The notion of multipliers in Hilbert space was introduced by Schatten in 1960 using orthonormal sequences and was generalized by Balazs in 2007 using Bessel sequences. This was extended to Banach spaces by Rahimi and Balazs in 2010 using p-Bessel sequences. In this paper, we further extend this by considering Lipschitz functions. On the way we define frames for metric spaces which extends the notion of frames and Bessel sequences for Banach spaces. We show that when the symbol sequence converges to zero, the multiplier is a Lipschitz compact operator. We study how the variation of parameters in the multiplier effects the properties of multiplier.
In this paper, we introduce (p,q)g-Bessel multipliers in Banach spaces and we show that under some conditions a (p,q)g-Bessel multiplier is invertible. Also, we show the continuous dependency of (p,q)g-Bessel multipliers on their parameters.
We prove that every unconditionally summable sequence in a Hilbert space can be factorized as the product of a square summable scalar sequence and a Bessel sequence. Some consequences on the representation of unconditionally convergent multipliers are obtained, thus providing positive answers to a conjecture by Balazs and Stoeva in some particular cases.
To achieve our main research goal, first we survey the approaches towards dual fusion frames existing in the literature and agree on the notion of duality for fusion frames in the sense of Kutyniok, Paternostro and Philipp ({\it Oper. Matrices} {\bf11} (2017), no. 2, 301--336). As a main result we show that different fusion frames have different dual fusion frames. Moreover, this duality notion leads to a new definition of Bessel fusion multipliers which is a slightly modified version of the commonly used definitions. Particularly, we show that with this definition in many cases Bessel fusion …
We investigate the problem of establishing bilateral continuous embeddings of the uniformly localized Bessel potential spaces $H^γ_{r, \: unif}(\mathbb{R}^n)$ into the multiplier spaces between Bessel potential spaces with positive smoothness indices. This problem is considered in the model situation when the natural norms of both of these Bessel potential spaces are generated by some inner product yet the description theorems for the corresponding multiplier space in terms of the spaces $H^γ_{r, \: unif}(\mathbb{R}^n)$ can not be established. The optimal character of the indices figuring in t…
Multipliers are operators that combine (frame-like) analysis, a multiplication with a fixed sequence, called the symbol, and synthesis. They are very interesting mathematical objects that also have a lot of applications for example in acoustical signal processing. It is known that bounded symbols and Bessel sequences guarantee unconditional convergence. In this paper we investigate necessary and equivalent conditions for the unconditional convergence of multipliers. In particular we show that, under mild conditions, unconditionally convergent multipliers can be transformed by shifting weights …
The notation of generalized Bessel multipliers is obtained by a bounded operator on $\ell^2$ which is inserted between the analysis and synthesis operators. We show that various properties of generalized multipliers are closely related to their parameters, in particular it will be shown that the membership of generalized Bessel multiplier in the certain operator classes requires that its symbol belongs in the same classes, in special sense. Also, we give some examples to illustrate our results. As we shall see, generalized multipliers associated with Riesz bases are well-behaved, more precisel…
In this paper we prove a randomized difference norm characterization for Bessel potential spaces with values in UMD Banach spaces. The main ingredients are $\mathcal{R}$-boundedness results for Fourier multiplier operators, which are of independent interest. As an application we characterize the pointwise multiplier property of the indicator function of the half-space on these spaces. All results are proved in the setting of weighted spaces.
In this paper we show that every g-frame for an \linebreak infinite dimensional Hilbert space $\mathcal{H}$ can be written as a sum of three g-orthonormal bases for $\mathcal{H}$. Also, we prove that every g-frame can be represented as a linear combination of two g-orthonormal bases if and only if it is a g-Riesz basis. Further, we show each g-Bessel multiplier is a Bessel multiplier and investigate the inversion of g-frame multipliers. Finally, we introduce the concept of controlled g-frames and weighted g-frames and show that the sequence induced by each controlled g-frame (resp. weighted g-…
In this paper we introduce the concept of von Neumann-Schatten Bessel multipliers and obtain some of their characterizations. Finally, special attention is devoted to the study of invertible Hilbert--Schmidt frame multipliers. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for diagonalization of matrices in finite dimensional setting as well as for dual $g$-frames. In particular, we show that a $g$-frame is uniquely determined by the set of its $g$-frames.
In this paper we study the concept of multipliers for continuous $g$-Bessel families in Hilbert spaces. We present necessary conditions for invertibility of multipliers for continuous $g$-Bessel families and sufficient conditions for invertibility of multipliers for continuous $g$-frames.
