Pekka Koskela, Zhuang Wang
In this paper, we study function spaces defined via dyadic energies on the boundaries of regular trees. We show that correct choices of dyadic energies result in Besov-type spaces that are trace spaces of (weighted) first order Sobolev spaces.
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We define distribution spaces of a sequence of convolutions of a set of distributions with smooth functions, the shearlet system. Then, we define associated sequence spaces and prove characterizations. We also show a reproducing identity in the class of distributions. Finally, we prove Sobolev-type embeddings within the shear anisotropic inhomogeneous spaces and embeddings between (classical dyadic) isotropic inhomogeneous spaces and shear anisotropic inhomogeneous spaces.
Motivated by a variety of representations of fractional powers of operators, we develop the theory of abstract Besov spaces $B^{ s, A }_{ q, X }$ for non-negative operators $A$ on Banach spaces $X$ with a full range of indices $s \in \mathbb{R}$ and $0 < q \leq \infty$. The approach we use is the dyadic decomposition of resolvents for non-negative operators, an analogue of the Littlewood-Paley decomposition in the construction of the classical Besov spaces. In particular, by using the reproducing formulas for fractional powers of operators and explicit quasi-norms estimates for Besov spaces we…
Haar frame characterizations of Besov-Sobolev spaces and optimal embeddings into their dyadic counterparts
本文研究了非无条件基参数下Besov和Triebel-Lizorkin空间的Haar系数行为，利用双重过采样Haar框架刻画了这些空间，并证明了经典空间与其二进对应空间之间的闭子集关系及端点情形下的等价范数。
Besov spaces and product BMO over polydisc
该研究证明了解析Besov空间$B_{p,p}^{1/p}(\mathbb{C}_+^n)$连续嵌入Chang–Fefferman乘积BMO，进而得出有限Schatten类小Hankel算子具有乘积BMO符号的结论，还给出Fejér均值的相关条件。
In this article, the authors establish the wavelet characterization of Besov and Triebel--Lizorkin spaces on a given space $(X,d,μ)$ of homogeneous type in the sense of Coifman and Weiss. Moreover, the authors introduce almost diagonal operators on Besov and Triebel--Lizorkin sequence spaces on $X$, and obtain their boundedness. Using this wavelet characterization and this boundedness of almost diagonal operators, the authors obtain the molecular characterization of Besov and Triebel--Lizorkin spaces. Applying this molecular characterization, the authors further establish the Littlewood--Paley…
In this article, the authors introduce the spaces of Lipschitz type on spaces of homogeneous type in the sense of Coifman and Weiss, and discuss their relations with Besov and Triebel-Lizorkin spaces. As an application, the authors establish the difference characterization of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type. A major novelty of this article is that all results presented in this article get rid of the dependence on the reverse doubling assumption of the considered measure of the underlying space ${\mathcal X}$ via using the geometrical property of ${\mathcal X}$ e…
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.
The objective of the paper is to describe Besov spaces on general compact Riemannian manifolds in terms of the best approximation by eigenfunctions of elliptic differential operators.
This article describes how the ideas promoted by the fundamental papers published by M. Frazier and B. Jawerth in the eighties have influenced subsequent developments related to the theory of atomic decompositions and Banach frames for function spaces such as the modulation spaces and Besov-Triebel-Lizorkin spaces. Both of these classes of spaces arise as special cases of two different, general constructions of function spaces: coorbit spaces and decomposition spaces. Coorbit spaces are defined by imposing certain decay conditions on the so-called voice transform of the function/distribution u…
In this paper, we extend the concept of continuous Bessel wavelet transform in $L^p$-space and derived the Parseval's as well as the inversion formulas. By using Bessel wavelet coefficients we characterized the Besov- Hankel space.
Mixed-norm Herz-type Besov-Triebel-Lizorkin spaces
基于混合范数Herz空间，引入混合范数Herz型Besov和Triebel-Lizorkin空间，建立其φ-变换刻画并证明Sobolev、Franke和Jawerth嵌入定理，推广了已知结果。
In this paper we obtain the non-asymptotic norm estimations of Besov's type between the norms of a functions in different Bilateral Grand Lebesgue spaces (BGLS). We also give some examples to show the sharpness of these inequalities.
We prove that the $\cal P$ norm estimate between a Hardy martingale and its cosine part are stable under dyadic perturbations, and show how dyadic stability of the $\cal P$ norm estimate is used in the proof that $L^1$ embeds into $L^1/H^1$.
