In this paper we study the asymptotic behavior of Kolmogorov, approximation, Bernstein and Weyl numbers of embeddings $ \mathcal{A}^{s,r}_{\rm mix}(\mathbb{T}^d) \to L_2(\mathbb{T}^d)$ and $\mathcal{A}^{s,r}_{\rm mix}(\mathbb{T}^d) \to \mathcal{A}(\mathbb{T}^d)$, where $\mathcal{A}^{s,r}_{\rm mix}(\mathbb{T}^d)$ is a weighted Wiener algebra of mixed smoothness $s$ and $\mathcal{A}(\mathbb{T}^d)$ is the Wiener algebra itself, both defined on the $d$-dimensional torus $\mathbb{T}^d$. Our main interest consists in the calculation of the associated asymptotic constants.
We study the operators T on the weighted space L^p commuting either with the right translations St or left translations P^+S_{-t} and we establish the existence of a symbol of T. We characterize completely the spectrum of St. We obtain a similar result for the spectrum of P^+S_{-t} and some spectral results for the bounded operators commuting with (St), t>0 or with (P^+St), t<0.
We define Wiener amalgam spaces of (quasi)analytic ultradistributions whose local components belong to a general class of translation and modulation invariant Banach spaces of ultradistributions and their global components are either weighted $L^p$ or weighted $\mathcal{C}_0$ spaces. We provide a discrete characterisation via so called uniformly concentrated partitions of unity. Finally, we study the complex interpolation method and we identify the strong duals for most of these Wiener amalgam spaces.
We establish the sharp conditions for the embedding between Wiener amalgam spaces $W_{p,q}^s$ and some classical spaces, including Sobolev spaces $L^{s,r}$, local Hardy spaces $h_{r}$, Besov spaces $B_{p,q}^s$, which partially improve and extend the main result obtained by Guo et al. in J. Funct. Anal., 273(1):404-443, 2017. In addition, we give the full characterization of inclusion between Wiener amalgam spaces $W_{p,q}^s$ and $α$-modulation spaces $M_{p,q}^{s,α}$. Especially, in the case of $α=0$ with $M_{p,q}^{s,α} = M_{p,q}^s$, we give the sharp conditions of the most general case of thes…
Inversion theorems of Wiener type are essential tools in analysis and number theory. We derive a weighted version of an inversion theorem of Wiener type for general Dirichlet series from that of Edwards from 1957, and we outline an alternative proof based on the duality theory of convex cones and extension techniques for characters of semigroups. Variants and arithmetical applications are described, including the case of multidimensional weighted generalized Dirichlet series.
Let $G$ be a locally compact, compactly generated group of polynomial growth and let $ω$ be a weight on $G$. Under proper assumptions on the weight $ω$, the Banach space $L^p(G,ω)$ is a Banach \ast-algebra. In this paper we give examples of such weighted $L^p$-algebras and we study some of their harmonic analysis properties, such as symmetry, existence of functional calculus, regularity, weak Wiener property, Wiener property, existence of minimal ideals of a given hull.
Let $μ$ be a measure on the real line $\mathbb{R}$ such that $\int_{\mathbb{R}}\frac{dμ(t)}{1+t^2} < \infty$ and let $a>0$. Assume that the norms $\|f\|_{L^2(\mathbb{R})}$ and $\|f\|_{L^2(μ)}$ are comparable for functions $f$ in the Paley-Wiener space $PW_{a}$ and that $PW_a$ is dense in $L^2(μ)$. We reconstruct the canonical Hamiltonian system $JX' = zHX$ such that $μ$ is the spectral measure for this system.
We study the Segal-Bargmann transform on $M(2).$ The range of this transform is characterized as a weighted Bergman space. In a similar fashion Poisson integrals are studied. Using a Gutzmer type formula we characterize the range as a class of functions extending holomorphically to an appropriate domain in the complexification of $M(2).$ We also prove a Paley-Wiener theorem for the inverse Fourier transform
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…
We study the bounded operators on weighted spaces Lw^2 on R^+ commuting either with the right translations St or left translations and we establish the existence of a symbol for these operators. We characterize completely the spectrum of St. We give also a spectral results for the left translations and for the bounded operators commuting with the semi-group (St) or with the semi-group of the left translations.
