Integral operators induced by the Fock kernel

Generalized Volterra companion operators on Fock spaces

We characterize the bounded and compact generalized Volterra companion integral operators on Fock spaces acting between the standard Fock spaces. As a special case, we prove that there exist no nontrivial compact Volterra companion integral and multiplication operators on Fock spaces. We also obtain asymptotic estimates for the norm of these operators.

Integral operators, embedding theorems and a Littlewood-Paley formula on weighted Fock spaces

We obtain a complete characterization of the entire functions $g$ such that the integral operator $(T_ g f)(z)=\int_{0}^{z}f(ζ)\,g'(ζ)\,dζ$ is bounded or compact, on a large class of Fock spaces $\mathcal{F}^ϕ_p$, induced by smooth radial weights that decay faster than the classical Gaussian one. In some respects, these spaces turn out to be significantly different than the classical Fock spaces. Descriptions of Schatten class integral operators are also provided. En route, we prove a Littlewood-Paley formula for $||\cdot||_{\mathcal{F}^ϕ_p}$ and we characterize the positive Borel measures for…

Volterra type and weighted composition operators on Fock spaces

Bounded and compact product of Volterra type integral and composition operators acting between weighted Fock spaces are described. We also estimate the norms of these operators in terms of Berezin type integral transforms on the complex plan $\mathbb{C}$. All our results are valid for weighted composition operators acting on the class of Fock spaces considered under appropriate interpretation of the weights.

Spectral properties of Volterra-type integral operators on Fock--Sobolev spaces

We study some spectral properties of Volterra-type integral operators $V_g$ and $I_g$ with holomorphic symbol $g$ on the Fock--Sobolev spaces $\mathcal{F}_{ψ_m}^p$. We showed that $V_g$ is bounded on $\mathcal{F}_{ψ_m}^p$ if and only if $g$ is a complex polynomial of degree not exceeding two, while compactness of $V_g$ is described by degree of $g$ being not bigger than one. We also identified all those positive numbers $p$ for which the operator $V_g$ belongs to the Schatten $\mathcal{S}_p$ classes. Finally, we characterize the spectrum of $V_g$ in terms of a closed disk of radius twice the c…

Integral Operators on Fock-Sobolev Spaces via Multipliers on Gauss-Sobolev Spaces

In this paper, we obtain an isometry between the Fock-Sobolev space and the Gauss-Sobolev space. As an application, we use multipliers on the Gauss-Sobolev space to characterize the boundedness of an integral operator on the Fock-Sobolev space.

A Fock space approach to the theory of strictly positive kernels

In this paper, we give a new approach to the theory of strictly positive kernels. Our method is based on the structure of Fock spaces. As its applications, various examples of strictly positive kernels are given. Moreover, we give a new proof of the universal approximation theorem for the Gauss kernel.

多复变量加权Fock空间上的Bergman核与Hankel算子

Bergman Kernels and Hankel Operators on Weighted Fock Spaces in Several Complex Variables

本文在多复变量加权Fock空间中，通过正则化曲率假设建立Bergman核估计，并构造\bar\partial解算子，进而用加权IDA空间刻画Hankel算子的有界性与紧性。

基于Fock空间的高斯RBF核：四元数与多复变量情形

Gaussian RBF-kernels via Fock spaces: quaternionic and several complex variables settings

该研究探讨复值高斯RBF核的两种扩展，分别构造基于切片超全纯函数的四元数高斯RBF核、研究多复变量下的高斯RBF核，并分析二者与Fock空间的关联。

Hankel operators on Fock spaces and related Bergman kernel estimates

Hankel operators with anti-holomorphic symbols are studied for a large class of weighted Fock spaces on $\cn$. The weights defining these Hilbert spaces are radial and subject to a mild smoothness condition. In addition, it is assumed that the weights decay at least as fast as the classical Gaussian weight. The main result of the paper says that a Hankel operator on such a Fock space is bounded if and only if the symbol belongs to a certain BMOA space, defined via the Berezin transform. The latter space coincides with a corresponding Bloch space which is defined by means of the Bergman metric.…

Path connected components of the space of Volterra-type integral operators

We study the topological structure of the space of Volterra-type integral operators on Fock spaces endowed with the operator norm. We proved that the space has the same connected and path connected components which is the set of all compact operators acting on the Fock spaces. We also obtained a characterization of isolated points of the space of the operators and showed that there exists no essentially isolated Volterra-type integral operator.