We determine upper asymptotic estimates of Kolmogorov and linear $n$-widths of unit balls in Sobolev and Besov norms in $L_{p}$-spaces on smooth compact Riemannian manifolds. For compact homogeneous manifolds, we establish estimates which are asymptotically exact, for the natural ranges of indices. The proofs heavily rely on our previous results such as: estimates for the near-diagonal localization of the kernels of elliptic operators, Plancherel-Polya inequalities on manifolds of bounded geometry, cubature formulas with positive coefficients and uniform estimates on Clebsch-Gordon coefficient…
Besov-Lipschitz norm and $p$-energy measure on scale-irregular Vicsek sets
该研究针对可能缺乏自相似性的尺度不规则Vicsek集，证明了其上$p$-能量范数与对应$p$-能量测度的存在性，并刻画了$p$-能量范数与Besov-Lipschitz范数的关联及相关收敛性质。
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…
We construct a new bounded functional calculus for the generators of bounded semigroups on Hilbert spaces and generators of bounded holomorphic semigroups on Banach spaces. The calculus is a natural (and strict) extension of the classical Hille-Phillips functional calculus, and it is compatible with the other well-known functional calculi. It satisfies the standard properties of functional calculi, provides a unified and direct approach to a number of norm-estimates in the literature, and allows improvements of some of them.
Equivalent norms and $φ$-transform of matrix-weighted anisotropic Besov-type and Triebel-Lizorkin-type spaces
本文引入与扩张矩阵A相关的矩阵加权各向异性Besov型和Triebel-Lizorkin型空间，利用矩阵权重性质证明其与对应平均空间等价、建立离散φ变换，还研究了p=∞时对应Triebel-Lizorkin空间的φ变换及相关空间关系。
The dyadic paraproduct is bounded in weighted Lebesgue spaces $L_p(w)$ if and only if the weight $w$ belongs to the Muckenhoupt class $A_p^d$. However, the sharp bounds on the norm of the dyadic paraproduct are not known even in the simplest $L_2(w)$ case. In this paper we prove the linear bound on the norm of the dyadic paraproduct in the weighted Lebesgue space $L_2(w)$ using Bellman function techniques and extrapolate this result to the $L_p(w)$ case.
We obtain sharp estimates for the quasi norm of the maximal function of f when it satisfies certain conditions.
In this article, the authors introduce Besov and Triebel-Lizorkin spaces on spaces of homogeneous type in the sense of Coifman and Weiss, prove that these (in)homogeneous Besov and Triebel-Lizorkin spaces are independent of the choices of both exp-ATIs (or exp-IATIs) and underlying spaces of distributions, and give some basic properties of these spaces. As applications, the authors show that some known function spaces coincide with certain special cases of Besov and Triebel-Lizorkin spaces and, moreover, obtain the boundedness of Calderón-Zygmund operators on these Besov and Triebel-Lizorkin s…
In this paper we investigate some basic results on the slice regular Besov spaces of hyperholomorphic functions on the unit ball $\mathbb{B}.$ We also characterize the boundedness, compactness and find the essential norm estimates of composition operators between these spaces.
We obtain sharp estimates for the localized distribution function of Mϕ, when ϕbelongs to Lp,\inf where M is the dyadic maximal operator. We obtain these estimates given the L1 and Lq norm, q < p and certain weak Lp-conditions.
We introduce atoms for dyadic atomic $H^1$ for which the equivalence between atomic and maximal function defnitions is dimension independent. We give the sharp, up to $\log(d)$ factor, estimates for the $H^1 \to L^1$ norm estimates for the special maximal function.
We investigate the isometric composition operators on the analytic Besov spaces. For $1<p<2$ we show that an isometric composition operator is induced only by a rotation of the disk. For $p>2$, we extend previous work on the subject. Finally, we analyze this same problem for the Besov spaces with an equivalent norm.