We give necessary and sufficient conditions for the embeddings $Λ\text{BV}^{(p)}\subseteq Γ\text{BV}^{(q_n\uparrow q)}$ and $Φ\text{BV}\subseteq\text{BV}^{(q_n\uparrow q)}$. As a consequence, a number of results in the literature, including a fundamental theorem of Perlman and Waterman, are simultaneously extended.
In this paper we first prove the metric approximation property for weighted mixed-norm $L_w^{(p_1,\dots ,p_n)}$ spaces. Using Gabor frame representation this implies that the same property holds in weighted modulation and Wiener amalgam spaces. As a consequence, Grothendieck's theory becomes applicable, and we give criteria for nuclearity and $r$-nuclearity for operators acting on these space as well as derive the corresponding trace formulae. Finally, we apply the notion of nuclearity to functions of the harmonic oscillator on modulation spaces.
This article contains a characterization of when certain weighted Sobolev spaces on $\Bbb R^n$ embed compactly into $L^2(\mathbb R^n, φ)$. The characterization is in terms of derivatives of the weight function $φ$ and involves the Wiener capacity, as it is obtained from reformulating the problem in terms of resolvent properties of Schrödinger operators. This reformulation also works for general domains.
We investigate the group $\mathcal{H}_\mathbb{C}$ of complexified Heisenberg matrices with entries from an infinite-dimensional complex Hilbert space $H$. Irreducible representations of the Weyl--Schr{ö}dinger type on the space $L^2_χ$ of quadratically integrable $\mathbb{C}$-valued functions are described. Integrability is understood with respect to the projective limit $χ=\varprojlimχ_i$ of probability Haar measures $χ_i$ defined on groups of unitary $i\times i$-matrices $U(i)$. The measure $χ$ is invariant under the infinite-dimensional group $U(\infty)=\bigcup U(i)$ and satisfies the abstr…
In this paper, we give the characterization of the embeddings between weighted Cesàro function spaces. The proof is based on the duality technique, which reduces this problem to the characterizations of some direct and reverse Hardy-type inequalities and iterated Hardy-type inequalities.
We give the characterization of the embeddings between weighted Tandori and Cesàro function spaces using the combination of duality arguments for weighted Lebesgue spaces and weighted Tandori spaces with estimates for the iterated integral operators.
Let $Ω$ be a John domain, and let $Γ\subset \partial Ω$ be an $h$-set. For some functions $h$ and some weight functions depending on distance from $Γ$, embedding theorems for a weighted Sobolev class is obtained.
Let $0<p\leq 1$, $ω$ be a weight on $\mathbb Z$, and let $\mathcal A$ be a unital Banach algebra. If $f$ is a continuous function from the unit circle $\mathbb T$ to $\mathcal A$ such that $\sum_{n\in \mathbb Z} \|\widehat f(n)\|^p ω(n)^p<\infty$ and $f(z)$ is left invertible for all $z \in \mathbb T$, then there is a weight $ν$ on $\mathbb Z$ and a continuous function $g:\mathbb T \to \mathcal A$ such that $1\leq ν\leq ω$, $ν$ is constant if and only if $ω$ is constant, $g$ is a left inverse of $f$ and $\sum_{n\in \mathbb Z}\|\widehat g(n)\|^pν(n)^p<\infty$. We shall obtain a similar result w…
We give some necessary conditions and sufficient conditions for the compactness of the embedding of Sobolev spaces $W^{1,p}(Ω,w) \to L^p(Ω,w),$ where $w$ is some weight on a domain $Ω\subset \Real^n$.
