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 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.
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 i…
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 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.
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.
We prove that the space of pointwise multipliers between two distinct Musielak--Orlicz spaces is another Musielak-Orlicz space and the function defining it is given by an appropriately generalized Legendre transform. In particular, we obtain characterization of pointwise multipliers between Nakano spaces. We also discuss factorization problem for Musielak-Orlicz spaces and exhibit some differences between Orlicz and Musielak-Orlicz cases.
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 $S^r_{p,q}B(\mathbb{R}^d)$.
In the paper we find representation of the space of pointwise multipliers between two Orlicz function spaces, which appears to be another Orlicz space and the formula for the Young function generating this space is given. Further, we apply this result to find necessary and suffcient conditions for factorization of Orlicz function spaces.
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…
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.
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…
We prove that the class of Banach function lattices in which all relatively weakly compact sets are equi-integrable sets (i.e. spaces satisfying the Dunford-Pettis criterion) coincides with the class of 1-disjointly homogeneous Banach lattices. A new examples of such spaces are provided. Furthermore, it is shown that Dunford-Pettis criterion is equivalent to de la Vallee Poussin criterion in all rearrangement invariant spaces on the interval. Finally, the results are applied to characterize weakly compact pointwise multipliers between Banach function lattices.
In this paper the solution of the pointwise multiplier problem 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)$ is presented, where $p_1,\,p_2,\,q_1,\,q_2 \in (0,\infty)$, $p_2 \le q_2$ and $u_1,\,u_2,\,v_1,\,v_2$ are weights on $(0,\infty)$.
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 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.
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, \ …
In Chapter 4 of [25] Triebel proved two theorems concerning pointwise multipliers and diffeomorphisms in function spaces of Besov and Triebel-Lizorkin type. In each case he presented two approaches, one via atoms and one via local means. While the approach via atoms was very satisfactory concerning the length and simplicity, only the rather technical approach via local means proved the theorems in full generality. In this paper we generalize two extensions of these atomic decompositions, one by Skrzypczak (see [22]) and one by Triebel and Winkelvoss (see [30]) so that we are able to give a sho…
Several results concerning multipliers of symmetric Banach function spaces are presented firstly. Then the results on multipliers of Calderón-Lozanovskii spaces are proved. We investigate assumptions on a Banach ideal space E and three Young functions φ_1, φ_2 and φ, generating the corresponding Calderón-Lozanovskii spaces E_{φ_1}, E_{φ_2}, E_φ so that the space of multipliers M(E_{φ_1}, E_φ) of all measurable x such that x,y \in E_φ for any y \in E_{φ_1} can be identified with E_{φ_2}. Sufficient conditions generalize earlier results by Ando, O'Neil, Zabreiko-Rutickii, Maligranda-Persson and …
We introduce generalized Campanato spaces $\mathcal{L}_{p,ϕ}$ on a probability space $(Ω,\mathcal{F},P)$, where $p\in[1,\infty)$ and $ϕ:(0,1]\to(0,\infty)$. If $p=1$ and $ϕ\equiv1$, then $\mathcal{L}_{p,ϕ}=\mathrm{BMO}$. We give a characterization of the set of all pointwise multipliers on $\mathcal{L}_{p,ϕ}$.
In this paper, we develop a thorough analysis of the boundedness properties of the maximal operator for the Bochner-Riesz means related to the Fourier-Bessel expansions. For this operator, we study weighted and unweighted inequalities in the spaces L^p((0,1),x^{2ν+1}dx). Moreover, weak and restricted weak type inequalities are obtained for the critical values of p. As a consequence, we deduce the almost everywhere pointwise convergence of these means.
In this paper we prove $L^p$ estimates for Stein's square functions associated to Fourier-Bessel expansions. Furthermore we prove transference results for square functions from Fourier-Bessel series to Hankel transforms. Actually, these are transference results for vector-valued multipliers from discrete to continuous in the Bessel setting. As a consequence, we deduce the sharpness of the range of $p$ for the $L^p$-boundedness of Fourier-Bessel Stein's square functions from the corresponding property for Hankel-Stein square functions. Finally, we deduce $L^p$ estimates for Fourier-Bessel multi…
In this paper we study the forward integral of operator-valued processes with respect to a cylindrical Brownian motion. In particular, we provide conditions under which the approximating sequence of processes of the forward integral, converges to the stochastic integral process with respect to Sobolev norms of smoothness alpha < 1/2. This result will be used to derive a new integration by parts formula for the forward integral.
On a general Lie group $G$ endowed with a sub-Riemannian structure and of local dimension $d$, we characterize the pointwise multipliers of Triebel--Lizorkin spaces $F^{p,q}_α$ for $p,q\in (1,\infty)$ and $α>d/p$, and those of Besov spaces $B^{p,q}_α$ for $q\in [1,\infty]$, $p>d$ and $d/p< α<1$. When $G$ is stratified, we extend the latter characterization to all $p,q\in [1,\infty]$ and $α>d/p$.
