F. J. Perez Lazaro
We prove embedding theorems for fully anisotropic Besov spaces. More concrete, inequalities between modulus of continuity in different metrics and of Sobolev type are obtained. Our goal is to get sharp estimates for some anisotropic cases previously unconsidered.
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The embedding relations between Besov-Triebel-Sobolev spaces and modulation spaces are determined explicitly. We extend the results of Sugimoto[2007]; Wang[2007] and Kobayashi[2011] to the most general cases. And we give the sharp embedding between Fourier $L^{p}$ spaces and modulation spaces.
We show that the zero smoothness Besov space $B_{p,q}^{0,1}$ does not embed into the Lorentz space $L_{p,q}$ unless $p=q$; here $p,q\in (1,\infty)$. This answers negatively a question proposed by O. V. Besov.
We study the continuity and compactness of embeddings for radial Besov and Triebel-Lizorkin spaces with weights in the Muckenhoupt class $A_\infty$. The main tool is a discretization in terms of an almost orthogonal wavelet expansion adapted to the radial situation.
We construct generalised shift-invariant systems of functions of several real variables for anisotropic Besov spaces that can be generated by the decomposition method using any given expansive matrix and establish the conditions on those systems under which they will constitute Banach frames or sets of atoms for the anisotropic homo- or inhomogeneous Besov spaces.
We study the embeddings of (homogeneous and inhomogeneous) anisotropic Besov spaces associated to an expansive matrix $A$ into Sobolev spaces, with focus on the influence of $A$ on the embedding behaviour. For a large range of parameters, we derive sharp characterizations of embeddings.
In this article we construct affine systems that provide a simultaneous atomic decomposition for a wide class of functional spaces including the Lebesgue spaces $L^p(\Rdst)$, $1<p<+\infty$. The novelty and difficulty of this construction is that we allow for non-lattice translations. We prove that for an arbitrary expansive matrix $A$ and any set $Λ$ - satisfying a certain spreadness condition but otherwise irregular- there exists a smooth window whose translations along the elements of $Λ$ and dilations by powers of $A$ provide an atomic decomposition for the whole range of the anisotropic Tr…
In this article, via certain lower bound conditions on the measures under consideration, the authors fully characterize the Sobolev embeddings for the scales of Hajłasz-Triebel-Lizorkin and Hajłasz-Besov spaces in the general context of quasi-metric measure spaces for an optimal range of the smoothness parameter $s$. An interesting facet of this work is how the range of $s$ for which the above characterizations of these embeddings hold true is intimately linked (in a quantitative manner) to the geometric makeup of the underlying space. Moreover, although stated for Hajłasz-Triebel-Lizorkin and…
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.
In this paper we prove the Jawerth-Franke embeddings of Herz-type Besov and Triebel-Lizorkin spaces. Moreover, we obtain the Jawerth-Franke embeddings of Besov and Triebel-Lizorkin spaces equipped with power weights. An application we present new embeddings between Besov and Herz spaces.
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。
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…
In this paper, we study limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces, $id_τ: B_{p_1,q_1}^{s_1,τ_1}(Ω) \rightarrow B_{p_2,q_2}^{s_2,τ_2}(Ω)$ and $id_τ: F_{p_1,q_1}^{s_1,τ_1}(Ω) \rightarrow F_{p_2,q_2}^{s_2,τ_2}(Ω)$, where $Ω\subset \mathbb{R}^d$ is a bounded domain, obtaining necessary and sufficient conditions for the continuity of $id_τ$. This can also be seen as the continuation of our previous studies of compactness of the embeddings in the non-limiting case. Moreover, we also construct Rychkov's linear, bounded universal extension operator for these spaces.
In this article, for an optimal range of the smoothness parameter $s$ that depends (quantitatively) on the geometric makeup of the underlying space, the authors identify purely measure theoretic conditions that fully characterize embedding and extension domains for the scale of Hajłasz--Triebel--Lizorkin spaces $M^s_{p,q}$ and Hajłasz--Besov spaces $N^s_{p,q}$ in general spaces of homogeneous type. Although stated in the context of quasi-metric spaces, these characterizations improve related work even in the metric setting. In particular, as a corollary of the main results in this article, the…
We study embeddings between generalised Besov-Morrey spaces. Both sufficient and necessary conditions for the embeddings are proved. Embeddings of the Besov-Morrey spaces into the Lebesgue spaces are also considered. Our approach requires a wavelet characterisation of the spaces which we establish for the system of Daubechies wavelets.
