Ajay K. Sharma, Sei-Ichiro Ueki
The purpose of this paper is to describe the characterization for the compact difference of two composition operators acting between analytic Besov spaces and the weighted little Bloch type space over the unit disk.
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Let $S(\mathbb{D})$ be the collection of all holomorphic self-maps on $\mathbb{D}$ of the complex plane $\mathbb{C}$, and $C_φ$ the composition operator induced by $φ\in S(\mathbb{D})$. We obtain that there are no hypercyclic composition operators on the little Bloch space $\mathcal{B}_0$ and the Besov space $B_p$.
In this paper, the power boundedness and mean ergodicity of multiplication operators are investigated on the Bloch space B, the little Bloch space B0 and the Besov Space Bp. We completely characterize power bounded, mean ergodic and uniformly mean ergodic multiplication operators on B and B0.
Let $g$ be a holomorphic map of $B$, where $B$ is the unit ball of ${C}^n$. Let $0<p<+\infty, -n-1<q<+\infty$, $q>-1$ and $α>0$. This paper gives some necessary and sufficient conditions for the Extended Ces$\acute{a}$ro Operators induced by $g$ to be bounded or compact between generalized Besov space $B(p,q)$ and $α$- Bloch space ${\mathcal B}^α.$
We consider a class of two-parameter weighted integral operators induced by harmonic Bergman-Besov kernels on the unit ball of $\mathbb{R}^{n}$ and characterize precisely those that are bounded from Lebesgue spaces $L^{p}_α$ into Harmonic Bergman-Besov $b^{q}_β$ or weighted Bloch Spaces $b^{\infty}_β $, for $1\leq p\leq\infty$, $1\leq q< \infty$ and $α,β\in \mathbb{R}$. These operators can be viewed as generalizations of the harmonic Bergman-Besov projections. Also, our results remove the disturbing conditions $β>-1$ when $q<\infty$ and $β\geq 0$ when $q=\infty$ of Doğan (A Class of Integral O…
New 2-microlocal Besov and Triebel-Lizorkin spaces via the Littlewood-Paley decomposition
该研究借助 Littlewood-Paley 分解定义新型 2-微局部 Besov 与 Triebel-Lizorkin 空间，并建立多种分解及振荡、差分刻画，同时证明相关奇异积分与伪微分算子的有界性。
We apply spectral theoretic methods to obtain a Littlewood-Paley decomposition of abstract inhomogeneous Besov spaces in terms of "smooth" and "bandlimited" functions. Well-known decompositions in several contexts are as special examples and are unified under the spectral theoretic approach
In this paper we consider the generalized little Hankel and Berezin type operators on Besov spaces of holomorphic functions on polydiscs and give results about the boundedness of these operators.
We use Littlewood-Paley-Stein type g-functions (also called generalized square functions) associated to symmetric diffusion semigroups to obtain a characterization of inhomogeneous abstract Besov spaces on the abstract Hilbert spaces. Then we apply our results for the abstract Besov spaces defined through the Poisson and Gauss-Weierstrass semigroups.
In this paper we investigate Besov spaces on graded Lie groups. We prove a Nikolskii type inequality (or the Reverse Hölder inequality) on graded Lie groups and as consequence we obtain embeddings of Besov spaces. We prove a version of the Littlewood-Paley theorem on graded Lie groups. The results are applied to obtain embedding properties of Besov spaces and multiplier theorems for both spectral and Fourier multipliers in Besov spaces on graded Lie groups. In particular, we give a number of sufficient conditions for the boundedness of Fourier multipliers in Besov spaces.
A Cesàro-like operator from Besov spaces to some spaces of analytic functions
该研究针对p>1、s>1的情形，完整刻画了类Cesàro算子从Besov空间B_p到介于平均Lipschitz空间与Bloch空间之间的Banach空间的有界性与紧性。
We determine both the semigroup and spectral properties of a group of weighted composition operators on the Little Bloch space. It turns out that these are strongly continuous groups of invertible isometries on the Bloch space. We then obtain the norm and spectra of the infinitesimal generator as well as the resulting resolvents which are given as integral operators. As consequences, we complete the analysis of the adjoint composition group on the predual on the nonreflexive Bergman space, and a group of isometries associated with a specific automorphism of the upper half plane.
