Sofic群与卷积算子

Stochastic homogenization of convolution type operators

This paper deals with homogenization problem for convolution type non-local operators in random statistically homogeneous ergodic media. Assuming that the convolution kernel has a finite second moment and satisfies the uniform ellipticity and certain symmetry conditions, we prove the almost sure homogenization result and show that the limit operator is a second order elliptic differential operator with constant deterministic coefficients.

Algebrable sets of hypercyclic vectors for convolution operators

We show that several convolution operators on the space of entire functions, such as the MacLane operator, support a dense hypercyclic algebra that is not finitely generated. Birkhoff's operator also has this property on the space of complex-valued smooth functions on the real line.

自由群上的弱型径向卷积算子

Weak type radial convolution operators on free group

刻画了自由群上具有非负核的弱型$(2,2)$和限制弱型$(2,2)$的径向卷积算子.对于$1<p<2.$也得到了弱型$(p,p)$的估计

Orlicz空间插值及其在算子卷积中的应用

Orlicz Space Interpolation and Its Applications to Operator Convolution

本文研究了非交换Orlicz空间的插值理论，并应用该理论得到Weyl伪微分符号和Werner函数-算子卷积的Young型卷积估计。

Hypercyclicity of convolution operators on spaces of entire functions

In this paper we use Nachbin's holomorphy types to generalize some recent results concerning hypercyclic convolution operators on Fréchet spaces of entire functions of bounded type of infinitely many complex variables.

Ergodic properties of convolution operators

Let $G$ be a locally compact group and $μ$ be a measure on $G$. In this paper we find conditions for the convolution operators $λ_p(μ)$, defined on $L^p(G)$ and given by convolution by $μ$, to be mean ergodic and uniformly mean ergodic. The ergodic properties of the operators $λ_p(μ)$ are related to the ergodic properties of the measure $μ$ as well.

Optimal extension of the Fourier transform and convolution operator on compact groups

Let $G$ be a compact group (not necessarily abelian) and let $Φ$ be a Young function satisfying the $Δ_2$-condition. We determine the optimal domain and the associated extended operator for both Fourier transform and the convolution operator defined on the Orlicz spaces $L^Φ(G).$

关于卷积型算子在Fock空间上有界性的研究

On the boundedness of certain operators of convolution type on Fock spaces

本文研究了扭曲Fock空间上卷积型算子的有界性问题，给出了卷积核所需满足的充分必要条件。

On Mellin convolution operators in Bessel potential spaces

Mellin convolution equations acting in Bessel potential spaces are considered. The study is based upon two results. The first one concerns the interaction of Mellin convolutions and Bessel potential operators (BPOs). In contrast to the Fourier convolutions, BPOs and Mellin convolutions do not commute and we derive an explicit formula for the corresponding commutator in the case of Mellin convolutions with meromorphic symbols. These results are used in the lifting of the Mellin convolution operators acting on Bessel potential spaces up to operators on Lebesgue spaces. The operators arising belo…

核型算子卷积代数的双投射性和双平坦性

Biprojectivity and biflatness for convolution algebras of nuclear operators

对于局部紧群G，Neufang定义了核型算子在空间N（L^p（G））上的卷积积.研究了卷积代数N（L^p（G））的同调性质，并将它们与群G的一些性质，如紧性、有限性、离散性和顺从性联系起来.

Hypercyclic Algebras for convolution operators of unimodular constant term

We study the existence of hypercyclic algebras for convolution operators $Φ(D)$ on the space of entire functions whose symbol $Φ$ has unimodular constant term. In particular, we provide new eigenvalue criteria for the existence of densely strongly algebrable sets of hypercyclic vectors.

Hypercyclicity, existence and approximation results for convolution operators on spaces of entire functions

In this work we shall prove new results on the theory of convolution operators on spaces of entire functions. The focus is on hypercyclicity results for convolution operators on spaces of entire functions of a given type and order; and existence and approximation results for convolution equations on spaces of entire functions of a given type and order. In both cases we give a general method to prove new results that recover, as particular cases, several results of the literature. Applications of these more general results are given, including new hypercyclicity results for convolution operator…

The weak type $(1,p)$ for convolution operators on locally compact groups

In this paper we provide necessary and sufficient conditions for the $\textnormal{weak}(1,p)$ boundedness, $1< p<\infty,$ of convolution operators on locally compact (Hausdorff) topological groups. So, we generalize a classical result due to Sobolev-Hardy-Littlewood and Stepanov. Applications to Fourier multipliers on Lie groups also are given.

