Estimates of n ‐widths of Besov classes on two‐point homogeneous manifolds

Estimates of Kolmogorov and linear n ‐widths of Besov classes on compact globally symmetric spaces of rank 1 (i.e., on S^d, P^d (?), P^d (?), P^d (?), P¹⁶(Cay)) are established. It is shown that these estimates have sharp orders in different important cases. A new characterisation of Besov spaces is also given (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global existence for the multi-dimensional compressible viscoelastic flows

The global solutions in critical spaces to the multi-dimensional compressible viscoelastic flows are considered. The global existence of the Cauchy problem with initial data close to an equilibrium state is established in Besov spaces. Using uniform estimates for a hyperbolic-parabolic linear system with convection terms, we prove the global existence in the Besov space which is invariant with respect to the scaling of the associated equations. Several important estimates are achieved, including a smoothing effect on the velocity, and the L¹-decay of the density and deformation gradient.     

N-Widths and Average Widths of Besov Classes in Sobolev Spaces

In this paper, we consider the n-widths and average widths of Besov classes in the usual Sobolev spaces. The weak asymptotic results concerning the Kolmogorov n-widths, the linear n-widths, the Gel＇fand n-widths, in the Sobolev spaces on T^d, and the infinite-dimensional widths and the average widths in the Sobolev spaces on Ra are obtained, respectively.     

Trace ideals for pseudo-differential operators and their commutators with symbols in α-modulation spaces

That symbols in the modulation space M ^1,1 generate pseudo-differential operators of the trace class was first stated by Feichtinger and proved by Gröchenig in [13]. In this paper, we show that the same is true if we replace M ^1,1 by the more general α-modulation spaces, which include modulation spaces (α = 0) and Besov spaces (α = 1) as special cases. The result with α = 0 corresponds to that of Gröchenig, and the one with α = 1 is a new result which states the trace property of the operators with symbols in the Besov space. As an application, we discuss the trace property of the commutator [α (X, D), a], where; a(χ) is a Lipschitz function and σ(χ, ξ) belongs to an α-modulation space.     

Atomic decomposition of Besov-type and Triebel-Lizorkin-type spaces

Abstract With the help of the maximal function caracterizations of the Besov-type space Bs,τp,q and the TriebelLizorkin-type space Fs,τp,q,we present the atomic decomposition of these function spaces.Our results cover the results on classical Besov and Triebel-Lizorkin spaces by taking τ=0.     

Bessel Potentials in Ahlfors Regular Metric Spaces

In this paper we introduce Bessel potentials and the Sobolev potential spaces resulting from them in the context of Ahlfors regular metric spaces. The Bessel kernel is defined using a Coifman type approximation of the identity, and we show integration against it improves the regularity of Lipschitz, Besov and Sobolev-type functions. For potential spaces, we prove density of Lipschitz functions, and several embedding results, including Sobolev-type embedding theorems. Finally, using singular integrals techniques such as the T1 theorem, we find that for small orders of regularity Bessel potentials are inversible, its inverse in terms of the fractional derivative, and show a way to characterize potential spaces, concluding that a function belongs to the Sobolev potential space if and only if itself and its fractional derivative are in L^p. Moreover, this characterization allows us to prove these spaces in fact coincide with the classical potential Sobolev spaces in the Euclidean case.     

Wavelet Characterizations for Anisotropic Besov Spaces

The goal of this paper is to provide wavelet characterizations for anisotropic Besov spaces. Depending on the anisotropy, appropriate biorthogonal tensor product bases are introduced and Jackson and Bernstein estimates are proved for two-parameter families of finite-dimensional spaces. These estimates lead to characterizations for anisotropic Besov spaces by anisotropy-dependent linear approximation spaces and lead further on to interpolation and embedding results. Finally, wavelet characterizations for anisotropic Besov spaces with respect to L_p-spaces with 0<p<∞ are derived.     

Vanishing moment conditions for wavelet atoms in higher dimensions

We provide explicit criteria for wavelets to give rise to frames and atomic decompositions in L²(?^d), but also in more general Banach function spaces. We consider wavelet systems that arise by translating and dilating the mother wavelet, with the dilations taken from a suitable subgroup of GL(?^d), the so-called dilation group.The paper provides a unified approach that is applicable to a wide range of dilation groups, thus giving rise to new atomic decompositions for homogeneous Besov spaces in arbitrary dimensions, but also for other function spaces such as shearlet coorbit spaces. The atomic decomposition results are obtained by applying the coorbit theory developed by Feichtinger and Gröchenig, and they can be informally described as follows: Given a function ψ ∈ L²(?^d) satisfying fairly mild decay, smoothness and vanishing moment conditions, any sufficiently fine sampling of the translations and dilations will give rise to a wavelet frame. Furthermore, the containment of the analyzed signal in certain smoothness spaces (generalizing the homogeneous Besov spaces) can be decided by looking at the frame coefficients, and convergence of the frame expansion holds in the norms of these spaces. We motivate these results by discussing nonlinear approximation.     

