Chokri Abdelkefi, Feriel Sassi
In this paper, different characterizations of the Besov-Dunkl spaces are given. We provide equivalence between these characterizations, using the Dunkl translation, the Dunkl transform and the Peetre K-functional.
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In the present paper, we define for the Dunkl tranlation operators on the real line, the Besov-Dunkl space of functions for which the remainder in the generalized Taylor's formula has a given order. We provide characterization of these spaces by the Dunkl convolution.
In the present paper, we propose to prove some properties and estimates of the integral remainder in the generalized Taylor formula associated to the Dunkl operator on the real line and to describe the Besov-Dunkl spaces for which the remainder has a given order.
In this paper, we show the inclusion and the density of the Schwartz space in Besov-Dunkl spaces and we prove an interpolation formula for these spaces by the real method. We give another characterization for these spaces by convolution. Finally, we establish further results concerning integrability of the Dunkl transform of function in a suitable Besov-Dunkl space.
Fractional Leibniz rules for the Dunkl Laplacian in Besov and Triebel--Lizorkin spaces
针对归一化根系和多重函数下的 Dunkl Laplacian 的分数幂，在 Besov 和 Triebel--Lizorkin 空间中建立了 Leibniz 型法则，通过谱乘子与 Dunkl 变换的相互作用以及 Dunkl 平移相关分布的支持性质，将 L^p 空间上的估计推广到更广泛的函数空间。
In this paper we study the Sobolev inequality in the Dunkl setting using two new approaches which provide a simpler elementary proof of the classical case $p=2$, as well as an extension to the coefficient $p=1$ that was previously unknown. We also find estimates of the sharp constants for the Sobolev inequality for Dunkl gradient. Related inequalities and some improvements are also considered (Nash inequality, Besov space embeddings).
On Limit Formulas for Besov Seminorms and Nonlocal Perimeters in the Dunkl Setting
本文在 Dunkl 框架下推广了 Besov 半范数极限公式与非局部周长极限公式，提出了无维数依赖的 Maz'ya--Shaposhnikova 公式，并弱化了边界正则性条件，结果在经典拉普拉斯情形下亦为新发现。
In this paper we study various forms of the Hardy inequality for Dunkl operators, including the classical inequality, $L^p$ inequalities, an improved Hardy inequality, as well as the Rellich inequality and a special case of the Caffarelli-Kohn-Nirenberg inequality. As a consequence, one-dimensional many-particle Hardy inequalities for generalised root systems are proved, which in the particular case of root systems $A_{N-1}$ improve some well-known results.
The paper is concerned with Besov spaces of variable smoothness $B^{φ_{0}}_{p,q}(\mathbb{R}^{n},\{t_{k}\})$, in which the norms are defined in terms of convolutions with smooth functions. A relation is found between the spaces $B^{φ_{0}}_{p,q}(\mathbb{R}^{n},\{t_{k}\})$ and the spaces $\widetilde{B}^{l}_{p,q,r}(\mathbb{R}^{n},\{t_{k}\})$, which were introduced earlier by the author.