Fourier convolution operators with symbols equivalent to zero at infinity on Banach function spaces

We study Fourier convolution operators $W^0(a)$ with symbols equivalent to zero at infinity on a separable Banach function space $X(\mathbb{R})$ such that the Hardy-Littlewood maximal operator is bounded on $X(\mathbb{R})$ and on its associate space $X'(\mathbb{R})$. We show that the limit operators of $W^0(a)$ are all equal to zero.

Homogenization of biased convolution type operators

This paper deals with homogenization of parabolic problems for integral convolution type operators with a non-symmetric jump kernel in a periodic elliptic medium. It is shown that the homogenization result holds in moving coordinates. We determine the corresponding effective velocity and prove that the limit operator is a second order parabolic operator with constant coefficients. We also consider the behaviour of the effective velocity in the case of small antisymmetric perturbations of a symmetric kernel.

Convolution operators via orthogonal polynomials

In this paper we aim to generalize results obtained in the framework of fractional calculus by the way of reformulating them in terms of operator theory. In its own turn, the achieved generalization allows us to spread the obtained technique on practical problems that connected with various physical - chemical processes.

Mixing convolution operators on spaces of entire functions

We show that if $E$ is an arbitrary $(DFN)$-space, then every nontrivial convolution operator on the Fréchet nuclear space $\mathcal{H}(E)$ is mixing, in particular hypercyclic. More generally we obtain the same conclusion when $E=F^{\prime}_c,$ where $F$ is a separable Fréchet space with the approximation property. On the opposite direction we show that a translation operator on the space $\mathcal{H}(\mathbb{C}^{\mathbb{N}}) $ is never hypercyclic.

Convolution operators defined by singular measures on the motion group

This paper contains an $L^{p}$ improving result for convolution operators defined by singular measures associated to hypersurfaces on the motion group. This needs only mild geometric properties of the surfaces, and it extends earlier results on Radon type transforms on $\mathbb{R}^{n}$. The proof relies on the harmonic analysis on the motion group.

Spectra and ergodic properties of multiplication and convolution operators on the space $\mathcal{S}(\mathbb{R})$

In this paper we investigate the spectra and the ergodic properties of the multiplication operators and the convolution operators acting on the Schwartz space $\mathcal{S}(\mathbb{R})$ of rapidly decreasing functions, i.e., operators of the form $M_h: \mathcal{S}(\mathbb{R})\to\mathcal{S}(\mathbb{R})$, $f \mapsto h f $, and $C_T\colon \mathcal{S}(\mathbb{R})\to\mathcal{S}(\mathbb{R})$, $f\mapsto T\star f$. Precisely, we determine their spectra and characterize when those operators are power bounded and mean ergodic.

Noncompactness of Fourier Convolution Operators on Banach Function Spaces

Let $X(\mathbb{R})$ be a separable Banach function space such that the Hardy-Littlewood maximal operator $M$ is bounded on $X(\mathbb{R})$ and on its associate space $X'(\mathbb{R})$. Suppose $a$ is a Fourier multiplier on the space $X(\mathbb{R})$. We show that the Fourier convolution operator $W^0(a)$ with symbol $a$ is compact on the space $X(\mathbb{R})$ if and only if $a=0$. This result implies that nontrivial Fourier convolution operators on Lebesgue spaces with Muckenhoupt weights are never compact.

加权Hahn空间上的卷积算子

Convolution Operators on Weighted Hahn Spaces

研究加权Hahn序列空间上卷积算子的有界性、紧性及乘子代数，完整刻画了谱与精细谱，并给出示例。

On perturbation of a surjective convolution operator

Let $μ\in {\cal E}'({\mathbb R}^n)$ be a compactly supported distribution such that its support is a convex set with non-empty interior. Let $X_2$ be a convex domain in ${\mathbb R}^n$, $X_1 = X_2 + supp \ μ$. Assuming that a convolution operator $A: {\cal E}(X_1) \to {\cal E}(X_2)$ acting by the rule $(Af)(x) = (μ* f)(x)$ is surjective we provide a condition on a linear continuous operator $B: {\cal E}(X_1) \to {\cal E}(X_2)$ that guarantees surjectivity of the operator $A+B$.