We give a new description of classical Besov spaces in terms of a new modulus of continuity. Then a similar approach is used to introduce Besov classes on an infinite-dimensional space endowed with a Gaussian measure.
In this paper we investigate the asymptotic behaviour of Weyl numbers of embeddings of tensor product Besov spaces into Lebesgue spaces. These results will be compared with the known behaviour of entropy numbers.
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 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.
In this paper we shall give two-sided sharp estimates of Kolmogorov numbers of embeddings of the Besov spaces with dominating mixed smoothness $S^t_{p,q}B((0,1)^d)$ into $ L_{\infty}((0,1)^d)$.
Rearrangement estimates and limiting embeddings for anisotropic Besov spaces
本文研究了各向异性 Besov 空间到 Lorentz 空间的嵌入问题，求得了部分指数趋于1时嵌入常数的精确渐近行为，并提出了基于部分连续模的重排估计。
A Classification Theorem on Non-compact Embeddings between Besov Spaces
本文对全空间和有界域上Besov空间之间的嵌入映射，依据严格奇异和有限严格奇异条件，给出了其弱紧性的完整分类，并细化了非紧性描述，扩展了Sobolev嵌入的近期结果。
In this article, we study the relation between Sobolev-type embeddings for Sobolev spaces or Besov spaces or Triebel-Lizorkin spaces defined either on a doubling or on a geodesic metric measure space and lower bound for measure of balls either in the whole space or in a domain inside the space.
We study embeddings of Besov-type and Triebel-Lizorkin-type spaces, $id_τ: {B}_{p_1,q_1}^{s_1,τ_1}(Ω) \hookrightarrow {B}_{p_2,q_2}^{s_2,τ_2}(Ω)$ and $id_τ: {F}_{p_1,q_1}^{s_1,τ_1}(Ω) \hookrightarrow {F}_{p_2,q_2}^{s_2,τ_2}(Ω) $, where $Ω\subset {\mathbb R}^d$ is a bounded domain, and obtain necessary and sufficient conditions for the compactness of $id_τ$. Moreover, we characterise its entropy and approximation numbers. Surprisingly, these results are completely obtained via embeddings and the application of the corresponding results for classical Besov and Triebel-Lizorkin spaces as well as …
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.
The aim of the paper is to establish (local) optimal embeddings of Besov spaces $B^{0,b}_{p,r}$ involving only a slowly varying smoothness $b$. In general, our target spaces are outside of the scale of Lorentz-Karamata spaces and are related to small Lebesgue spaces. In particular, we improve results from [CGO11b], where the targets are (local) Lorentz-Karamata spaces. To derive such results, we apply limiting real interpolation techniques and weighted Hardy-type inequalities.
The purpose of this article is twofold. The first is to strengthen fractional Sobolev type inequalities in Besov spaces via the classical Lorentz space. In doing so, we show that the Sobolev inequality in Besov spaces is equivalent to the fractional Hardy inequality and the iso-capacitary type inequality. Secondly, we will strengthen fractional Sobolev type inequalities in Besov spaces via capacitary Lorentz spaces associated with Besov capacities. For this purpose, we first study the embedding of the associated capacitary Lorentz space to the classical Lorentz space. Then, the embedding of th…
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 …
It is shown that a Banach space $E$ has type $p$ if and only for some (all) $d\ge 1$ the Besov space $B_{p,p}^{(\frac1p-\frac12)d}(\R^d;E)$ embeds into the space $\g(L^2(\R^d),E)$ of $\g$-radonifying operators $L^2(\R^d)\to E$. A similar result characterizing cotype $q$ is obtained. These results may be viewed as $E$-valued extensions of the classical Sobolev embedding theorems.