We prove a general type description result for the multipliers acting between two periodic Bessel potential spaces, defined on the $n$--dimensional torus, in a case when their smoothness indices are of different signs. This is done through the detailed examination of a periodic analogue of the linear operator $J_s$, which is employed in the definition of the scale of the Bessel potential space defined on the whole space $\mathbb{R}^n$. Our method of defining this periodic analogue of $J_s$ uses the results about an asymptotic behaviour of the generalized Fourier coefficients and existence of a…
For $p > 1, γ\in \mathbb{R}$, denote by $H^γ_p(\mathbb{R}^n)$ the Bessel potential space, by $H^γ_{p, unif}(\mathbb{R}^n)$ the corresponding uniformly localized Bessel potential space and by $M[s, -t]$ the space of multipliers from $H^s_2(\mathbb{R}^n)$ into $H^{-t}_2(\mathbb{R}^n)$. Assume that $s, t \geqslant 0, n/2 > \max(s, t) > 0, r: = \min(s, t), p_1: = n/max(s, t)$. Then the following embeddings hold $$ H^{-r}_{p_1, unif}(\mathbb{R}^n) \subset M[s, -t] \subset H^{-r}_{2, unif}(\mathbb{R}^n). $$ The main result of the paper claims the sharpness of the left embedding in the following sens…
Consider the multidimensional Bessel operator $$B f(x) = -\sum_{j=1}^N \left(\partial_j^2 f(x) +\frac{α_j}{x_j} \partial_j f(x)\right), \quad x\in(0,\infty)^N. $$ Let $d = \sum_{j=1}^N \max(1,α_j+1)$ be the homogeneous dimension of the space $(0,\infty)^N$ equipped with the measure $x_1^{α_1}... x_N^{α_N} dx_1...dx_N$. In the general case $α_1,...,α_N >-1$ we prove multiplier theorems for spectral multipliers $m(B)$ on $L^{1,\infty}$ and the Hardy space $H^1$. We assume that $m$ satisfies the classical Hörmander condition $$\sup_{t>0} \left||η(\cdot) m(t\cdot)\right||_{W^{2,β}(\mathbb{R})}<\in…
Multipliers have been recently introduced as operators for Bessel sequences and frames in Hilbert spaces. These operators are defined by a fixed multiplication pattern (the symbol) which is inserted between the analysis and synthesis operators. In this paper, we will generalize the concept of Bessel multipliers for p-Bessel and p-Riesz sequences in Banach spaces. It will be shown that bounded symbols lead to bounded operators. Symbols converging to zero induce compact operators. Furthermore, we will give sufficient conditions for multipliers to be nuclear operators. Finally, we will show the c…
Let $\{λ_n\}_n \in \ell^\infty(\mathbb{N})$. In 1960, R. Schatten \cite{SCHATTEN} studied operators of the form $\sum_{n=1}^{\infty}λ_n (x_n\otimes \bar{y_n})$, where $\{x_n\}_n$, $\{y_n\}_n$ are orthonormal sequences in a Hilbert space. In 2007, P. Balazs \cite{BALAZS3} generalized this by replacing $\{x_n\}_n$ and $\{y_n\}_n$ by Bessel sequences. In this paper, we generalize this by studying the operators of the form $\sum_{n=1}^{\infty}λ_n (A^*_nx_n\otimes \bar{B^*_ny_n})$, where $\{A_n\}_n$ and $\{B_n\}_n$ are operator-valued Bessel sequences and $\{x_n\}_n$, $\{y_n\}_n$ are sequences in t…
This paper introduces the concept of Bessel multipliers. These operators are defined by a fixed multiplication pattern, which is inserted between the Analysis and synthesis operators. The proposed concept unifies the approach used for Gabor multipliers for arbitrary analysis/synthesis systems, which form Bessel sequences, like wavelet or irregular Gabor frames. The basic properties of this class of operators are investigated. In particular the implications of summability properties of the symbol for the membership of the corresponding operators in certain operator classes are specified. As a s…
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.
The objective of this paper is to describe the space of multipliers acting from a Bessel potential space $H^s_p(\mathbb R^n)$ into another space $H^{-t}_q(\mathbb R^n)$, provided that the smooth indices of these spaces have different signs, i.e. $s, t \geqslant 0$. This space of multipliers consists of distributions $u$, such that for all $φ\in H^s_p(\mathbb R^n)$ the product $φ\cdot u$ is well-defined and belongs to the space $H^{-t}_q(\mathbb R^n)$. We succeed to describe this space explicitly, provided that $p \leqslant q$ and one of the following conditions $$ s \geqslant t \geqslant 0, \ …
The Marcinkiewicz multipliers are $L^{p}$ bounded for $1<p<\infty $ on the Heisenberg group $\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ (Müller, Ricci and Stein \cite{MRS}). This is surprising in the sense that these multipliers are invariant under a two parameter group of dilations on $\mathbb{C}^{n}\times \mathbb{R}$, while there is \emph{no} two parameter group of \emph{automorphic} dilations on $\mathbb{H} ^{n}$. The purpose of this paper is to establish a theory of the flag Lipschitz space on the Heisenberg group $\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ in the sen…
Titchmarsh theorems for Hölder-Lipschitz functions on fundamental domains of lattices in $\mathbb{R}^{d}$ with applications to boundedness of Fourier multipliers
In this paper, we investigate the invertibility of generalized g-Bessel multipliers. We show that for semi-normalized symbols, the inverse of any invertible generalized g-frame multiplier can be represented as a generalized g-frame multiplier. Also we give several approaches for constructing invertible generalized g-frame multipliers from the given one. It is worth mentioning that some of our results are quite different from those studied in the previous literatures on this topic.