In this paper, we study a family of general fractional Sobolev spaces $\MsqpOm$ when $\Om=\Rn$ or $\Om$ is a bounded domain, having a compact, Lipschitz boundary $\Bdy$, in $\Rn$ for $n\geq2$. Among other results, some compact embedding results of $\MVsqpRn\hookrightarrow\LqRn$ and $\MVsqpRn\hookrightarrow\LlRn$ for suitable potential functions $V(x)$ are described.
Inspired by an extension of Wiener's lemma on the relation of measures $μ$ on the unit circle and their Fourier coefficients $\widehatμ(k_n)$ along subsequences $(k_n)$ of the natural numbers by Cuny, Eisner and Farkas [CEF19, arXiv:1701.00101], we study the validity of the lemma when the Fourier coefficients are weighted by a sequence of probability measures. By using convergence with respect to a filter derived from these measure sequences, we obtain similar results, now also allowing the consideration of locally compact abelian groups other than $\mathbb{T}$ and $\mathbb{R}$. As an applicat…
In this paper embeddings between weighted complementary local Morrey-type spaces ${\,^{^{\bf c}}\!}LM_{pθ,ω}({\mathbb R}^n,v)$ and weighted local Morrey-type spaces $LM_{pθ,ω}({\mathbb R}^n,v)$ are characterized. In particular, two-sided estimates of the optimal constant $c$ in the inequality \begin{equation*} \bigg( \int_0^{\infty} \bigg( \int_{B(0,t)} f(x)^{p_2}v_2(x)\,dx \bigg)^{\frac{q_2}{p_2}} u_2(t)\,dt\bigg)^{\frac{1}{q_2}} \le c \bigg( \int_0^{\infty} \bigg( \int_{\,^{^{\bf c}\!}B(0,t)} f(x)^{p_1} v_1(x)\,dx\bigg)^{\frac{q_1}{p_1}} u_1(t)\,dt\bigg)^{\frac{1}{q_1}} \end{equation*} are o…
A characterisation is given of bounded embeddings from weighted $L^2$ spaces on bounded intervals into $L^2$ spaces on the half-plane, induced by isomorphisms given by the Laplace transform onto weighted Hardy and Bergman spaces (Zen spaces). As an application necessary and sufficient conditions are given for the boundedness of truncated Hankel and Toeplitz integral operators, including the weighted case.
Let $\mathbb{H}$ be a separable Hilbert space. In this paper we establish a generalization of Walnut's representation and Janssen's representation of the $\mathbb{H}-$valued Gabor frame operator on $\mathbb{H}-$valued weighted amalgam spaces $W_{\mathbb{H}}(L^p,L^q_v)$, $1 \leq p, q \leq \infty$. Also we show that the frame operator is invertible on $W_{\mathbb{H}}(L^p,L^q_v)$, $1 \leq p, q \leq \infty$, if the window function is in the Wiener amalgam space $W_{\mathbb{H}}(L^{\infty},L^1_w)$. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to …
We study compactness and boundedness of embeddings from Sobolev type spaces on metric spaces into $L^q$ spaces with respect to another measure. The considered Sobolev spaces can be of fractional order and some statements allow also nondoubling measures. Our results are formulated in a general form, using sequences of covering families and local Poincaré type inequalities. We show how to construct such suitable coverings and Poincaré inequalities. For locally doubling measures, we prove a self-improvement property for two-weighted Poincaré inequalities, which applies also to lower-dimensional m…
In this paper embeddings between weighted Copson function spaces ${\operatorname{Cop}}_{p_1,q_1}(u_1,v_1)$ and weighted Cesàro function spaces ${\operatorname{Ces}}_{p_2,q_2}(u_2,v_2)$ are characterized. In particular, two-sided estimates of the optimal constant $c$ in the inequality \begin{equation*} \bigg( \int_0^{\infty} \bigg( \int_0^t f(τ)^{p_2}v_2(τ)\,dτ\bigg)^{\frac{q_2}{p_2}} u_2(t)\,dt\bigg)^{\frac{1}{q_2}} \le c \bigg( \int_0^{\infty} \bigg( \int_t^{\infty} f(τ)^{p_1} v_1(τ)\,dτ\bigg)^{\frac{q_1}{p_1}} u_1(t)\,dt\bigg)^{\frac{1}{q_1}}, \end{equation*} where $p_1,\,p_2,\,q_1,\,q_2 \in…
In this paper we will establish necessary and sufficient conditions for a Laplace-Carleson embedding to be bounded for certain spaces of functions on the positive half-line. We will use these results to characterise weighted (infinite-time) admissibility of control and observation operators.
