The main goal of this paper is to give a complete proof of the trace theorem for Besov-type spaces of generalized smoothness associated with complete Bernstein functions satisfying certain scaling conditions on $d$-sets $D\subset\mathbb R^n$, $d\leq n$. The proof closely follows the classical approach by Jonsson and Wallin and the trace theorem for classical Besov spaces. Here, the trace space is defined by means of differences. When $d=n$, as an application of the trace theorem, we give a condition under which the test functions $C_c^\infty(D)$ are dense in the trace space on $D$.
Under certain restrictions we describe the set of all pointwise multipliers in case of Sobolev and Besov spaces of dominating mixed smoothness. In addition we shall give necessary and sufficient conditions for the case that these spaces form algebras with respect to pointwise multiplication.
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)$.
Kolmogorov widths of an intersection of Besov classes with dominating mixed smoothness in a Besov space
Faculty of Mechanics and Mathematics, Lomonosov Moscow State University（莫斯科国立大学力学与数学系）；Moscow Center for Fundamental and Applied Mathematics（莫斯科基础与应用数学中心）
The aim of this paper is to study properties of Besov-type spaces with variable smoothness. We show that these spaces are characterized by the phi-transforms in appropriate sequence spaces and we obtain atomic decompositions for these spaces.
We continue our investigations on pointwise multipliers for Besov spaces of dominating mixed smoothness. This time we study the algebra property of the classes $S^r_{p,q}B(\mathbb{R}^d)$ with respect to pointwise multiplication. In addition if $p\leq q$, we are able to describe the space of all pointwise multipliers for $S^r_{p,q}B(\mathbb{R}^d)$.
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.
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.
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.
We compare Besov spaces with isotropic smoothness with Besov spaces of dominating mixed smoothness. Necessary and sufficient conditions for continuous embeddings will be given.
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.
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…
The paper puts forward new Besov spaces of variable smoothness $B^{φ_{0}}_{p,q}(G,\{t_{k}\})$ and $\widetilde{B}^{l}_{p,q,r}(Ω,\{t_{k}\})$ on rough domains. A~domain~$G$ is either a~bounded Lipschitz domain in~$\mathbb{R}^{n}$ or the epigraph of a~Lipschitz function, a~domain~$Ω$ is an $(\varepsilon,δ)$-domain. These spaces are shown to be the traces of the spaces $B^{φ_{0}}_{p,q}(\mathbb{R}^{n},\{t_{k}\})$ and $\widetilde{B}^{l}_{p,q,r}(\mathbb{R}^{n},\{t_{k}\})$ on domains $G$ and~$Ω$, respectively. The extension operator $\operatorname{Ext}_{1}:B^{φ_{0}}_{p,q}(G,\{t_{k}\}) \to B^{φ_{0}}_{p,…
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 …
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.
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. $
In this paper, we study the smoothness of restrictions of Besov functions. It is known that for any $f\in B\_{p,q}^s(\mathbb{R}^N)$ with $q\leq p$ we have $f(\cdot,y)\in B\_{p,q}^s(\mathbb{R}^d)$ for a.e. $y\in \mathbb{R}^{N-d}$. We prove that this is no longer true when $p\<q$. Namely, we construct a function $f\in B\_{p,q}^s(\mathbb{R}^N)$ such that $f(\cdot,y)\notin B\_{p,q}^s(\mathbb{R}^d)$ for a.e. $y\in \mathbb{R}^{N-d}$. We show that, in fact, $f(\cdot,y)$ belong to $B\_{p,q}^{(s,Ψ)}(\mathbb{R}^d)$ for a.e. $y\in\mathbb{R}^{N-d}$, a Besov space of generalized smoothness, and, when $q=\i…
This paper is concerned with a boundedness of trace and extension operators for Besov spaces and Triebel-Lizorkin spaces on upper half space with variable exponents. To define trace and extension operators, we introduce a quarkonial decomposition for Besov spaces and Triebel-Lizorkin spaces with variable exponents on ${\mathbb R}^n$. Furthermore, we study trace and extension operators for Besov spaces and Triebel-Lizorkin spaces with variable exponents on upper half spaces ${\mathbb R}^n_+$.
We give a new characterization of Nikolskii-Besov classes of functions of fractional smoothness by means of a nonlinear integration by parts formula in the form of a nonlinear inequality. A similar characterization is obtained for Nikolskii-Besov classes with respect to Gaussian measures on finite- and infinite-dimensional spaces.
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.
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.
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 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 introduce Besov-type spaces with variable smoothness and integrability. We show that these spaces are characterized by the $φ$-transforms in appropriate sequence spaces and we obtain atomic decompositions for these spaces. Moreover the Sobolev embeddings for these function spaces are obtained.
The present paper is concerned with new Besov-type space of variable smoothness. Nonlinear spline-approximation approach is used to give atomic decomposition of such space. Characterization of the trace space on hyperplane is also obtained.