The Fourier transform and convolutions generated by a differential operator with boundary condition on a segment

We introduce the concepts of the Fourier transform and convolution generated by an arbitrary restriction of the differentiation operator in the space $L_{2}(0,b).$ In contrast to the classical convolution, the introduced convolution explicitly depends on the boundary condition that defines the domain of the operator $L.$ The convolution is closely connected to the inverse operator or to the resolvent. So, we first find a representation for the resolvent, and then introduce the required convolution.

Convolution operators on Banach lattices with shift-invariant norms

Let G be a locally compact abelian group and let μbe a complex valued regular Borel measure on G. In this paper we consider a generalisation of a class of Banach lattices introduced in [6]. We use Laplace transform methods to show that the norm of a convolution operator with symbol μon such a space is bounded below by the L_\infty norm of the Fourier-Stieltjes transform of μ. We also show that for any Banach lattice of locally integrable functions on G with a shift-invariant norm, the norm of a convolution operator with symbol μis bounded above by the total variation of μ.

测度-算子卷积及其在混合态Gabor乘子中的应用

Measure-operator convolutions and applications to mixed-state Gabor multipliers

本研究利用Feichtinger的最新成果，将Weyl-Heisenberg群上的函数-算子卷积推广为测度-算子卷积，借此证明混合态Gabor乘子的特征值分布新结果、格点版Berezin-Lieb不等式，还得到Gabor乘子关于参数的连续性及对局域算子的逼近性质。

Continuity of Pseudo-Differential Operator $h_{μ,a}$ Involving Hankel Translation and Hankel Convolution on Some Gevrey Spaces

The Pseudo-Differential Operator (p.d.o.) $h_{μ,a}$ associated with the Bessel Operator involving the symbol $a(x,y)$ whose derivatives satisfy certain growth conditions depending on some increasing sequences is studied on certain Gevrey spaces. The p.d.o. $h_{μ,a}$ on Hankel translation $τ$ and Hankel convolution of Gevrey functions is continuous linear map into another Gevrey spaces.

球体上的伯格曼空间上的函数-算子卷积代数及其盖尔范德理论

The function-operator convolution algebra over the Bergman space of the ball and its Gelfand theory

本文研究了球体上伯格曼空间的函数-算子卷积代数结构及其盖尔范德理论，探讨了该代数的性质及相关算子的傅里叶变换。

Operators that coerce the surjectivity of convolution

Considered are operators that leave the set of non-invertible (in the sense of Ehrenpreis) distributions stable. They simultaneously generalise the operation of convolution by a distribution with compact support and the operation of multiplication by a real analytic function; they are here called pseudo-convolutions since they also generalise pseudo-differential operators. (It is shown that the elliptic real analytic pseudo-differential operators leave both the non-invertible and the invertible distributions invariant.) But when the condition of real-analyticity is relaxed, such operators may …

Hypercyclic algebras for convolution and composition operators

We provide an alternative proof to those by Shkarin and by Bayart and Matheron that the operator $D$ of complex differentiation supports a hypercyclic algebra on the space of entire functions. In particular we obtain hypercyclic algebras for many convolution operators not induced by polynomials, such as $\mbox{cos}(D)$, $De^D$, or $e^D-aI$, where $0<a\le 1$. In contrast, weighted composition operators on function algebras of analytic functions on a plane domain fail to support supercyclic algebras. Non-trivial translations on the space of complex-valued, smooth functions on the real line do su…

度量测度空间上若干卷积型算子的有界性

Boundedness of some convolution-type operators on metric measure spaces

本文研究度量测度空间上源于计算机视觉变换的卷积型算子的有界性，提出并分析其性质。

Strongly mixing convolution operators on Fréchet spaces of holomorphic functions

A theorem of Godefroy and Shapiro states that non-trivial convolution operators on the space of entire functions on $\mathbb{C}^n$ are hypercyclic. Moreover, it was shown by Bonilla and Grosse-Erdmann that they have frequently hypercyclic functions of exponential growth. On the other hand, in the infinite dimensional setting, the Godefroy-Shapiro theorem has been extended to several spaces of entire functions defined on Banach spaces. We prove that on all these spaces, non-trivial convolution operators are strongly mixing with respect to a gaussian probability measure of full support. For the …