In this paper order estimates for entropy numbers of embeddings of weighted Sobolev spaces on a John domain are obtained. In addition, we obtain order estimates for entropy numbers of summation operators on trees.
In the present paper we study embedding operators for weighted Sobolev spaces whose weights satisfy the well-known Muckenhoupt A_p-condition. Sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities. The proposed method is based on the concept of 'generalized' quasiconformal homeomorphisms (homeomorphisms with bounded mean distortion.) The choice of the homeomorphism type depends on the choice of the corresponding weighted Sobolev space. Such classes of homeomorphisms induce bounded composition oper…
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.
In this paper certain $n$-dimensional inequalities are shown to be equivalent to the inequalities in the one-dimensional setting. By this means, embeddings between weighted local Morrey-type spaces are characterized for some ranges of parameters.
We study nuclear embeddings for weighted spaces of Besov and Triebel-Lizorkin type where the weight belongs to some Muckenhoupt class and is essentially of polynomial type. Here we can extend our previous results [17,19] where we studied the compactness of corresponding embeddings. The concept of nuclearity goes back to Grothendieck who defined it in [14]. Recently there is a refreshed interest to study such questions [5-8,49]. This led us to the investigation in the weighted setting. We obtain complete characterisations for the nuclearity of the corresponding embedding. Essential tools are a …
We consider spaces of holomorphic functions which are square-integrable against a Gaussian weight, which appear in the theory of metaplectic FBI--Bargmann transforms. We identify the operator norm of embeddings between two such spaces, by relating these embeddings to Fourier integral operators with complex phase.
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…
In this study, we define double weighted variable exponent Sobolev spaces $W^{1,q(.),p(.)}\left( Ω,\vartheta _{0},\vartheta \right) $ with respect to two different weight functions. Also, we investigate the basic properties of this spaces. Moreover, we discuss the existence of weak solutions for weighted Dirichlet problem of $p(.)$-Laplacian equation \begin{equation*} \left\{ \begin{array}{cc} -\text{div}\left( \vartheta (x)\left\vert \nabla f\right\vert ^{p(x)-2}\nabla f\right) =\vartheta _{0}(x)\left\vert f\right\vert ^{q(x)-2}f & x\in Ω\\ f=0 & x\in \partial Ω\end{array} \right. \end{equati…
In this paper we obtain order estimates for entropy numbers of embeddings of weighted Sobolev spaces into weighted Lebesgue spaces and of weighted summation operators on trees. Here we consider some critical conditions on the parameters.
We characterize all the real numbers a,b,c and 1<= p,q,r<infty such that the weighted Sobolev space W_{a,b}^(q,p)(R^N\{0}) with power weights |x|^a and |x|^b is continuously embedded into L^{r}(R^N;|x|^cdx). Furthermore, we show that this embedding is (almost) always characterized by a multiplicative norm inequality. When a,b,c>-N and attention is confined to smooth functions with compact support, these inequalities coincide with the Caffarelli-Kohn-Nirenberg inequalities. Variants for radially symmetric functions, or when r<=min{p,q}, are also obtained along the way.
Gelfand and Kolmogorov numbers of Sobolev embeddings of weighted function spaces II
School of Computer Science and Technology, Anhui University（安徽大学计算机科学与技术学院）；School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）；School of Mathematical Sciences, Anhui University（安徽大学数学科学学院）