We obtain exhaustive results and treat in a unified way the question of boundedness, compactness, and weak compactness of composition operators from the Bloch space into any space from a large family of conformally invariant spaces that includes the classical spaces like $BMOA$, $Q_α$, and analytic Besov spaces $B^p$. In particular, by combining techniques from both complex and functional analysis, we prove that in this setting weak compactness is equivalent to compactness. For the operators into the corresponding "small" spaces we also characterize the boundedness and show that it is equivale…
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…
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 space $\dot{B}_{p,q}^s$ 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 th…
We introduce a general definition of homogeneous Besov spaces on a stratified Lie group $G$, based on a Littlewood-Paley-type decomposition of Schwartz functions with all moments vanishing. We show that under mild and intuitive conditions the spaces thus defined are independent of the decomposition employed. A corollary of this is that previously constructed versions of homogeneous Besov spaces on $G$, relying on the spectral calculus of a sub-Laplacian of the group, are consistent, i.e., independent of the choice of sub-Laplacian. We further prove characterizations of homogeneous Besov spaces…
Let $B_p(s)$ be an analytic Besov type space. Let $M(B_p(s))$ be the class of multipliers of $B_p(s)$ and let $F(p, p-2, s)$ be the Möbius invariant subspace generated by $B_p(s)$. In this paper, when $0<s<1$ and $\max\{s, 1-s\}<p\leq 1$, we give a completed description of interpolating sequences for $M(B_p(s))$ and $F(p, p-2, s)\cap H^\infty$. We also consider certain condition appeared in this description by an $L^p$ characterization and the closure of $F(p, p-2, s)$ in the Bloch space.
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…
We present some Caffarelli-Kohn-Nirenberg-type inequalities on Herz-type Besov-Triebel-Lizorkin spaces, Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces. More Precisely, we investigate the inequalities \begin{equation*} \big\|f\big\|_{\dot{k}_{v,σ}^{α_{1},r}}\leq c\big\|f\big\|_{\dot{K}_{u}^{α_{2},δ}}^{1-θ}\big\|f\big\|_{\dot{K}_{p}^{α_{3},δ_{1}}A_{β}^{s}}^{θ}, \end{equation*} and \begin{equation*} \big\|f\big\|_{\mathcal{E}_{p,2,u}^{σ}}\leq c\big\|f\big\|_{\mathcal{M}_{μ}^{δ}}^{1-θ}\big\|f\big\|_{\mathcal{N}_{q,β,v}^{s}}^{θ}, \end{equation*} with some appropriate assumptions on the para…
In this note, we establish characterizations for the homogeneous Besov-type spaces $\dot{B}^{s,τ}_{p,q}(\mathbb{R}^n)$ and Triebel-Lizorkin-type spaces $\dot{F}^{s,τ}_{p,q}(\mathbb{R}^n)$, introduced by Yang and Yuan, through fractional Hajłasz-type gradients for suitable values of the parameters $p$, $q$ and $τ$ when $0 < s < 1$, and through grand Littlewood-Paley-type maximal functions for all admissible values of the parameters. These characterizations extend the characterizations obtained by Koskela, Yang and Zhou for the standard homogeneous Besov and Triebel-Lizorkin spaces.
Peetre conjecture on real interpolation spaces of Besov spaces and Grid K functional
本研究利用小波系数与网格结构定义的长方体顶点分类等新技术，彻底解决Besov空间实插值空间的Peetre猜想，弥补了Lions-Peetre迭代定理处理强非线性的不足，给出K泛函的具体小波表达式。
In this paper we introduce new function spaces which we call anisotropic hyperbolic Besov and Triebel-Lizorkin spaces. Their definition is based on a hyperbolic Littlewood-Paley analysis involving an anisotropy vector only occurring in the smoothness weights. Such spaces provide a general and natural setting in order to understand what kind of anisotropic smoothness can be described using hyperbolic wavelets (in the literature also sometimes called tensor-product wavelets), a wavelet class which hitherto has been mainly used to characterize spaces of dominating mixed smoothness. A centerpiece …
In this paper, we completely characterize nuclear Volterra composition operators $T^ϕ_g : \mathcal H^\infty_ν\longrightarrow \mathcal H^\infty_μ$ and $S^ϕ_g : \mathcal H^\infty_ν\longrightarrow \mathcal H^\infty_μ$ acting between weighted type spaces in terms of the symbols $g$ and $ϕ$ of $T^ϕ_g$ and $S^ϕ_g$ and weights $ν$ and $μ$, when the weights $ν$ and $μ$ are normal weights in the sense of Shields and Williams. Moreover, nuclear Volterra composition operators acting between little weighted type spaces and Bloch spaces of order $β$ are also characterized.