Calkin images of Fourier convolution operators with slowly oscillating symbols

Let $Φ$ be a $C^*$-subalgebra of $L^\infty(\mathbb{R})$ and $SO_{X(\mathbb{R})}^\diamond$ be the Banach algebra of slowly oscillating Fourier multipliers on a Banach function space $X(\mathbb{R})$. We show that the intersection of the Calkin image of the algebra generated by the operators of multiplication $aI$ by functions $a\inΦ$ and the Calkin image of the algebra generated by the Fourier convolution operators $W^0(b)$ with symbols in $SO_{X(\mathbb{R})}^\diamond$ coincides with the Calkin image of the algebra generated by the operators of multiplication by constants.

Orlicz空间与Orlicz Schatten类的卷积，及其在Toeplitz算子中的应用

Convolutions of Orlicz spaces and Orlicz Schatten classes, with applications to Toeplitz operators

本文研究由Young函数Φ定义的符号类s_{A,Φ}的卷积性质，证明其Young型结果，并将所得结论应用于Toeplitz算子，建立相应的Orlicz Schatten类性质。

Oscillating convolution operators on the Heisenberg group

In this paper, we consider oscillating convolution operotors on the Heisenberg group $H^n_a$ with respect to the norm $ρ(x,t) = ρ_1(b x, b t)$ with $ρ_1(x,t)= (|x|^4 + t^2)^{1/4}$. We obtain $L^2$ boundedness properties using the oscillatory integral estimates for degenerate phases in the Euclidean setting. Our result contains an improvement of the Lyall's result for the cases $\frac{a^2}{b^2} \geq C_β$.

扭曲卷积、伪微分算子和傅立叶调制空间

Twisted convolution, pseudo-differential operators and Fourier modulation spaces

讨论了（加权）傅立叶调制空间上扭卷积的连续性.我们使用这些结果来建立连续性结果的扭曲卷积Lebesgue空间。例如，我们证明了如果$ω$是一个适当的权，而$1\le p\le 2$是，那么$L^p_{(ω)}$是扭卷积下的一个代数.

Invertibility of convolution operators on homogeneous groups

We say that a tempered distribution $A$ belongs to the class $S^m(\Ge)$ on a homogeneous Lie algebra $\Ge$ if its Abelian Fourier transform $a=\hat{A}$ is a smooth function on the dual $\Ges$ and satisfies the estimates $$ |D^αa(ξ)|\le C_α(1+|ξ|)^{m-|α|}. $$ Let $A\in S^0(\Ge)$. Then the operator $f\mapsto f\star\widetilde{A}(x)$ is bounded on $L^2(\Ge)$. Suppose that the operator is invertible and denote by $B$ the convolution kernel of its inverse. We show that $B$ belongs to the class $S^0(\Ge)$ as well. As a corollary we generalize Melin's theorem on the parametrix construction for Rocklan…

Point-wise estimates for nonlocal heat kernel of convolution type operators

The paper deals with point-wise estimates for the heat kernel of a nonlocal convolution type operator with a kernel that decays at least exponentially at infinity. It is shown that the large time behaviour of the heat kernel depends essentially on whether $|x|\ll t$, or $|x|\sim t$, or $|x|\gg t$. We obtain sharp point-wise upper bounds for the heat kernel in all these regions. In the first region the nonlocal heat kernel behaves like the classical one, while in the other regions we observe an essential difference.

单边Grand Lebesque空间中的张量、Sobolev、乘法和卷积算子

Tensor, Sobolev, Multiplicative and Convolution Operators in the Bide - Side Grand Lebesque Spaces

本文研究了某些巴拿赫空间，即所谓的Bide-Side Grand Lebesque空间中的乘法不等式、张量不等式、Sobolev不等式和卷积不等式，并举例说明它们的尖锐性。

On spectra of convolution operators with potentials

This paper focuses on the spectral properties of a bounded self-adjoint operator in $L_2(\mathds R^d)$ being the sum of a convolution operator with an integrable convolution kernel and an operator of multiplication by a continuous potential converging to zero at infinity. We study both the essential and the discrete spectra of this operator. It is shown that the essential spectrum of the sum is the union of the essential spectrum of the convolution operator and the image of the potential. We then provide a number of sufficient conditions for the existence of discrete spectrum and obtain lower …

