Entropy numbers of Sobolev embeddings of radial Besov spaces

Let be the radial subspace of the Besov space . We prove the independence of the asymptotic behavior of the entropy numbers

from the difference s₀−s₁ as long as the embedding itself is compact. In fact, we shall show that

This is in a certain contrast to earlier results on entropy numbers in the context of Besov spaces B_p,q^s(Ω) on bounded domains Ω.     

Approximation and entropy numbers in Besov spaces of generalized smoothness

We determine the exact asymptotic behaviour of entropy and approximation numbers of the limiting restriction operator , defined by J(f)=f|_Ω. Here Ω is a non-empty bounded domain in , ψ is an increasing slowly varying function, , and is the Besov space of generalized smoothness given by the function t^sψ(t). Our results improve and extend those established by Leopold [Embeddings and entropy numbers in Besov spaces of generalized smoothness, in: Function Spaces, Lecture Notes in Pure and Applied Mathematics, vol. 213, Marcel Dekker, New York, 2000, pp. 323–336].     

Wavelet Frames,Sobolev Embeddings and Negative Spectrum of Schrödinger Operators on Manifolds with Bounded Geometry

We consider weighted function spaces of Sobolev-Besov type and Schrödinger type operators on noncompact Riemannian manifolds with bounded geometry. First we give characterization of the spaces in terms of wavelet frames. Then we describe the necessary and sufficient conditions for the compactness of Sobolev embeddings between the spaces. An asymptotic behavior of the corresponding entropy numbers is calculated. At the end we use the asymptotic behavior to estimate the number of negative eigenvalues of the Schrödinger type operators.     

Entropy Numbers of Trudinger-Strichartz Embeddings of Radial Besov Spaces and Applications

The asymptotic behaviour of entropy numbers of Trudinger–Strichartzembeddings of radial Besov spaces on Rⁿ into exponential Orliczspaces is calculated. Estimates of the entropy numbers as wellas estimates of entropy numbers of Sobolev embeddings of radialBesov spaces are applied to spectral theory of certain pseudo-differentialoperators.     

Embeddings of anisotropic Besov spaces into Sobolev spaces

We study the embeddings of (homogeneous and inhomogeneous) anisotropic Besov spaces associated to an expansive matrix A into Sobolev spaces, with a focus on the influence of A on the embedding behavior. For a large range of parameters, we derive sharp characterizations of embeddings.     

Continuous embeddings of Besov-Morrey function spaces

We study embeddings of spaces of Besov-Morrey type, M Bp1,q1s1,r1(Rd ) → M Bp2 ,q2s2 ,r2 (R d ), and obtain necessary and sufficient conditions for this. Moreover, we can also characterise the special weighted situation Bp1 ,r1s1 (R d , w) → M Bp2 ,q2s2 ,r2 (Rd ) for a Muckenhoupt A ∞ weight w, with wα(x) = |x|α , α -d1, as a typical example.     

Lagrange Interpolation in Weighted Besov Spaces

The behavior of the Lagrange polynomial L _m (w,f) , based on the zeros of the orthogonal polynomials, is studied in some weighted Besov spaces B ^p _r,q (u) . It is proved that L _m (w) is a uniformly bounded map under suitable conditions on the weight functions and the parameters p , r , and q . December 11, 1996. Date revised: October 29, 1997. Date accepted: June 15, 1998.     

Interpolating Sequences for Besov Spaces

We characterise the interpolating sequences for the Besov spaces B_p and for their multiplier spaces. We also construct linear operators of interpolation.     

Besov空间上的H?rmander乘子定理

本文在紧Ｌｉｅ群上建立了一个Ｂｅｓｏｖ空间上的Ｈ?ｒｍａｎｄｅｒ乘子定理．     

Weighted Composition Operators Between Weighted Bloch Type Spaces and Weighted Banach Spaces of Holomorphic Functions

Let φ: 𝔻 → 𝔻 and ψ: 𝔻 → ? be analytic maps. They induce a weighted composition operator ψ C _φ acting between weighted Bloch type spaces and weighted Banach spaces of holomorphic functions. Under some assumptions on the weights, we give a necessary as well as a sufficient condition when such an operator is bounded resp. compact.     