We prove that the sequence spaces $\ell_p\oplus\ell_q$ and the spaces of infinite matrices $\ell_p(\ell_q)$, $\ell_q(\ell_p)$ and $(\bigoplus_{n=1}^\infty \ell_p^n)_{\ell_q}$, which are isomorphic to certain Besov spaces, have an almost greedy basis whenever $0<p<1<q<\infty$. More precisely, we custom-build almost greedy bases in such a way that the Lebesgue parameters grow in a prescribed manner. Our arguments critically depend on the extension of the Dilworth-Kalton-Kutzarova method from [S. J. Dilworth, N. J. Kalton, and D. Kutzarova, On the existence of almost greedy bases in Banach spaces…
In this paper we will focus on understanding the relation between Sobolev embedding theorems for Hajłasz-Besov spaces defined on a doubling metric measure space $(Ω,d,μ)$ and the non-collapsing condition of the measure, i.e. \[ \inf_{x\inΩ}μ(B(x,1))>0. \] We will also obtain embedding results for Hajłasz-Besov spaces whose modulus of smoothness is generated by a rearrangement invariant quasi-norm.
In this paper, we show that Besov and Triebel-Lizorkin functions can be approximated by a Hölder continuous function both in the Lusin sense and in norm. The results are proven in metric measure spaces for Hajłasz-Besov and Hajłasz-Triebel-Lizorkin functions defined by a pointwise inequality. We also prove new inequalities for medians, including a Poincaré type inequality, which we use in the proof of the main result.
A Dyadic Approach to Weak Characterizations of Function Spaces
本文利用二进立方体上的平均振荡或均值定义弱型拟范数，将其作为上半空间相应拟范数的离散变体，通过与已知函数空间比较，获得了新嵌入与刻画，并举例证明了结果的精确性。
In this paper, we study the traces of Orlicz-Sobolev spaces on a regular rooted tree. After giving a dyadic decomposition of the boundary of the regular tree, we present a characterization on the trace spaces of those first order Orlicz-Sobolev spaces whose Young function is of the form $t^p\log^λ(e+t)$, based on integral averages on dyadic elements of the dyadic decomposition.
In this paper, we study the traces and the extensions for weighted Sobolev spaces on upper half spaces when the weights reach to the borderline cases. We first give a full characterization of the existence of trace spaces for these weighted Sobolev spaces, and then study the trace parts and the extension parts between the weighted Sobolev spaces and a new kind of Besov-type spaces (on hyperplanes) which are defined by using integral averages over selected layers of dyadic cubes.
Besov and Triebel--Lizorkin Spaces on Filtered Lie Groups, II: Pointwise Multiplication, Localization, Differences
本文在实连通李群上发展加权次强制算子相关的Besov与Triebel-Lizorkin空间理论，研究其代数性质、逐点乘子、局部化范数及差分刻画。
We consider BMO spaces of operator-valued functions, among them the space of operator-valued functions $B$ which define a bounded paraproduct on $L^2(H)$. We obtain several equivalent formulations of $\|π_B\|$ in terms of the norm of the "sweep" function of $B$ or of averages of the norms of martingales transforms of $B$ in related spaces. Furthermore, we investigate a connection between John-Nirenberg type inequalities and Carleson-type inequalities via a product formula for paraproducts and deduce sharp dimensional estimates for John-Nirenberg type inequalities.
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.
Using partial derivatives $\partial_zf$ and $\partial_{\ol z}f$, we introduce Besov spaces of polyanalytic functions on the unit disk and on the upper half-plane. We then prove that the dilatations of each function in polyanalytic Besov spaces converge to the same function in norm. This opens the way for the norm approximation of functions in polyanalytic Besov spaces by polyanalytic polynomials.
We give a sufficient (and, in the case of a compact domain, a necessary) condition for the embedding of Sobolev space of functions with integrable gradient into Besov-Orlicz spaces to be bounded. The condition has a form of a simple integral inequality involving Young and weight functions. We provide an example with Matuszewska-Orlicz indices of involved Orlicz norm equal to one. The main tool is the molecular decomposition of functions from a BV space.
We consider a dilation operator on Besov spaces $(B^s_{r,t}(K))$ over local fields and estimate an operator norm on such a field for $s > σ_r = \text{max}\big(\frac{1}{r} -1,~0\big)$ which depends on the constant $k$ unlike the case of Euclidean spaces. In $\mathbb{R}^n$, it is independent of constant. A constant $k$ appears for liming case $s=0$ and $s=σ_r$. In case of local fields, the limig case is still open. Further we also estimate the localization property of Besov spaces over local fields.
In this paper the authors obtain a new equivalent norms of the Besov spaces of variable smoothness and integrability. Our main tools are the continuous version of Calderon reproducing formula, maximal inequalities and variable exponent technique, but allowing the parameters to vary from point to point will raise extra difficulties which, in general, are overcome by imposing regularity assumptions on these exponents.