调和NA群上卷积算子的满射性

Surjectivity of convolution operators on harmonic $NA$ groups

本文刻画了调和NA群上径向紧支撑分布诱导的右卷积算子在光滑径向函数空间上满射的充要条件，即其球面傅里叶变换缓慢递减，并由此证明了球面平均算子的满射性。

Two Families of Hypercyclic Non-Convolution Operators

Let $H(\mathbb{C})$ be the set of all entire functions endowed with the topology of uniform convergence on compact sets. Let $λ,b\in\mathbb{C}$, let $C_{λ,b}:H(\mathbb{C})\to H(\mathbb{C})$ be the composition operator $C_{λ,b} f(z)=f(λz+b)$, and let $D$ be the derivative operator. We extend results on the hypercyclicity of the non-convolution operators $T_{λ,b}=C_{λ,b} \circ D$ by showing that whenever $|λ|\geq 1$, the collection of operators \begin{align*} \{ψ(T_{λ,b}): ψ(z)\in H(\mathbb{C}), ψ(0)=0 \text{ and } ψ(T_{λ,b}) \text{ is continuous}\} \end{align*} forms an algebra under the usual …

Algebra of convolution type operators with continuous data on Banach function spaces

We show that if the Hardy-Littlewood maximal operator is bounded on a reflexive Banach function space $X(\mathbb{R})$ and on its associate space $X'(\mathbb{R})$, then the space $X(\mathbb{R})$ has an unconditional wavelet basis. As a consequence of the existence of a Schauder basis in $X(\mathbb{R})$, we prove that the ideal of compact operators $\mathcal{K}(X(\mathbb{R}))$ on the space $X(\mathbb{R})$ is contained in the Banach algebra generated by all operators of multiplication $aI$ by functions $a\in C(\dot{\mathbb{R}})$, where $\dot{\mathbb{R}}=\mathbb{R}\cup\{\infty\}$, and by all Fouri…

Chaos in convolution operators on the space of entire functions of infinitely many complex variables

A classical result of Godefroy and Shapiro states that every nontrivial convolution operator on the space $\mathcal{H}(\mathbb{C}^n)$ of entire functions of several complex variables is hypercyclic. In sharp contrast with this result Fávaro and Mujica show that no translation operator on the space $\mathcal{H}(\mathbb{C}^\mathbb{N})$ of entire functions of infinitely many complex variables is hypercyclic. In this work we study the linear dynamics of convolution operators on $\mathcal{H}(\mathbb{C}^\mathbb{N})$. First we show that no convolution operator on $\mathcal{H}(\mathbb{C}^\mathbb{N})$ …

On Convolution Dominated Operators

For a locally compact group $G$ we consider the algebra $CD(G)$ of convolution dominated operators on $L^{2}(G)$: An operator $A:L^2(G)\to L^2(G)$ is called convolution dominated if there exists $a\in L^1(G)$ such that for all $f \in L^2(G)$ $ |Af(x)| \leq a * |f| (x)$, for almost all $x \in G$. In the case of discrete groups those operators can be dealt with quite sufficiently if the group in question is rigidly symmetric. For non-discrete groups we investigate the subalgebra of regular convolution dominated operators $CD_{reg}(G)$. For amenable $G$ which is rigidly symmetric as a discrete gr…

Remarks on some results of G. Pisier and P. Saab on convolutions

A result of G. Pisier says that a convolution operator $\star f : M(G) \to C(G),$ where $G$ is a compact Abelian group, can be factored through a Hilbert space if and only if $f$ has the absolutely summable set of Fourier coefficients. P. Saab (2010) generalized this result in some directions in the vector-valued cases. We give some further generalizations of the results of G. Pisier and P. Saab, considering, in particular, the factorizations of the operators through the operators of Schatten classes in Hilbert spaces. Also, some related theorem on the factorization of operators through the op…

卷积与贝塞尔算子在幂级数空间上的拓扑化与幂有界性

Topologizability and Power Boundedness of Convolutions and Toeplitz Operators on Power Series Spaces