We examine the surjectivity of isometries between weighted spaces of holomorphic functions. We show that for certain classical weights on the open unit disc all isometries of the weighted space of holomorphic functions, ${ \mathcal H}_{v_o}( Δ)$, are surjective. Criteria for surjectivity of isometries of ${ \mathcal H}_v(U)$ in terms of a separation condition on points in the image of ${ \mathcal H}_{v_o}(U)$ are also given for $U$ a bounded open set in $\mathbb{C}$. Considering the weight $v(z)= 1-|z|^2$ and the isomorphism $f\mapsto f'$ we are able to show that all isometries of the little B…
Sharp embeddings between quasi-Banach Besov spaces and shallow ReLU variation spaces
本文在拟Banach范围内建立了Besov空间与浅层ReLU^k变分空间之间的尖锐嵌入，确定了最优光滑性阈值，并应用于PDE解的正则性到逼近误差和贪婪算法收敛性的转化。
The inclusions between the Besov spaces $B^q$, the Bloch space $\mathcal{B}$ and the standard weighted Bergman spaces $A^p_α$ are completely understood, but the norms of the corresponding inclusion operators are in general unknown. In this work, we compute or estimate asymptotically the norms of these inclusions.
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…
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。
The properties of Volterra-composition operators on the weighted Bergman space with exponential type weights are investigated in this paper. We state some necessary and sufficient conditions that a Volterra-composition operator from the weighted Bergman space to Bloch type space ( or little Bloch type space ) must satisfy for the Volterra-composition operator to be bounded or compact.
Shift invariant subspaces of large index in the Bloch space
本文研究Bloch空间与小Bloch空间上移位算子的不变子空间格，构造大指标闭不变子空间，并利用二次对偶中弱星闭包的指标稳定性证明任意指标弱星闭不变子空间的存在性。
In this paper, the authors propose a new framework under which a theory of generalized Besov-type and Triebel-Lizorkin-type function spaces is developed. Many function spaces appearing in harmonic analysis fall under the scope of this new framework. Among others, the boundedness of the Hardy-Littlewood maximal operator or the related vector-valued maximal function on any of these function spaces is not required to construct these generalized scales of smoothness spaces. Instead of this, a key idea used in this framework is an application of the Peetre maximal function. This idea originates fro…
Hörmander type Fourier multiplier theorem and Nikolskii inequality on quantum tori, and applications
本文在量子环上研究Hörmander型傅里叶乘子定理和Nikolskii不等式，并推导了相关空间的嵌入定理和插值性质。
Let $U^{n}$ be the unit polydisc of ${\Bbb C}^{n}$ and $ϕ=(ϕ_1, >..., ϕ_n)$ a holomorphic self-map of $U^{n}.$ By ${\cal B}^p(U^{n})$, ${\cal B}^p_{0}(U^{n})$ and ${\cal B}^p_{0*}(U^{n})$ denote the $p$-Bloch space, Little $p$-Bloch space and Little star $p$-Bloch space in the unit polydisc $U^n$ respectively, where $p, q>0$. This paper gives the estimates of the essential norms of bounded composition operators $C_ϕ$ induced by $ϕ$ between ${\cal B}^p(U^n)$ (${\cal B}^p_{0}(U^n)$ or ${\cal B}^p_{0*}(U^n)$) and ${\cal B}^q(U^n)$ (${\cal B}^q_{0}(U^n)$ or ${\cal B}^q_{0*}(U^n)$). As their applic…
In this paper we prove structural and topological characterizations of the screened Sobolev spaces with screening functions bounded below and above by positive constants. We generalize a method of interpolation to the case of seminormed spaces. This method, which we call the truncated method, generates the screened Sobolev subfamily and a more general screened Besov scale. We then prove that the screened Besov spaces are equivalent to the sum of a Lebesgue space and a homogeneous Sobolev space and provide a Littlewood-Paley frequency space characterization.