We discuss the growth envelopes of Fourier-analytically defined Besov and Triebel-Lizorkin spaces $B^s_{p,q}(\R^n)$ and $F^s_{p,q}(\R^n)$ for $s=σ_p=n\max(\frac 1p-1,0)$. These results may be also reformulated as optimal embeddings into the scale of Lorentz spaces $L_{p,q}(\R^n)$. We close several open problems outlined already by H. Triebel in [H. Triebel, The structure of functions, Birkhäuser, Basel, 2001.] and explicitly formulated by D. D. Haroske in [D. D. Haroske, Envelopes and sharp embeddings of function spaces, Chapman & Hall / CRC, Boca Raton, 2007.].
The purpose of the present paper is to investigate the decay of Bernstein numbers of the embedding from $B^t_{p_1,q}((0,1)^d)$ into the space $L_{p_2}((0,1)^d) $. The asymptotic behaviour of Bernstein numbers of the identity $id: S_{p_1,p_1}^tB((0,1)^d)\rightarrow L_{p_2}((0,1)^d)$ will be also considered.
There are two main aims of the paper. The first one is to extend the criterion for the precompactness of sets in Banach function spaces to the setting of quasi-Banach function spaces. The second one is to extend the criterion for the precompactness of sets in the Lebesgue spaces $L_p(\mathbb R^n)$, $1 \leq p < \infty$, to the so-called power quasi-Banach function spaces. These criteria are applied to establish compact embeddings of abstract Besov spaces into quasi-Banach function spaces. The results are illustrated on embeddings of Besov spaces $B^s_{p,q}(\mathbb R^n)$, $0<s<1$, $0<p,q\leq \in…
For an interpolation pair $(E_0,E_1)$ of Banach spaces with $E_1 \hookrightarrow E_0$ we use vectors $b_1,b_2,\ldots \in E_1$ that satisfy an extremal property with respect to the $J$- and $K$-functional to construct sub-spaces that are isometric to $\ell_q^{(θ)}$. The construction is based on a randomisation using independent Rademacher variables. We verify that systems obtained by re-scaling a function with a certain periodicity property share this extreme property. This implies the sharpness of natural embeddings of Hölder spaces obtained by the real interpolation into the corresponding Gau…
Mixed-norm Herz-type Besov 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。
Compact embeddings of Sobolev, Besov, and Triebel-Lizorkin spaces
本文在拟度量测度空间的一般框架下，建立了保证分数阶Sobolev、Besov与Triebel-Lizorkin空间嵌入紧性的充分必要条件，并针对依赖于底层空间几何结构的指数最优范围给出了刻画。
In this article, the authors introduce Besov-type spaces with variable smoothness and integrability. The authors then establish their characterizations, respectively, in terms of $φ$-transforms in the sense of Frazier and Jawerth, smooth atoms or Peetre maximal functions, as well as a Sobolev-type embedding. As an application of their atomic characterization, the authors obtain a trace theorem of these variable Besov-type spaces.
Compact embeddings of variable exponent Sobolev, Besov, and Triebel-Lizorkin spaces on metric measure spaces
研究可变指数Sobolev、Besov和Triebel-Lizorkin空间在有界和无界度量测度空间上的紧嵌入条件，证明了紧性条件的充分性，并探讨了等距群作用对嵌入紧性的影响。
Orlicz-Sobolev embeddings and heat kernel based Besov classes
针对具有不同局部和全局结构的度量测度空间，本文研究Besov空间与有界变差函数相关泛函不等式，并发展$L^1$理论以刻画有限周界集和等周不等式中的结构差异。
In this paper, we investigate the relation between Sobolev-type embeddings of Hajłasz-Besov spaces (and also Hajłasz-Triebel-Lizorkin spaces) defined on a metric measure space $(X,d,μ)$ and lower bound for the measure $μ.$ We prove that if the measure $μ$ satisfies $μ(B(x,r))\geq cr^Q$ for some $Q>0$ and for any ball $B(x,r)\subset X,$ then the Sobolev-type embeddings hold on balls for both these spaces. On the other hand, if the Sobolev-type embeddings hold in a domain $Ω\subset X,$ then we prove that the domain $Ω$ satisfies the so-called measure density condition, i.e., $μ(B(x,r)\capΩ)\geq …
In this paper, we introduce a class of homeomorphisms between metric spaces, which are locally biHölder continuous mappings. Then an embedding result between Besov spaces induced by locally biHölder continuous mappings between Ahlfors regular spaces is established, which extends the corresponding result of Björn-Björn-Gill-Shanmugalingam (J. Reine Angew. Math. 725: 63-114, 2017). Furthermore, an example is constructed to show that our embedding result is more general. We also introduce a geometric condition, named as uniform boundedness, to characterize when a quasisymmetric mapping between un…
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 consider the classical Besov and Triebel-Lizorkin spaces defined via differences and prove a homogeneity property for functions with bounded support in the frame of these spaces. As the proof is based on compact embeddings between the studied function spaces we present also some results on the entropy numbers of these embeddings. Moreover, we derive some applications in terms of pointwise multipliers.