We introduce Besov spaces with variable smoothness and integrability by using the continuous version of Calderòn reproducing formula. We show that our space is well-defined, i.e., independent of the choice of basis functions. We characterize these function spaces by so-called Peetre maximal functions and we obtain the Sobolev embeddings for these function spaces. We use these results to prove the atomic decomposition for these spaces.
There are known trace and extension theorems relating functions in a weighted Sobolev space in a domain U to functions in a Besov space on the boundary bU. We extend these theorems to the case where the Sobolev exponent p is less than one by modifying our Sobolev spaces to consider averages of functions in Whitney balls. Averaged Sobolev spaces are also of interest in the applications in the case where p>1, and so we also provide trace and extension results in that case. Finally, we provide some comparable results for Neumann traces and extensions.
We give characterizations for homogeneous and inhomogeneous Besov-Lizorkin-Triebel spaces in terms of continuous local means for the full range of parameters. In particular, we prove characterizations in terms of Lusin functions and spaces involving the Peetre maximal function to apply the classical coorbit space theory due to Feichtinger and Gröchenig. This results in atomic decompositions and wavelet bases for homogeneous spaces. In particular we give sufficient conditions for suitable wavelets in terms of moment, decay and smoothness conditions.
We study the existence of traces of Besov spaces on fractal $h$-sets $Γ$ with the special focus laid on necessary assumptions implying this existence, or, in other words, present criteria for the non-existence of traces. In that sense our paper can be regarded as an extension of [Br4] and a continuation of the recent paper [Ca2]. Closely connected with the problem of existence of traces is the notion of dichotomy in function spaces: We can prove that -- depending on the function space and the set $Γ$ -- there occurs an alternative: either the trace on $Γ$ exists, or smooth functions compactly …
Motivated by the results of Korry and Kinnunen and Saksman, we study the behaviour of the discrete fractional maximal operator on fractional Hajlasz spaces, Hajlasz-Besov and Hajlasz-Triebel-Lizorkin spaces on metric measure spaces. We show that the discrete fractional maximal operator maps these spaces to the spaces of the same type with higher smoothness. Our results extend and unify aforementioned results. We present our results in general setting, but they are new already in the Euclidean case.
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 a new characterization of the homogeneous Besov spaces $\dot{\mathcal B}^{s}_{p,q}(Z)$ with smoothness $s \in (0,1)$ in the setting of doubling metric measure spaces $(Z,d,μ)$. The characterization is given in terms of a hyperbolic filling of the metric space $(Z,d)$, a construction which has previously appeared in the context of other function spaces in [3,1,2]. We use the characterization to obtain results concerning the density of Lipschitz functions in the spaces $\dot{\mathcal B}^{s}_{p,q}(Z)$ and a general complex interpolation formula in the smoothness range $0 < s < 1$.
In this paper, we consider the trace theorem for modulation spaces, alpha modulation spaces and Besov spaces. For the modulation space, we obtain the sharp results.
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…
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 …
The Hausdorff dimension of the graphs of the functions in Hölder and Besov spaces (in this case with integrability p \geq 1) on fractal d-sets is studied. Denoting by s \in (0,1] the smoothness parameter, the sharp upper bound min{d+1-s,d/s} is obtained. In particular, when passing from d \geq s to d<s there is a change of behaviour from d+1-s to d/s which implies that even highly nonsmooth functions defined on cubes in R^n have not so rough graphs when restricted to, say, \emp{rarefied} fractals.
We introduce a family of quasi-Banach spaces - which we call wave packet smoothness spaces - that includes those function spaces which can be characterised by the sparsity of their expansions in Gabor frames, wave atoms, and many other frame constructions. We construct Banach frames for and atomic decompositions of the wave packet smoothness spaces and study their embeddings in each other and in a few more classical function spaces such as Besov and Sobolev spaces.
We consider the problem of determining the asymptotic order of the Gelfand numbers of mixed-(quasi-)norm embeddings $\ell^b_p(\ell^d_q) \hookrightarrow \ell^b_r(\ell^d_u)$ given that $p \leq r$ and $q \leq u$, with emphasis on cases with $p\leq 1$ and/or $q\leq 1$. These cases turn out to be related to structured sparsity. We obtain sharp bounds in a number of interesting parameter constellations. Our new matching bounds for the Gelfand numbers of the embeddings of $\ell_1^b(\ell_2^d)$ and $\ell_2^b(\ell_1^d)$ into $\ell_2^b(\ell_2^d)$ imply optimality assertions for the recovery of block-spar…
In this paper we consider smoothness and decay properties of radial functions belonging to smoothness spaces related to Morrey spaces (Sobolev-Morrey spaces, Besov-type spaces and Besov-Morrey spaces). Within this framework we prove generalizations of the Radial Lemma of Strauss.
We establish trace theorems for function spaces defined on general Ahlfors regular metric spaces $Z$. The results cover the Triebel-Lizorkin spaces and the Besov spaces for smoothness indices $s<1,$ as well as the first order Hajłasz-Sobolev space $M^{1,p}(Z)$. They generalize the classical results from the Euclidean setting, since the traces of these function spaces onto any closed Ahlfors regular subset $F \subset Z$ are Besov spaces defined intrinsically on $F$. Our method employs the definitions of the function spaces via hyperbolic fillings of the underlying metric space.