We prove Paley-Littlewood decompositions for the scales of fractional powers of $0$-sectorial operators $A$ on a Banach space which correspond to Triebel-Lizorkin spaces and the scale of Besov spaces if $A$ is the classical Laplace operator on $L^p(\mathbb{R}^n).$We use the $H^\infty$-calculus, spectral multiplier theorems and generalized square functions on Banach spaces and apply our results to Laplace-type operators on manifolds and graphs, Schrödinger operators and Hermite expansion.We also give variants of these results for bisectorial operators and for generators of groups with a bounded…
Let p,q>0. We extend to the n-polydisk previous one-variable characterization results of K. Madigan on the $p$-Lipschitz space and K. Madigan/A. Matheson on the Bloch space by obtaining function-theoretic conditions on a holomorphic self-map of the polydisk such that the induced composition operator is bounded or compact between p- and q-Bloch spaces of the polydisk. These conditions turn out to be different in the cases when p is in (0,1) and when p is at least 1. We also obtain corresponding characterization results for composition operators between generalized little p- and q-Bloch spaces o…
Matrix-Weighted Besov-Triebel-Lizorkin Spaces of Optimal Scale: Real-Variable Characterizations, Invariance on Integrable Index, and Sobolev-Type Embedding
本文利用增长函数引入广义矩阵加权Besov-Triebel-Lizorkin型空间，通过多种函数刻画该空间并建立几乎对角算子有界性以获得分子和小波刻画，证明了该框架的最优性，并给出关于可积指数不变性和Sobolev型嵌入的充要条件。
This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative $d$-torus $\mathbb{T}^d_θ$ (with $θ$ a skew symmetric real $d\times d$-matrix). These spaces share many properties with their classical counterparts. We prove, among other basic properties, the lifting theorem for all these spaces and a Poincaré type inequality for Sobolev spaces. We also show that the Sobolev space $W^k_\infty(\mathbb{T}^d_θ)$ coincides with the Lipschitz space of order $k$, already studied by Weaver in the case $k=1$. We establish the embedding inequalities of all these spac…
Regularity of Weighted Tensorized Fractional Brownian Fields and associated function spaces
本文研究新型自相似高斯场 WTFBS 的局部正则性，并提出加权张量化 Besov 空间，借助 Littlewood-Paley 理论与双曲小波分析刻画其函数空间结构。
In this paper we describe the surjective linear isometries on a vector valued little Bloch space with range space a strictly convex and smooth complex Banach space. We also describe the hermitian operators and the generalized bi-circular projections supported by these spaces.
We consider the space $\bk^1_{\log^α}$, of analytic functions on the unit disk $\D,$ defined by the requirement $\int_\D|f'(z)|ϕ(|z|)\,dA(z)<\infty,$ where $ϕ(r)=\log^α(1/(1-r))$ and show that it is a predual of the "$\log^α$-Bloch" space and the dual of the corresponding little Bloch space. We prove that a function $f(z)=\sum_{n=0}^\infty a_nz^n$ with $a_n\downarrow 0$ is in $\bk^1_{\log^α}$ iff $\sum_{n=0}^\infty \log^α(n+2)/(n+1)<\infty$ and apply this to obtain a criterion for membership of the Libera transform of a function with positive coefficients in $\bk^1_{\log^α}.$ Some properties o…
Composition Semigroups on the Besov Spaces
研究贝索夫空间\(B_p\)上复合算子半群，对比其与经典空间特性差异，给出\(p \geq 2\)及\(1 < p < 2\)时不同情况，给出反例，在算子范数有界假设下刻画使\([\varphi_t, B_p] = B_p\)的半群\(\{\varphi_t\}\)。
Multiplicative operators on analytic function spaces
本文研究了解析函数空间上算子的乘法性质，证明复合算子在特定空间中是几乎乘法的，并完善了Schwartz的结论。
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。
We identify the predual of the nonreflexive Bergman space of the upper half plane, $L_a^1(\uP,μ_{\al})$, with the little Bloch space of the upper half plane consisting of functions vanishing at $i$. We then investigate both the semigroup and spectral properties of the adjoint groups of composition operators which are naturally obtained from the duality pairing and are therefore defined on the identified predual.