Some results on the Dunkl-Williams constant
本文通过推导公式计算了实赋范线性空间中Dunkl-Williams常数DW(X)及与Birkhoff正交相关的常数DW_B(X)，并证明了ℓ₂-ℓ₁空间和十二面体单位球空间的DW(X)值分别为2√2和8(2−√3)。
Besov regularity in non-linear generalized functions
本文通过构造与经典Besov空间相关的新广义函数模与空间，将各类Schwartz分布空间自然嵌入其中，并给出检测分布Besov正则性的精确判据。
We investigate Besov spaces of self-affine tilings of ${\Bbb R}^{n}$ and discuss various characterizations of those Besov spaces. We see what is a finite set of functions which generates the Besov spaces from a view of multiresolution approximation on self-affine lattice tilings of ${\Bbb R}^{n}$. Using this result we give a generalization of already known characterizations of Besov spaces given by wavelet expansion and we apply to study the pointwise H${\ddot {\rm o}}$lder space. Furthermore we give descriptions of scaling exponents measured by Besov spaces, and estimations of a pointwise H${…
In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: Sharp embeddings between the Besov spaces defined by differences and by Fourier-analytical decompositions as well as between Besov and Sobolev/Triebel-Lizorkin spaces; Various new characterizations for Besov norms in terms of different K-functionals. For instance, we derive characterizations via ball averages, approximation methods, heat kernels, and Bianchini-type norms; Sharp estimates for Besov norms of derivatives and pote…
Coorbit space theory is an abstract approach to function spaces and their atomic decompositions. The original theory developed by Feichtinger and Gr{ö}chenig in the late 1980ies heavily uses integrable representations of locally compact groups. Their theory covers, in particular, homogeneous Besov-Lizorkin-Triebel spaces, modulation spaces, Bergman spaces, and the recent shearlet spaces. However, inhomogeneous Besov-Lizorkin-Triebel spaces cannot be covered by their group theoretical approach. Later it was recognized by Fornasier and the first named author that one may replace coherent states …
On Besov and Triebel-Lizorkin spaces associated with the Grushin operator
本文针对Grushin算子（含或不含漂移）定义经典与非经典Besov和Triebel-Lizorkin空间，建立其多种刻画，并研究复插值、嵌入性质及分数阶Leibniz法则。
We establish convolution inequalities for Besov spaces $B_{p,q}^s$ and Triebel--Lizorkin spaces $F_{p,q}^s$. As an application, we study the mapping properties of convolution semigroups, considered as operators on the function spaces $A_{p,q}^s$, $A \in \{B,F\}$. Our results apply to a wide class of convolution semigroups including the Gauß--Weierstraß semigroup, stable semigroups and heat kernels for higher-order powers of the Laplacian $(-Δ)^m$, and we can derive various caloric smoothing estimates.
We study Schauder basis properties for the Haar system in Besov spaces $B^s_{p,q}(\mathbb{R}^d)$. We give a complete description of the limiting cases, obtaining various positive results for $q\leq \min\{1,p\}$, and providing new counterexamples in other situations. The study is based on suitable estimates of the dyadic averaging operators $\mathbb{E}_N$; in particular we find asymptotically optimal growth rates for the norms of these operators in global and local situations.
Besov spaces in operator theory
本综述探讨了Besov类在算子理论中的多种应用，包括描述Schatten-von Neumann类的Hankel算子、Hilbert空间上幂有界算子多项式的范数估计，以及在算子扰动理论中的重要作用。
In this paper we study various key properties for $2$-microlocal Besov and Triebel--Lizorkin spaces with all exponents variable, including the lifting property, embeddings and Fourier multipliers. We also clarify and improve some statements recently published.