We study properties of the classical fractional Sobolev spaces (or Bessel potential spaces) on non-Lipschitz subsets of $\mathbb{R}^n$. We investigate the extent to which the properties of these spaces, and the relations between them, that hold in the well-studied case of a Lipschitz open set, generalise to non-Lipschitz cases. Our motivation is to develop the functional analytic framework in which to formulate and analyse integral equations on non-Lipschitz sets. In particular we consider an application to boundary integral equations for wave scattering by planar screens that are non-Lipschit…
Multipliers have been recently introduced by P. Balazs as operators for Bessel sequences and frames in Hilbert spaces. These are operators that combine (frame-like) analysis, a multiplication with a fixed sequence (called the symbol) and synthesis. Weighted and controlled frames have been introduced to improve the numerical efficiency of iterative algorithms for inverting the frame operator Also g-frames are the most popular generalization of frames that include almost all of the frame extensions. In this manuscript the concept of the controlled g-frames will be defined and we will show that c…
We study the pointwise multiplier property of the characteristic function of the half-space on weighted mixed-norm anisotropic vector-valued function spaces of Bessel potential and Triebel-Lizorkin type.
We investigate pointwise multipliers on vector-valued function spaces over $\mathbb{R}^d$, equipped with Muckenhoupt weights. The main result is that in the natural parameter range, the characteristic function of the half-space is a pointwise multiplier on Bessel-potential spaces with values in a UMD Banach space. This is proved for a class of power weights, including the unweighted case, and extends the classical result of Shamir and Strichartz. The multiplication estimate is based on the paraproduct technique and a randomized Littlewood-Paley decomposition. An analogous result is obtained fo…
In our companion paper (S.N. Chandler Wilde, D.P. Hewett, A. Moiola, Sobolev spaces on non-Lipschitz subsets of $\mathbb{R}^n$ with application to boundary integral equations on fractal screens, 2016) we studied a number of different Sobolev spaces on a general (non-Lipschitz) open subset $Ω$ of $\mathbb{R}^n$, defined as closed subspaces of the classical Bessel potential spaces $H^s(\mathbb{R}^n)$ for $s\in\mathbb{R}$. These spaces are mapped by the restriction operator to certain spaces of distributions on $Ω$. In this note we make some observations about the relation between these spaces of…
Inspired by the work of Hedenmalm, Lindqvist and Seip, we consider different properties of dilations systems of a fixed function $φ\in L^2(0,1)$. More precisely, we study when the system $\{φ(nx)\}_n$ is a Bessel sequence, a Riesz sequence, or it satisfies the lower frame bound. We are able to characterize these properties in terms of multipliers of the Hardy space $\mathcal{H}^2$ of Dirichtet series and, also, in terms of Hardy spaces on the infinite polytorus. We also address the multivariate case.
A rather complete investigation of anisotropic Bessel potential, Besov, and Hölder spaces on cylinders over (possibly) noncompact Riemannian manifolds with boundary is carried out. The geometry of the underlying manifold near its 'ends' is determined by a singularity function which leads naturally to the study of weighted function spaces. Besides of the derivation of Sobolev-type embedding results, sharp trace theorems, point-wise multiplier properties, and interpolation characterizations particular emphasize is put on spaces distinguished by boundary conditions. This work is the fundament for…
Consider the Schrödinger operator $\mathcal{L}=-Δ+V$ in $\mathbb{R}^n, n\ge 3,$ where $V$ is a nonnegative potential satisfying a reverse Hölder condition of the type \begin{equation*} \left( \frac{1}{|B|}\int_B V(y)^qdy\right)^{1/q}\le \frac{C}{|B|}\int_B V(y)dy, \, \text{ for some }q>n/2. \end{equation*} We define $Λ^α_{\mathcal{L}},\, 0<α<2,$ the class of measurable functions such that $$ \|ρ(\cdot)^{-α}f(\cdot)\|_\infty<\infty \quad \, \, \text{and}\:\: \quad \sup_{|z|>0}\frac{\|f(\cdot+z)+f(\cdot-z)-2f(\cdot)\|_\infty}{|z|^α}<\infty, $$ where $ρ$ is the critical radius function associated…