Pointwise lower bounds in growth spaces with little o conditions
本文研究了开单位圆上解析函数模之和的逐点下界问题，针对具有 little o 条件的函数，补充了 Ramey-Ullrich 和 Abakumov-Doubtsov 在 Bloch 空间等空间中的已有结果。
For the classical space of functions with bounded mean oscillation, it is well known that VMO** = BMO and there are many characterizations of the distance from a function f in BMO to VMO. When considering the Bloch space, results in the same vein are available with respect to the little Bloch space. In this paper such duality results and distance formulas are obtained by pure functional analysis. Applications include general Möbius invariant spaces such as Q_K-spaces, weighted spaces, Lipschitz-Hölder spaces and rectangular BMO of several variables.
The pointwise multiscale texture operator: analytical foundations and functional characterization
该研究从高斯尺度空间视角提出多尺度纹理算子，建立其函数分析性质，将经典贝索夫与索伯列夫正则性等价表征为多尺度纹理持续性，搭建了相关理论间的严谨桥梁。
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。
Generalized tri-circular projections on some vector-valued spaces of analytic functions
本文刻画了开单位圆盘上三个解析函数向量值空间（哈代空间、有界解析函数空间及小布洛赫空间）上的广义三圆投影，给出了其完整分类。
For analytic functions $g$ on the unit disc with non-negative Maclaurin coefficients, we describe the boundedness and compactness of the integral operator $T_g(f)(z)=\int_0^zf(ζ)g'(ζ)\,dζ$ from a space $X$ of analytic functions in the unit disc to $H^\infty$, in terms of neat and useful conditions on the Maclaurin coefficients of $g$. The choices of $X$ that will be considered contain the Hardy and the Hardy-Littlewood spaces, the Dirichlet-type spaces $D^p_{p-1}$, as well as the classical Bloch and BMOA spaces.
Metrically bounded Volterra-type operators on area Nevanlinna spaces
本文针对面积Nevanlinna空间上的度量有界Volterra型算子的符号类精确刻画问题，通过建立该空间的Littlewood-Paley型刻画，得出$J_g$对应$g$属于Bloch空间、$I_g$对应$g$属于$H^\infty$的结论，解决了该领域的公开问题。
Contractive Hardy--Littlewood inequalities in the Dirichlet range
本文研究 Dirichlet 范围（0<α<1）内的解析函数空间 A_α^p，证明压缩不等式 ||f||_{β,q} ≤ ||f||_{α,p} 在 α/p=β/q 时成立，并确定等号条件，推广了经典 Bergman 空间的结果。
Square function characterization of Hardy-type spaces
研究在特定假设下将哈代空间的刻画推广到哈代型空间\(X_A\)，并得出复合算子\(C_\varphi: \mathcal{B}^\omega \to X_A\)有界性和紧性的精确准则，推广了一维相关结果。
Reverse Carleson measures for spaces of analytic functions
本文定义了拟巴拿赫解析函数空间的$q$-逆Carleson测度，完全刻画了Hardy空间、BMOA空间、Bloch空间等的$q$-逆Carleson测度，并给出了全纯Triebel-Lizorkin空间和Besov空间的相关结果。
We consider operators T : M_0 -> Z and T : M -> Z, where Z is a Banach space and (M_0, M) is a pair of Banach spaces belonging to a general construction in which M is defined by a "big-O" condition and M_0 is given by the corresponding "little-o" condition. Prototype examples of such spaces M are given by $\ell^\infty$, weighted spaces of functions or their derivatives, bounded mean oscillation, Lipschitz-Hölder spaces, and many others. The main result characterizes the weakly compact operators T in terms of a certain norm naturally attached to M, weaker than the M-norm, and shows that weakly …
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