Grand variable Herz-Morrey type Besove spaces and Triebel-Lizorkin spaces
研究向量值次线性算子在大变量Herz-Morrey空间上的有界性，并定义相应Besov与Triebel-Lizorkin空间，证明基于Peetre极大算子的等价拟范数。
Characterization of non-linear Besov spaces
本文利用原子分解与度量嵌入证明非线性 Besov 空间中两类经典范数推广等价，并建立其与非线性 $p$-变差空间的相互嵌入，且无需 UMD 或可分性假设。
Analytic Besov functional calculus for several commuting operators
本文研究n个变量的解析Besov函数作用于Banach空间上n个交换$C_0$-半群生成元的条件，采用新方法处理$n=1$情形并扩展至$n\ge2$，同时阐明谱映射性质并提供算子范数估计。
We show that, for $0<s<1$, $0<p<\infty$, $0<q<\infty$, Hajłasz-Besov and Hajłasz-Triebel-Lizorkin functions can be approximated in the norm by discrete median convolutions. This allows us to show that, for these functions, the limit of medians, \[ \lim_{r\to 0}m_u^γ(B(x,r))=u^*(x), \] exists quasieverywhere and defines a quasicontinuous representative of $u$. The above limit exists quasieverywhere also for Hajłasz functions $u\in M^{s,p}$, $0<s\le 1$, $0<p<\infty$, but approximation of $u$ in $M^{s,p}$ by discrete (median) convolutions is not in general possible.
An RD-space $\mathcal X$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in $\mathcal X$. In this paper, the authors first give several equivalent characterizations of RD-spaces and show that the definitions of spaces of test functions on $\mathcal X$ are independent of the choice of the regularity $ε\in (0,1)$; as a result of this, the Besov and Triebel-Lizorkin spaces on $\mathcal X$ are also independent of the choice of the underlying distribution space. Then the authors characterize the norms of inhomogene…
We study (homogeneous and inhomogeneous) anisotropic Besov spaces associated to expansive dilation matrices $A \in {\rm GL}(d,\mathbb{R})$, with the goal of clarifying when two such matrices induce the same scale of Besov spaces. For this purpose, we first establish that anisotropic Besov spaces have an alternative description as decomposition spaces. This result allows to relate properties of function spaces to combinatorial properties of the underlying coverings. This principle is applied to the question of classifying dilation matrices. It turns out the scales of homogeneous and inhomogeneo…
Although numerous studies have focused on normal Besov spaces, limited studies have been conducted on exponentially weighted Besov spaces. Therefore, we define exponentially weighted Besov space $VB_{p,q}^{δ,w}(\mathbb{R}^d)$ whose smoothness includes normal Besov spaces, Besov spaces with dominating mixed smoothness, and their interpolation. Furthermore, we obtain wavelet characterization of $VB_{p,q}^{δ,w}(\mathbb{R}^d)$. Next, approximation formulas such as sparse grids are derived using the determined formula. The results of this study are expected to provide considerable insight into the …
Some Intrinsic Characterizations of Besov-Triebel-Lizorkin-Morrey-type Spaces on Lipschitz Domains
本文在特殊 Lipschitz 域上建立了 Besov-Triebel-Lizorkin 型空间和 Besov-Morrey 空间的 Littlewood-Paley 型刻画，并证明其范数可通过导数等价表示。
Within the new concept of a local iterated function system (local IFS), we consider a class of attractors of such IFSs, namely those that are graphs of functions. These new functions are called local fractal functions and they extend and generalize those that are currently found in the fractal literature. For a class of local fractal functions, we derive explicit conditions for them to be elements of Besov and Triebel--Lizorkin spaces. These two scales of functions spaces play an important role in interpolation theory and for certain ranges of their defining parameters describe many classical …
Starting from a Whitney decomposition of a symmetric cone $Ω$, analog to the dyadic partition $[2^j, 2^{j+1})$ of the positive real line, in this paper we develop an adapted Littlewood-Paley theory for functions with spectrum in $Ω$. In particular, we define a natural class of Besov spaces of such functions, $B^{p,q}_ν$, where the role of usual derivation is now played by the generalized wave operator of the cone $Δ(\frac{\partial}{\partial x})$. Our main result shows that $B^{p,q}_ν$ consists precisely of the distributional boundary values of holomorphic functions in the Bergman space $A^{p,q…
In this paper we show how to compute the $Λ_α$ norm, $α\ge 0$, using the dyadic grid. This result is a consequence of the description of the Hardy spaces $H^p(R^N)$ in terms of dyadic and special atoms.