Let $Γ$ be a fractal $h$-set and ${\mathbb{B}}^{σ}_{p,q}(Γ)$ be a trace space of Besov type defined on $Γ$. While we dealt in [9] with growth envelopes of such spaces mainly and investigated the existence of traces in detail in [12], we now study continuous embeddings between different spaces of that type on $Γ$. We obtain necessary and sufficient conditions for such an embedding to hold, and can prove in some cases complete characterisations. It also includes the situation when the target space is of type $L_r(Γ)$ and, as a by-product, under mild assumptions on the $h$-set $Γ$ we obtain the e…
Generalised Hajłasz-Besov spaces on $RD$-spaces
本文在具有反向加倍性质的$RD$-空间上引入了一类新的广义Hajłasz-Besov空间，并成功将经典理论中的嵌入与Sobolev型嵌入等结果推广至该空间。
We compare Besov spaces with isotropic smoothness with Besov spaces of dominating mixed smoothness. Necessary and sufficient conditions for continuous embeddings will be given.
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。
Real-Variable Characterizations and Their Applications of Anisotropic Besov Spaces with Matrix $\mathcal A_\infty$ Weights
本文建立带$\mathcal A_{p,\infty}$矩阵权的各向异性Besov空间理论，给出其$\varphi$变换与分子刻画，构造反例说明权条件的最优性，引入临界重标指数并得到相关算子的精确有界性。
In this paper we prove Nikolskii's inequality on general compact Lie groups and on compact homogeneous spaces with the constant interpreted in terms of the eigenvalue counting function of the Laplacian on the space, giving the best constant for certain indices, attained on the Dirichlet kernel. Consequently, we establish embedding theorems between Besov spaces on compact homogeneous spaces, as well as embeddings between Besov spaces and Wiener and Beurling spaces. We also analyse Triebel-Lizorkin spaces and $β$-versions of Wiener and Beurling spaces and their embeddings, and interpolation prop…
This paper is concerned with proving some embeddings of the form \begin{equation*} F_{p_{1},q}^{s_{1}}\cdot B_{p_{2},\infty }^{s_{2}}\cdot ...\cdot B_{p_{m},\infty }^{s_{m}}\hookrightarrow F_{p,q}^{s_{1}},\quad m\geq 2. \end{equation*} The different embeddings obtained here are under certain restrictions on the parameters. In particular, we improve some results of pointwise multiplication on Triebel-Lizorkin spaces. Franke-Jawerth embeddings, the $\pm $ - method of Gustaffson-Peetre and the relation between Hardy spaces and Triebel-Lizorkin spaces are the main tools.
Widths of embeddings of 2-microlocal Besov spaces
研究带权重的2-微局部Besov空间之间紧嵌入的逼近数、Gelfand数和Kolmogorov数的渐近行为，其中权重由到$d$-集$U\subset \mathbb{R}^n$的距离定义，并在大多数情况下给出了精确估计，包括拟Banach空间情形。
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。
We establish conditions on the parameters which are both necessary and sufficient in order that Besov and Triebel-Lizorkin spaces of generalized smoothness contain only regular distributions. We also connect this with the possibility of embedding such spaces in some particular Lebesgue spaces.