RESTRICTED NONLINEAR APPROXIMATION AND SINGULAR SOLUTIONS OF BOUNDARY INTEGRAL EQUATIONS

This paper studies several problems, which are potentially relevant for the construction of adaptive numerical schemes. First, biorthogonal spline wavelets on [0,1] are chosen as a starting point for characterizations of functions in Besov spaces B _r,r ⁶ (0.1) with 0<σ<∞ and (1+σ)⁻¹相似文献   

Local Solarity of Suns in Normed Linear Spaces

The paper is concerned with solarity of intersections of suns with bars (in particular, with closed balls and extreme hyperstrips) in normed linear spaces. A sun in a finite-dimensional (BM)-space (in particular, in ? ¹(n)) is shown to be monotone path connected. A nonempty intersection of an m-connected set (in particular, a sun in a two-dimensional space or in a finite-dimensional (BM)-space) with a bar is shown to be a monotone path-connected sun. Similar results are obtained for boundedly compact subsets of infinite-dimensional spaces. A nonempty intersection of a monotone path-connected subset of a normed space with a bar is shown to be a monotone path-connected α-sun.     

Generalized variational inequalities with generalized coercivity conditions

Some results due to Fang and Peterson on generalized variational inequalities in the space ? ⁿ are extended to infinite-dimensional spaces. Theorems on the existence of solutions of such inequalities under generalized coercivity conditions are obtained.     

Unitary representations of unimodular Lie groups in Bergman spaces

For an arbitrary unimodular Lie group G, we construct strongly continuous unitary representations in the Bergman space of a strongly pseudoconvex neighborhood of G in the complexification of its underlying manifold. These representation spaces are infinite-dimensional and have compact kernels. In particular, the Bergman spaces of these natural manifolds are infinite-dimensional.     

Function Spaces Associated with Schrödinger Operators: The Pöschl-Teller Potential

We address the function space theory associated with the Schrödinger operator H = ?d²/dx² + V. The discussion is featured with potential V (x) = ?n(n + 1) sech²x, which is called in quantum physics Pöschl-Teller potential. Using a dyadic system, we introduce Triebel-Lizorkin spaces and Besov spaces associated with H. We then use interpolation method to identify these spaces with the classical ones for a certain range of p, q > 1. A physical implication is that the corresponding wave function ψ(t, x) = e^?itHf(x) admits appropriate time decay in the Besov space scale.     

Intrinsic characterizations of Besov spaces on Lipschitz domains

The aim of this paper is to study the equivalence between quasi‐norms of Besov spaces on domains. We suppose that the domain Ω ? ?ⁿ is a bounded Lipschitz open subset in ?ⁿ. First, we define Besov spaces on Ω as the restrictions of the corresponding Besov spaces on ?ⁿ. Then, with the help of equivalent and intrinsic characterizations (the Peetre‐type characterization 3.10 and the characterization via local means 3.13) of these spaces, we get another equivalent and intrinsic quasi‐norm using, this time, generalized differences and moduli of smoothness. We extend the well‐known characterization of Besov spaces on ?ⁿ described in Theorem 2.4 to the case of Lipschitz domains.     

Function spaces on fractals

We construct function spaces, analogs of Hölder-Zygmund, Besov and Sobolev spaces, on a class of post-critically finite self-similar fractals in general, and the Sierpinski gasket in particular, based on the Laplacian and effective resistance metric of Kigami. This theory is unrelated to the usual embeddings of these fractals in Euclidean space, and so our spaces are distinct from the function spaces of Jonsson and Wallin, although there are some coincidences for small orders of smoothness. We show that the Laplacian acts as one would expect an elliptic pseudodifferential operator of order d+1 on a space of dimension d to act, where d is determined by the growth rate of the measure of metric balls. We establish some Sobolev embedding theorems and some results on complex interpolation on these spaces.     

Besov spaces and Bergman projections on the ballLes espaces de Besov et les projections de Bergman dans la boule

A class of radial differential operators are investigated yielding a natural classification of diagonal Besov spaces on the unit ball of $\text{C}{}^{\mathrm{N}}$. Precise conditions are given for the boundedness of Bergman projections from certain L^p spaces onto Besov spaces. Right inverses for these projections are also provided. Applications to complex interpolation are presented. To cite this article: H.T. Kaptano?lu, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 729–732.     

A fixed point theorem for the infinite-dimensional simplex

We define the infinite-dimensional simplex to be the closure of the convex hull of the standard basis vectors in R^∞, and prove that this space has the fixed point property: any continuous function from the space into itself has a fixed point. Our proof is constructive, in the sense that it can be used to find an approximate fixed point; the proof relies on elementary analysis and Sperner's lemma. The fixed point theorem is shown to imply Schauder's fixed point theorem on infinite-dimensional compact convex subsets of normed spaces.     

Nonlinear Neumann–Transmission Problems for Stokes and Brinkman Equations on Euclidean Lipschitz Domains

The purpose of this paper is to use a layer potential analysis and the Leray–Schauder degree theory to show an existence result for a nonlinear Neumann–transmission problem corresponding to the Stokes and Brinkman operators on Euclidean Lipschitz domains with boundary data in L ^p spaces, Sobolev spaces, and also in Besov spaces.     

Best m-Term Approximation and Sobolev–Besov Spaces of Dominating Mixed Smoothness—the Case of Compact Embeddings

We shall investigate the asymptotic behavior of the widths of best m-term approximation with respect to tensor products of Sobolev as well as Besov spaces in case of compact embeddings. Furthermore, we compare best m-term approximation with optimal linear approximation and entropy numbers.     

Imbedding theorems of Besov spaces in Banach lattices

The paper examines imbeddings of Besov spaces B _{E, θ} ^ω in ideal spaces (Banach lattices) given that ω ∈ S_k). In particular, the symmetric hull of the space B _{E, θ} ^ω is described (E is a symmetric space), an inequality of different metrics is obtained, and imbeddings in Orlicz and Lorentz spaces and in some weighted spaces are studied. Most of the results are easily extended to the anisotropic case. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 159, pp. 69–82, 1987.