Besov and Triebel-Lizorkin spaces on homogeneous groups
该研究在任意齐次群上建立了全参数范围的Besov空间与Triebel-Lizorkin空间理论，证明其与Littlewood-Paley分解选取无关并给出多种刻画，涵盖多种经典函数空间特例。
In the last decade, methods based on various kinds of spherical wavelet bases have found applications in virtually all areas where analysis of spherical data is required, including cosmology, weather prediction, and geodesy. In particular, the so-called needlets (=band-limited Parseval frames) have become an important tool for the analysis of Cosmic Microwave Background (CMB) temperature data. The goal of the present paper is to construct band-limited and highly localized Parseval frames on general compact homogeneous manifolds. Our construction can be considered as an analogue of the well-kno…
Wavelet bases and frames consisting of band limited functions of nearly exponential localization on Rd are a powerful tool in harmonic analysis by making various spaces of functions and distributions more accessible for study and utilization, and providing sparse representation of natural function spaces (e.g. Besov spaces) on Rd. Such frames are also available on the sphere and in more general homogeneous spaces, on the interval and ball. The purpose of this article is to develop band limited well-localized frames in the general setting of Dirichlet spaces with doubling measure and a local sc…
Let $B^σ_{2, \infty}$ denote the Besov space defined on a compact set $K \subset {\Bbb R}^d$ which is equipped with an $α$-regular measure $μ$. The {\it critical exponent} $σ^*$ is the supremum of the $σ$ such that $B^σ_{2, \infty} \cap C(K)$ is dense in $C(K)$. It is well-known that for many standard self-similar sets $K$, $B^{σ^*}_{2, \infty}$ are the domain of some local regular Dirichlet forms. In this paper, we explore new situations that the underlying fractal sets admit inhomogeneous resistance scalings, which yield two types of critical exponents. We will restrict our consideration on …
This paper gives a survey of methods for the construction of space-frequency concentrated frames on Riemannian manifolds with bounded curvature, and the applications of these frames to the analysis of function spaces. In this general context, the notion of frequency is defined using the spectrum of a distinguished differential operator on the manifold, typically the Laplace-Beltrami operator. Our exposition starts with the case of the real line, which serves as motivation and blueprint for the material in the subsequent sections. After the discussion of the real line, our presentation starts o…
We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space $\dot W^{1,p}$. The resulting spaces are identified as a special class of real interpolation spaces of Sobolev-Slobodecki\uı spaces. We establish the equivalence between Fourier analytic definitions and definitions via difference operators acting on measurable functions. We prove various new results on embeddings and non-embeddings, and give applications to harmonic and caloric extensions. For suitable wavelet bases we obtain a characterization of the approximation …
Matrix-Weighted Besov-Type and Triebel--Lizorkin-Type Spaces II: Sharp Boundedness of Almost Diagonal Operators
本文在矩阵加权Besov型与Triebel--Lizorkin型序列空间上建立了几乎对角算子的锐界有界性，覆盖全部指数范围，改进了已有结果。
In this paper we propose a general coorbit space theory suitable to define coorbits of quasi-Banach spaces using an abstract continuous frame, indexed by a locally compact Hausdorff space, and an associated generalized voice transform. The proposed theory realizes a further step in the development of a universal abstract theory towards various function spaces and their atomic decompositions which has been initiated by Feichtinger and Gr{ö}chenig in the late 1980ies. We combine the recent approaches in Rauhut, Ullrich and Rauhut to identify, in particular, various inhomogeneous (quasi-Banach) s…
Variable Matrix-Weighted Besov Spaces
本文引入了可变矩阵加权贝索夫空间，并通过一系列算子和分解特征研究了其性质。
This article focuses on Chemin's space $\mathcal{S}'_h$ of homogeneous distributions, which was introduced to serve as a basis for realizations of subcritical homogeneous Besov spaces. We will discuss how this construction fails in multiple ways for supercritical spaces. In particular, we study its intersection $X_h := \mathcal{S}'_h \cap X$ with various Banach spaces $X$, namely supercritical homogeneous Besov spaces and the Lebesgue space $L^\infty$. For each $X$, we find out if the intersection $X_h$ is dense in $X$. If it is not, then we study its closure $C = {\rm clos}(X_h)$ and prove th…
Given a metric measure space $X$, we consider a scale of function spaces $T^{p,q}_s(X)$, called the weighted tent space scale. This is an extension of the tent space scale of Coifman, Meyer, and Stein. Under various geometric assumptions on $X$ we identify some associated interpolation spaces, in particular certain real interpolation spaces. These are identified with a new scale of function spaces, which we call $Z$-spaces, that have recently appeared in the work of Barton and Mayboroda on elliptic boundary value problems with boundary data in Besov spaces. We also prove Hardy--Littlewood--Sob…
Change of variable in spaces of mixed smoothness and numerical integration of multivariate functions on the unit cube
本文研究了在混合光滑性空间中多变量函数的最优数值积分问题，提出两种改进的Frolov求积公式，通过变量替换和点乘操作，保持了最小最坏误差率，并扩展了经典结果。
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