constructed to form unconditional and even greedy bases for the α -modulation spaces. We study m -term nonlinear approximation with brushlet bases, and give complete characterizations of the associated approximation spaces in terms of α -modulation spaces. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
TL;DR: In this article, the authors decrivons ici explicitement le lien entre ces deux procedes, i.e., associer, a chaque caractere regulier de Ex, a classe de representations admissibles irreductible supercuspidales de GLn(F), and an extension non ramifiee de F, de degre n.
Abstract: Soit F un corps local non archimedien a corps residuel fini, et soit E une extension non ramifiee de F, de degre n. On dispose de plusieurs procedes pour associer, a chaque caractere regulier de Ex, une classe de representations admissibles irreductibles supercuspidales de GLn(F). L'un est totalement explicite et est du a P. Gerardin. Un autre, obtenu par l'utilisation globale de formules des traces, est du a D. Kazhdan. Nous decrivons ici explicitement le lien entre ces deux procedes.
TL;DR: A construction of a one-dimensional Schrodinger operator with an inner structure defined on a set of Lebesgue measure zero and an interaction given on such a set is given in this article.
Abstract: A construction of a one-dimensional Schrodinger operator that has an inner structure defined on a set of Lebesgue measure zero and an interaction given on such a set. General Krein–Feller operators are constructed and the spectrum of a Schrodinger operator with a δ′-interaction given on a Cantor set is studied. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
TL;DR: In this article, the class of convex univalent functions f in the unit disc normalized by f (0) = f ∈, f (1) = 2λ ∈ and |λ | ≤ 1 were determined explicitly the regions of variability.
Abstract: Let be the class of convex univalent functions f in the unit disc normalized by f (0) = f ′(0) – 1 = 0 For z0 ∈ and |λ | ≤ 1 we shall determine explicitly the regions of variability {log f ′(z0): f ∈ , f ″(0) = 2λ } (© 2006 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim)
TL;DR: In this paper, it is shown that every irreducible system of Okubo normal form of semi-simple type whose monodromy representation is rigid is obtained from a rank 1 system of O-normal form by a finite iteration of the operations.
Abstract: For systems of differential equations of the form (xIn – T )dy /dx = Ay (systems of Okubo normal form), where A is an n × n constant matrix and T is an n × n constant diagonal matrix, two kinds of operations (extension and restriction) are defined It is shown that every irreducible system of Okubo normal form of semi-simple type whose monodromy representation is rigid is obtained from a rank 1 system of Okubo normal form by a finite iteration of the operations Moreover, an algorithm to calculate the generators of monodromy groups for rigid systems of Okubo normal form is given (© 2006 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim)
TL;DR: In this article, the Hardy-type Hardy operator is used to measure the Kolmogorov width of a positive function with respect to the n-th approximation number of the function.
Abstract: Let I = [a , b ] ⊂ ℝ, let 1 < q ≤ p < ∞, let u and v be positive functions with u ∈ Lp ′ (I ) and v ∈ Lq(I ), and let T : Lp(I ) Lq(I ) be the Hardy-type operator given by
Given any n ∈ ℕ, let sn stand for either the n -th approximation number of T or the n -th Kolmogorov width of T . We show that
where cpq is an explicit constant depending only on p and q . (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
TL;DR: In this article, conditions on the weight w in order to characterize functions in weighted Besov spaces Bp.q/w.φ in terms of differences Δxf were given.
Abstract: We find conditions on the weight w in order to characterize functions in weighted Besov spaces Bp.q/w.φ in terms of differences Δxf.