In this paper, we study the Besov regularity of d-dimensional Lévy white noises. More precisely, we describe new sample paths properties of a given white noise in terms of weighted Besov spaces. In particular, the smoothness and integrability properties of Lévy white noises are characterized using the Blumenthal-Getoor indices. Our techniques rely on wavelet methods and generalized moments estimates for Lévy white noises.
For each integrability parameter $p \in (0,\infty]$, the critical smoothness of a periodic generalized function $f$, denoted by $s_f(p)$ is the supremum over the smoothness parameters $s$ for which $f$ belongs to the Besov space $B_{p,p}^s$ (or other similar function spaces). This paper investigates the evolution of the critical smoothness with respect to the integrability parameter $p$. Our main result is a simple characterization of all the possible critical smoothness functions $p\mapsto s_f(p)$ when $f$ describes the space of generalized periodic functions. We moreover characterize the com…
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…
We study traces of weighted Triebel-Lizorkin spaces $F^s_{p,q}({\mathbb R}^n,w)$ on hyperplanes ${\mathbb R}^{n-k}$, where the weight is of Muckenhoupt type. We concentrate on the example weight $w_α(x) = |x_n|^α$ when $|x_n|\leq 1$, $x\in{\mathbb R}^n$, and $w_α(x)=1$ otherwise, where $α>-1$. Here we use some refined atomic decomposition argument as well as an appropriate wavelet representation in corresponding (unweighted) Besov spaces. The second main outcome is the description of the real interpolation space $(B^{s_1}_{p_1,p_1}({\mathbb R}^{n-k}), B^{s_2}_{p_2,p_2}({\mathbb R}^{n-k}))_{θ,r…
In this paper we provide non-smooth atomic decompositions of 2-microlocal Besov-type and Triebel-Lizorkin-type spaces with variable exponents $B^{\mathrm{\boldsymbolω}, ϕ}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$ and $F^{\mathrm{\boldsymbolω}, ϕ}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$. Of big importance in general, and an essential tool here, are the characterizations of the spaces via maximal functions and local means, that we also present. These spaces were recently introduced by Wu at al. and cover not only variable 2-microlocal Besov and Triebel-Lizorkin spaces $B^{\mathrm{\boldsymbolω}}_{p(\cdot),q…
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…
Let $\ell\in\mathbb{N}$ and $p\in(1,\infty]$. In this article, the authors prove that the sequence $\{f-B_{\ell,2^{-k}}f\}_{k\in\mathbb{Z}}$ consisting of the differences between $f$ and the ball average $B_{\ell,2^{-k}}f$ characterizes the Besov space $\dot B^α_{p,q}(\rn)$ with $q\in (0, \infty]$ and the Triebel-Lizorkin space $\dot F^α_{p,q}(\rn)$ with $q\in (1,\infty]$ when the smoothness order $α\in(0,2\ell)$. More precisely, it is proved that $f-B_{\ell,2^{-k}}f$ plays the same role as the approximation to the identity $φ_{2^{-k}}\ast f$ appearing in the definitions of $\dot B^α_{p,q}(\rn…
In this article, we establish a characterization of the set $M(B^{0,b}_{p,\infty}(\mathbb{R}^n))$ of all pointwise multipliers of Besov spaces $B^{0,b}_{p,\infty}(\mathbb{R}^n)$ with only logarithmic smoothness $b\in\mathbb{R}$ in the special cases $p=1$ and $p=\infty$. As applications of these two characterizations, we clarify whether or not the three concrete examples, namely characteristic functions of open sets, continuous functions defined by differences, and the functions $e^{ik\cdot x}$ with $k\in\mathbb{Z}^n$ and $x\in\mathbb{R}^n$, are pointwise multipliers of $B^{0,b}_{1,\infty}(\mat…
We provide non-smooth atomic decompositions for Besov spaces $\Bd(\rn)$, $s>0$, $0<p,q\leq \infty$, defined via differences. The results are used to compute the trace of Besov spaces on the boundary $Γ$ of bounded Lipschitz domains $Ω$ with smoothness $s$ restricted to $0<s<1$ and no further restrictions on the parameters $p,q$. We conclude with some more applications in terms of pointwise multipliers.
We study the embedding $\text{id}: \ell_p^b(\ell_q^d) \to \ell_r^b(\ell_u^d)$ and prove matching bounds for the entropy numbers $e_k(\text{id})$ provided that $0<p<r\leq \infty$ and $0<q\leq u\leq \infty$. Based on this finding, we establish optimal dimension-free asymptotic rates for the entropy numbers of embeddings of Besov and Triebel-Lizorkin spaces of small dominating mixed smoothness which settles an open question in the literature. Both results rely on a novel covering construction recently found by Edmunds and Netrusov.
