Wavelet decomposition and embeddings of generalised Besov–Morrey spaces (original) (raw)

Embeddings of Besov–Morrey spaces on bounded domains

Studia Mathematica, 2013

We study embeddings of spaces of Besov-Morrey type, idΩ : N s 1 p 1 ,u 1 ,q 1 (Ω) → N s 2 p 2 ,u 2 ,q 2 (Ω), where Ω ⊂ R d is a bounded domain, and obtain necessary and sufficient conditions for the continuity and compactness of idΩ. This continues our earlier studies relating to the case of R d. Moreover, we also characterise embeddings into the scale of Lp spaces or into the space of bounded continuous functions. 1. Introduction. In recent years smoothness spaces related to Morrey spaces, in particular Besov-Morrey and Triebel-Lizorkin-Morrey spaces, attracted some attention. The classical Morrey spaces M p,u (R d), 0 < u ≤ p < ∞, were introduced by Ch. B. Morrey [Mo] and are part of the wider class of Morrey-Campanato spaces (cf. [Pe]). They can be considered as a complement to L p spaces, since M p,p (R d) = L p (R d). However, on the one hand the Morrey spaces with u < p consist of locally u-integrable functions, but on the other hand the spaces scale with d/p instead of d/u, that is, f (λ •) | M p,u (R d) = λ −d/p f | M p,u (R d) , λ > 0. This property is very useful for some partial differential equations. Built upon this basic family M p,u (R d), different spaces of Besov-Triebel-Lizorkin type were defined in the last years. H. Kozono and M. Yamazaki [KY] and A. Mazzucato [Ma] introduced the Besov-Morrey N s p,u,q spaces and used them in the theory of Navier-Stokes equations. As before, if u = p, then these spaces coincide with the classical ones, i.e., N s p,p,q (R d) = B s p,q (R d). Some of their properties including wavelet characterisations were proved by Y. Sawano [S1, S4, S3], Y. Sawano and H. Tanaka [ST2, ST1] and L. Tang and J. Xu [TX]. The most systematic and general approach to spaces of this type can be found in the recent book [YSY] of W. Yuan, W. Sickel and D. Yang or in the very recent survey papers by W. Sickel [Si1, Si2]. We recommend the monograph and the survey for further up-to-date references 2010 Mathematics Subject Classification: Primary 46E35.

Wavelet bases in generalized Besov spaces

Journal of Mathematical Analysis and Applications, 2005

In this paper we obtain a wavelet representation in (inhomogeneous) Besov spaces of generalized smoothness via interpolation techniques. As consequence, we show that compactly supported wavelets of Daubechies type provide an unconditional Schauder basis in these spaces when the integrability parameters are finite.  2004 Elsevier Inc. All rights reserved.

Wavelet Decompositions of Anisotropic Besov Spaces

Mathematische Nachrichten, 2002

In this paper we develop the natural multiresolution analysis framework related to anisotropic Besov spaces B α p,q (ℝ n ). We prove two new Jackson and Bernstein type inequalities for these spaces, and obtain from well-known techniques [12, 7] new norm equivalences in terms ...

Wavelet characterizations for anisotropic Besov spaces: case

2001

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 pspaces with 0 < p < ∞ are derived.

Wavelet Characterizations for Anisotropic Besov Spaces

Applied and Computational Harmonic Analysis, 2002

ABSTRACT We provide wavelet characterizations for anisotropic Besov spaces B p;q (R n ) allowing the summability index p to be &lt; 1. We use a well-known technique (see [2] and [5]) which is based upon the two fundamental inequalities of Bernstein and of Jackson. Math Subject Classication: 42C15 Keywords: anisotropic Besov spaces, wavelet decompositions, Bernstein and Jackson inequalities Corresponding author: Anita Tabacco Dipartimento di Matematica Politecnico di Torino Corso Duca degli Abruzzi, 24 10129 Torino, Italy E-mail: tabacco@polito.it Research supported by the European Commission, within the TMR Network Harmonic Analysis 1998-2001&quot;. 1 2 1

A.Mayeli, Homogeneous Besov spaces on stratified Lie groups and their wavelet characterization

2012

We establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, both in terms of continuous and discrete wavelet systems. We first introduce a notion of homogeneous Besov spaceḂ s p,q in terms of a Littlewood-Paley-type decomposition, in analogy to the well-known characterization of the Euclidean case. Such decompositions can be defined via the spectral measure of a suitably chosen sub-Laplacian. We prove that the scale of Besov spaces is independent of the precise choice of Littlewood-Paley decomposition. In particular, different sub-Laplacians yield the same Besov spaces. We then turn to wavelet characterizations, first via continuous wavelet transforms (which can be viewed as continuous-scale Littlewood-Paley decompositions), then via discretely indexed systems. We prove the existence of wavelet frames and associated atomic decomposition formulas for all homogeneous Besov spacesḂ s p,q , with 1 ≤ p, q < ∞ and s ∈ R.

The Generalized Wavelet Transform on Sobolev Type Spaces

Advances in Mathematics: Scientific Journal, 2020

The Generalized Wavelet transform is studied on the Sobolev type spaceB ω k (R n) Boundedness results in this Sobolev space is obtained. Compactly supported wavelets on distribution space are also studied. Approximation properties of the generalized wavelet transform will also be discussed. R n

Embedding theorems for Besov–Morrey spaces of many groups of variables

Georgian Mathematical Journal, 2017

In this paper we introduce a new function space B p , θ , a , ϰ , τ 〈 l 〉 ⁢ ( s , G ) {B_{p,\theta,a,\varkappa,\tau}^{\langle l\rangle}(s,G)} with parameters of many groups of variables of Besov–Morrey type. In view of the embedding theorems we study some properties of the functions which belong to these spaces.

Characterization of Local Besov Spaces via Wavelet Basis Expansions

Frontiers in Applied Mathematics and Statistics, 2017

In this paper we deal with local Besov spaces of periodic functions of one variable. We characterize these spaces in terms of summability conditions on the coefficients in series expansions of their elements with respect to an orthogonal Schauder basis of trigonometric polynomials. We consider a Schauder basis that was constructed by using ideas of a periodic multiresolution analysis and corresponding wavelet spaces. As an interim result we obtain a characterization of local Besov spaces via operators of the orthogonal projection on the corresponding scaling and wavelet spaces. In order to achieve our new results, we substantially use a theorem on the discretization of scaling and wavelet spaces as well as a connection between local and usual classical Besov spaces. The corresponding characterizations are also given for the classical Besov spaces.

Characterization of weighted function spaces in terms of wavelet transforms

Boletim da Sociedade Paranaense de Matemática, 2018

In this paper, we have characterized a weighted function space $ B_{\omega,\psi}^{p,q}, ~ 1\leq p,q<\infty$ in terms of wavelet transform and shown that the norms on the spaces Bomega,psip,qB_{\omega,\psi}^{p,q}Bomega,psip,q and bigwedgeomegap,q\bigwedge_\omega^{p,q}bigwedgeomegap,q (the space defined in terms of differences trianglex\triangle_xtrianglex) are equivalent.