The present paper, is devoted to investigation of operator--valued Fourier multiplier theorems from $B_{q_{1},r}^{s}$ to $B_{q_{2},r}^{s}$, optimal embedding of Besov spaces, the separability and positivity of differential operators. Here, we show that these differential operators generate analytic semigroup.
Gelfand numbers represent a measure for the information complexity which is given by the number of information needed to approximate functions in a subset of a normed space with an error less than $\varepsilon$. More precisely, Gelfand numbers coincide up to the factor 2 with the minimal error $ e^{\rm wor}(n,Λ^{\rm all})$ which describes the error of the optimal (non-linear) algorithm that is based on $n$ arbitrary linear functionals. This explain the crucial role of Gelfand numbers in the study of approximation problems. Let $S^t_{p_1,p_1}B((0,1)^d)$ be the Besov spaces with dominating mixed…
The classical Jawerth and Franke embeddings $$ F^{s_0}_{p_0,q}({\mathbb R}^n)\hookrightarrow B^{s_1}_{p_1,p_0}({\mathbb R}^n) \quad \mbox{and} \quad B^{s_0}_{p_0,p_1}({\mathbb R}^n)\hookrightarrow F^{s_1}_{p_1,q}({\mathbb R}^n) $$ are versions of Sobolev embedding between the scales of Besov and Triebel-Lizorkin function spaces for $s_0>s_1$ and $$ s_0-\frac{n}{p_0} = s_1-\frac{n}{p_1}.$$ We prove Jawerth and Franke embeddings for the scales of Besov and Triebel-Lizorkin spaces with all exponents variable $$ F^{s_0(\cdot)}_{p_0(\cdot),q(\cdot)}\hookrightarrow B^{s_1(\cdot)}_{p_1(\cdot),p_0(\cd…
Classification of anisotropic local Hardy spaces and inhomogeneous Triebel-Lizorkin spaces
本文通过拟范数的粗等价性刻画了膨胀矩阵产生相同各向异性局部Hardy和非齐次Triebel-Lizorkin空间的条件，完成了相关空间的分类。
In this paper we give a thorough study of Lipschitz spaces. We obtain the following new results: (1) Sharp Jawerth-Franke-type embeddings between the Besov and Lipschitz spaces extending the classical results for Besov and Sobolev spaces; (2) Sharp embeddings between Lipschitz spaces with different parameters extending the Brézis-Wainger result; (3) Characterizations for Lipschitz spaces norms via Fourier transforms and wavelets; (4) Sharp embeddings from Lipschitz spaces into Lebesgue/Lorentz-Zygmund spaces.
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.
Embeddings of variable Sobolev, Besov, and Triebel-Lizorkin spaces on metric measure spaces
本文研究了在度量测度空间上可变指数Hajłasz型光滑空间的嵌入理论，提出新的可变指数Sobolev、Triebel-Lizorkin和Besov空间，并建立了到可变指数Lebesgue和Hölder空间的Sobolev、Morrey和Moser-Trudinger型嵌入。
Continuing previous work, this paper provides maximal characterizations of anisotropic Triebel-Lizorkin spaces $\dot{\mathbf{F}}^α_{p,q}$ for the endpoint case of $p = \infty$ and the full scale of parameters $α\in \mathbb{R}$ and $q \in (0,\infty]$. In particular, a Peetre-type characterization of the anisotropic Besov space $\dot{\mathbf{B}}^α_{\infty,\infty} = \dot{\mathbf{F}}^α_{\infty,\infty}$ is obtained. As a consequence, it is shown that there exist dual molecular frames and Riesz sequences in $\dot{\mathbf{F}}^α_{\infty,q}$.
We show that a domain is an extension domain for a Hajłasz-Besov or for a Hajłasz-Triebel-Lizorkin space if and only if it satisfies a measure density condition. We use a modification of the Whitney extension where integral averages are replaced by median values, which allows us to handle also the case $0<p<1$. The necessity of the measure density condition is derived from embedding theorems; in the case of Hajłasz-Besov spaces we apply an optimal Lorentz-type Sobolev embedding theorem which we prove using a new interpolation result. This interpolation theorem says that Hajłasz-Besov spaces ar…
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