本文研究了卷积与贝塞尔算子在幂级数空间上的拓扑化与幂有界性，确定了 Toeplitz 算子的必要条件及其在不同空间上的应用。

紧量子群上卷积算子的遍历性

On ergodic properties of convolution operators associated with compact quantum groups

Junge和Q.Xu关于非交换L^p-空间中核平均的遍历性质的最新结果被应用于分析紧量子群上卷积诱导的算子的几乎一致收敛性。

Homogenization of non-autonomous operators of convolution type in periodic media

The paper deals with periodic homogenization problem for a para\-bo\-lic equation whose elliptic part is a convolution type operator with rapidly oscillating coefficients. It is assumed that the coefficients are rapidly oscillating periodic functions both in spatial and temporal variables and that the scaling is diffusive that is the scaling factor of the temporal variable is equal to the square of the scaling factor of the spatial variable. Under the assumption that the convolution kernel has a finite second moment and that the operator is symmetric in spatial variables we show that the studi…

Hypercyclic behavior of some non-convolution operators on $H(\mathbb{C}^N)$

We study hypercyclicity properties of a family of non-convolution operators defined on spaces of holomorphic functions on $\mathbb{C}^N$. These operators are a composition of a differentiation operator and an affine composition operator, and are analogues of operators studied by Aron and Markose on $H(\mathbb{C})$. The hypercyclic behavior is more involved than in the one dimensional case, and depends on several parameters involved.

离散群上的卷积控制算子

Convolution-Dominated Operators on Discrete Groups

我们研究了由离散群$G$索引的无限矩阵$A$，这些矩阵被卷积算子控制，在这个意义上，$|(Ac)(x)| \leq (a \ast |c|)(x)$对于$x\in G$和一些$a\in \ell ^1(G)$。这类“卷积支配”矩阵形成了包含在$\ell ^2(G)$上有界算子代数中的Banach-*-代数。我们的主要结果表明，卷积支配矩阵的逆也是卷积支配的，只要$G$是顺从的和刚性对称的。对于阿贝尔群，这一结果可以追溯到Gohberg，Baskakov，和其他人，对于非阿贝尔群完全不同的技术是必需的，如广义$L^1$ -代数和对称的群代数。

${\ell}^{\infty}(Γ)$上的内射卷积算子是满射的

Injective convolution operators on ${\ell}^{\infty}(Γ)$ are surjective

设$Γ$是一个离散群，设$f \in \ell^1(Γ)$。我们观察到，如果自然卷积算子$ρ_f:\ell^{\infty}(Γ)\to \ell^{\inf ty}(Γ)$是单射的，那么f在$\ell^1(Γ)$中是可逆的。我们的证明简化和推广计算预印本Deninger和施密特，呼吁直接有限的代数$\ell^1(Γ)$。我们给出简单的例子来说明，对于非离散G，一般不能用$\ell^p$、$1\leq p< \infty$或$L^{\infty}(G)$代替$\ell^{\infty}$.最后，我们考虑了将主要结果推广到$Γ$上加权卷积算子的情形，并给出了部分结果。

冯·诺依曼代数上群作用的算子卷积的结合性

Associativity of operator convolutions for group actions on von Neumann algebras

本文针对局部紧幺模群在半有限冯·诺依曼代数上的特定作用，证明了算子卷积的结合性，为量子调和分析一般版本奠定了基础。

On the R-boundedness of stochastic convolution operators

The $R$-boundedness of certain families of vector-valued stochastic convolution operators with scalar-valued square integrable kernels is the key ingredient in the recent proof of stochastic maximal $L^p$-regularity, $2<p<\infty$, for certain classes of sectorial operators acting on spaces $X=L^q(μ)$, $2\le q<\infty$. This paper presents a systematic study of $R$-boundedness of such families. Our main result generalises the afore-mentioned $R$-boundedness result to a larger class of Banach lattices $X$ and relates it to the $\ell^{1}$-boundedness of an associated class of deterministic convolu…

非可分序列空间上卷积算子的谱性质

Spectral Properties of Convolution Operators on Non-Separable Sequence Spaces

本文针对非可分序列空间上的卷积算子，提出线性泛函方法结合单侧移位算子精细谱分解，完整刻画其各类谱，并推广至算子多项式，统一了带状三角矩阵的谱理论。