We show that if the Hilbert transform with values in a Banach space is $L^p$ bounded, then so is the dyadic Hilbert transform, with a linear relation of the norms.
Operator-valued Fourier multipliers of bounded s-variation
本文研究了具有有限s-变差的算子值傅里叶乘子在加权空间中的有界性，通过加权向量值变分Carleson不等式和Littlewood-Paley-Rubio de Francia类型的估计，建立了相关定理。
In this paper we find the orders of decay for Kolmogorov widths of some Besov classes related to $W^1_1$ (the behaviour of the widths for $W^1_1$ remains unknown): $$ d_n(B^1_{1,θ}[0,1],L_q[0,1])\asymp n^{-1/2}\log^{\max(\frac12,1-\frac{1}θ)}n,\quad 2<q<\infty. $$ The proof relies on the lower bound for widths of product of octahedra in a special norm (maximum of two weighted $\ell_q$ norms). This bound generalizes the theorem of B.S.~Kashin on widths of octahedra in $\ell_q^N$.
Characterizations of Sobolev functions via Besov-type energy functionals in fractals
本文在度量测度空间（含Vicsek集等分形）中建立了 Sobolev 函数的刻画，给出了 Korevaar--Schoen--Sobolev 空间的等价范数，并证明了在适当条件下 $p$-能量形式的定义域可等同于 Besov 型函数空间。
Approximation of partial differential operators on Nikolskii-Besov functions classes of mixed smoothness in domains of a certain kind
研究在特定区域上具有混合光滑性的Nikolskii - Besov函数类上偏微分算子的逼近问题，通过用“$L_p$平均”混合连续性模定义范数，给出最佳逼近精度量的上下估计，拓宽了相关空间类。
A Direct Approach to Rough Paths in Time-Weighted Besov Spaces
本文扩展Besov粗糙路径框架至时间加权空间，建立加权缝补引理并构造积分，证明Itô-Lyons映射的Lipschitz连续性，实现位置敏感的误差控制，并以奇异系数扩散为例。
The article examines Nikolskii and Besov spaces with norms defined using "$L_p$-averaged" mixed moduli of continuity of functions of appropriate orders, instead of mixed moduli of continuity of known orders for certain mixed derivative functions. The author builds continuous linear mappings of such spaces of functions defined on cube $ I^d, $ to ordinary Nikolskii and Besov spaces of mixed smoothness in $ \mathbb R^d, $ that are function extension operators, thus incurring coincidence of both kinds of spaces on the cube $ I^d. $
We give several conditions for pregaussianity of norm balls of Besov spaces defined over $\mathbb{R}^d$ by exploiting results in Haroske and Triebel (2005). Furthermore, complementing sufficient conditions in Nickl and Pötscher (2005), we give necessary conditions on the parameters of the Besov space to obtain the Donsker property of such balls. For certain parameter combinations Besov balls are shown to be pregaussian but not Donsker.
We consider the dilation property of the modulation spaces $M^{p,q}$. Let $D_λ:f(t)\mapsto f(λt)$ be the dilation operator, and we consider the behavior of the operator norm $\|D_λ\|_{M^{p,q}\to M^{p,q}}$ with respect to $λ$. Our result determines the best order for it, and as an application, we establish the optimality of the inclusion relation between the modulation spaces and Besov space, which was proved by Toft.
In this paper we develop a theory of Besov and Triebel--Lizorkin spaces on general noncompact Lie groups endowed with a sub-Riemannian structure. Such spaces are defined by means of hypoelliptic sub-Laplacians with drift, and endowed with a measure whose density with respect to a right Haar measure is a continuous positive character of the group. We prove several equivalent characterizations of their norms, we establish comparison results also involving Sobolev spaces of recent introduction, and investigate their complex interpolation and algebra properties.
Approximation in weighted holomorphic Besov spaces in C^n
本文研究多圆柱和单位球中全纯函数的加权 Bergman 与 Besov 空间，寻求非径向权函数的 Mergelyan 型条件以保证函数伸缩按范数收敛且解析多项式在空间中稠密。
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