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.
In this paper we investigate the compact and weakly compact multipliers of the Herz-algebras $A_p(G)$. Let $B_p(G)$ be the space of pointwise multipliers of $A_p(G)$. We show that there is a topological invariant mean on $B^*_p (G)$. Furthermore, we show that if $B^*_p(G)$ is separable, then $G$ is a discrete group.
We investigate relations between symmetrizations of quasi-Banach function spaces and constructions such as Calderon-Lozanovskii spaces, pointwise product spaces and pointwise multipliers. We show that under reasonable assumptions the symmetrization commutes with these operations. We determine also the spaces of pointwise multipliers between Lorentz spaces and Cesaro spaces. Developed methods may be regarded as an arithmetic of quasi-Banach function spaces and proofs of Theorems 3, 4 and 6 give a kind of tutorial for these methods. Finally, the above results will be used in proofs of some facto…
We study multipliers of Hardy-Orlicz spaces $\mH_Φ$ which are strictly contained between $\bigcup_{p>0}H^p$ and so-called ``big'' Hardy-Orlicz spaces. Big Hardy-Orlicz spaces, carrying an algebraic structure, are equal to their multiplier algebra, whereas in classical Hardy spaces $H^p$, the multipliers reduce to $H^{\infty}$. For Hardy-Orlicz spaces $\mH_Φ$ between these two extremal situations and subject to some conditions, we exhibit multipliers that are in Hardy-Orlicz spaces the defining functions of which are related to $Φ$. Even if the results do not entirely characterize the multiplie…
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 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 prove a factorization theorem for reproducing kernel Hilbert spaces whose kernel has a normalized complete Nevanlinna-Pick factor. This result relates the functions in the original space to pointwise multipliers determined by the Nevanlinna-Pick kernel and has a number of interesting applications. For example, for a large class of spaces including Dirichlet and Drury-Arveson spaces, we construct for every function $f$ in the space a pluriharmonic majorant of $|f|^2$ with the property that whenever the majorant is bounded, the corresponding function $f$ is a pointwise multiplier.
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…
The well-known factorization theorem of Lozanovski{\u ı} may be written in the form $L^{1}\equiv E\odot E^{\prime}$, where $\odot $ means the pointwise product of Banach ideal spaces. A natural generalization of this problem would be the question when one can factorize $F$ through $E$, i.e., when $F\equiv E\odot M(E, F) \,$, where $M(E, F) $ is the space of pointwise multipliers from $E$ to $F$. Properties of $M(E, F) $ were investigated in our earlier paper [KLM12] and here we collect and prove some properties of the construction $E\odot F$. The formulas for pointwise product of Calderón-Loza…
We prove a sharp multiplier theorem of Mihlin-Hörmander type for the Grushin operator on the unit sphere in $\mathbb{R}^3$, and a corresponding boundedness result for the associated Bochner-Riesz means. The proof hinges on precise pointwise bounds for spherical harmonics.
This paper is devoted to characterizing the Riemann-Stieltjes operators and pointwise multipliers acting on M${\rm \ddot{o}}$bius invariant spaces $Q_p$, which unify BMOA and Bloch space in the scale of $p$. The boundedness and compactness of these operators on $Q_p$ spaces are determined by means of an embedding theorem, i.e. $Q_p$ spaces boundedly embedded in the non-isotropic tent type spaces $T_q^\infty$.
Let $\mathbb B_n$ be the open unit ball in $\mathbb C^n$. We characterize the spectra of pointwise multipliers $M_u$ acting on Banach spaces of analytic functions on $\mathbb B_n$ satisfying some general conditions. These spaces include Bergman-Sobolev spaces $A^p_{α,β}$, Bloch-type spaces $\mathcal B_α$, weighted Hardy spaces $H^p_w$ with Muckenhoupt weights and Hardy-Sobolev Hilbert spaces $H^2_β$. Moreover, we describe the essential spectra of multipliers in most of the aforementioned spaces, in particular, in those spaces for which the set of multipliers is a subset of the ball algebra.
For a pointwise multiplier $φ$ of the Hardy-Sobolev space $H^2_β$ on the open unit ball $\bn$ in $\cn$, we study spectral properties of the multiplication operator $M_φ: H^2_β\to H^2_β$. In particular, we compute the spectrum and essential spectrum of $M_φ$ and develop the Fredholm theory for these operators.
We prove a general sparse domination theorem in a space of homogeneous type, in which a vector-valued operator is controlled pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination in which the usual $\ell^1$-sum in the sparse operator is replaced by an $\ell^r$-sum. This sparse domination theorem is applicable to various operators from both harmonic analysis and (S)PDE. Using our main theorem, we prove the $A_2$-theorem for vector-valued Calderón--Zygmund operators in a space of homogeneous type, from which we deduce an…