Entropy numbers of Besov classes of generalized smoothness on the sphere

We investigate the asymptotic behavior of the entropy numbers of Besov classes BBΩp,θ(Sd 1)of generalized smoothness on the sphere inL q(Sd 1)for 1≤p,q,θ≤∞,and get their asymptotic orders.We also obtain the exact orders of entropy numbers of Sobolev classesBWr p(Sd 1)inL q(Sd 1)whenpand/orqis equal to 1 or∞.This provides the last piece as far as exact orders of entropy numbers ofBWr p(Sd 1)inL q(Sd 1)are concerned.     

Complex Interpolation of Weighted Besov and Lizorkin–Triebel Spaces

We study complex interpolation of weighted Besov and Lizorkin–Triebel spaces.The used weights w0,w1 are local Muckenhoupt weights in the sense of Rychkov.As a first step we calculate the Calder′on products of associated sequence spaces.Finally,as a corollary of these investigations,we obtain results on complex interpolation of radial subspaces of Besov and Lizorkin–Triebel spaces on Rd.     

Embeddings of weighted Morrey spaces

We consider embedding theorems within the scale of weighted Morrey spaces , where the weight w belongs to the Muckenhoupt class and . This includes, in particular, the classical setting of weighted Lebesgue spaces. We study some typical examples for the weight like , but also deal with quite general assumptions.     

d-集上的Besov空间的一个迹定理

In the paper we give a trace theorem of Besov spaces Bαp,p(Rn) on a d-set.It is a kind of extension of related results of Jonsson and Wallin,which has important applications on PDE theory.     

Bilipschitz Embeddings of Metric Spaces into Space Forms

The paper describes some basic geometric tools to construct bilipschitz embeddings of metric spaces into (finite-dimensional) Euclidean or hyperbolic spaces. One of the main results implies the following: If X is a geodesic metric space with convex distance function and the property that geodesic segments can be extended to rays, then X admits a bilipschitz embedding into some Euclidean space iff X has the doubling property, and X admits a bilipschitz embedding into some hyperbolic space iff X is Gromov hyperbolic and doubling up to some scale. In either case the image of the embedding is shown to be a Lipschitz retract in the target space, provided X is complete.     

Littlewood-Paley Functions and Triebel-Lizorkin Spaces,Besov Spaces

We establish Littlewood-Paley characterizations of Triebel-Lizorkin spaces and Besov spaces in Euclidean spaces using several square functions defined via the spherical average, the ball average, the Bochner-Riesz means and some other well-known operators. We provide a simple proof so that we are able to extend and improve many results published in recent papers.     

双线性Fourier乘子在Triebel-Lizorkin和Besov空间的有界性

本文利用Littlewood-Paley分解,Fourier变换和逆变换等方法,研究了双线性Fourier乘子在非齐次正光滑性Triebel-Lizorkin空间和Besov空间的有界性.     

Sharp Estimates in Bergman and Besov Spaces on Bounded Symmetric Domains

Sharp estimates of the point-evaluation functional in weighted Bergman spaces L ^p _a (Ω, dν _α) and for the point-evaluation derivalive functional in Besov spaces B ^p (Ω) are obtained for bounded symmetric domains Ω in ℂ ⁿ . Received October 25, 1999, Accepted December 6, 2000     

Some Applications of Besov Spaces on Fractals

Let F be a compact d-set in R^n with 0 〈 d ≤ n, which includes various kinds of fractals. The author establishes an embedding theorem for the Besov spaces Bpq^s（F） of Triebel and the Sobolev spaces W^1,P（F,d,μ） of Hajtasz when s 〉 1, 1 〈 p 〈∞ and 0 〈 q ≤ ∞. The author also gives some applications of the estimates of the entropy numbers in the estimates of the eigenvalues of some fractal pseudodifferential operators in the spaces Bpq^0（F） and Fpq^0（F）.     

Beyond Besov Spaces,Part 2: Oscillation Spaces

We study several extensions of Besov spaces; these extensions include the oscillation spaces O_p^s,swhich take into account correlations between the positions of large wavelet coefficients through the scales and, more generally, spaces defined through the distributions of suprema of wavelet coefficients on wavelet subtrees. These spaces are independent of the particular wavelet basis chosen. Examples of applications will be taken from image processing and multifractal analysis.