Small Hankel operators on generalized Fock spaces

We consider Fock spaces $F^{p,\ell}_α$ of entire functions on ${\mathbb C}$ associated to the weights $e^{-α|z|^{2\ell}}$, where $α>0$ and $\ell$ is a positive integer. We compute explicitly the corresponding Bergman kernel associated to $F^{2,\ell}_α$ and, using an adequate factorization of this kernel, we characterize the boundedness and the compactness of the small Hankel operator $\mathfrak{h}^{\ell}_{b,α}$ on $F^{p,\ell}_α$. Moreover, we also determine when $\mathfrak{h}^{\ell}_{b,α}$ is a Hilbert-Schmidt operator on $F^{2,\ell}_α$.

Localization and Compactness in Bergman and Fock spaces

In this paper we study the compactness of operators on the Bergman space of the unit ball and on very generally weighted Bargmann-Fock spaces in terms of the behavior of their Berezin transforms and the norms of the operators acting on reproducing kernels. In particular, in the Bergman space setting we show how a vanishing Berezin transform combined with certain (integral) growth conditions on an operator $T$ are sufficient to imply that the operator is compact. In the weighted Bargmann-Fock space setting we show that the reproducing kernel thesis for compactness holds for operators satisfying…

Integral, differential and multiplication operators on generalized Fock spaces

Volterra companion integral and multiplication operators with holomorphic symbols are studied for a large class of generalized Fock spaces on the complex plane $\CC$. The weights defining these spaces are radial and subject to a mild smoothness condition. In addition, we assumed that the weights decay faster than the classical Gaussian weight. One of our main results show that there exists no nontrivial holomorphic symbols $g$ which induce bounded Volterra companion integral $I_g$ and multiplication operators $M_g$ acting between the weighted spaces. We also describe the bounded and compact Vo…

Fock 空间上的典范积分算子 II

Canonical integral operators on the Fock space II

本文继续研究 Fock 空间上的典范线性算子，证明其紧性蕴含属于所有 Schatten 类，并给出奇异值、范数与迹公式，同时建立 $F^p$ 与 $F^2$ 上有界性和紧性的等价关系。

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。

Feynman integral for functional Schrödinger equations

We consider functional Schrödinger equations associated with a wide class of Hamiltonians in all Fock representations of the bosonic canonical commutation relations, in particular the Cook-Fock, Friedrichs-Fock, and Bargmann-Fock models. An infinite-dimensional symbolic calculus allows us to prove the convergence of the corresponding Hamiltonian Feynman integrals for propagators of coherent states.

A Boundedness Criterion for Singular Integral Operators of convolution type on the Fock Space

We show that for an entire function $φ$ belonging to the Fock space ${\mathscr F}^2(\mathbb{C}^n)$ on the complex Euclidean space $\mathbb{C}^n$, the integral operator \begin{eqnarray*} S_φF(z)=\int_{\mathbb{C}^n} F(w) e^{z \cdot\bar{w}} φ(z- \bar{w})\,dλ(w), \ \ \ \ \ z\in \mathbb{C}^n, \end{eqnarray*} is bounded on ${\mathscr F}^2(\mathbb{C}^n)$ if and only if there exists a function $m\in L^{\infty}(\mathbb{R}^n)$ such that $$ φ(z)=\int_{\mathbb{R}^n} m(x)e^{-2\left(x-\frac{i}{2} z \right)\cdot \left(x-\frac{i}{2} z \right)} dx, \ \ \ \ \ \ z\in \mathbb{C}^n. $$ Here $dλ(w)= π^{-n}e^{-\left…

多解析Fock空间的量子调和分析

Quantum harmonic analysis for polyanalytic Fock spaces

本研究构建可约情形下的量子调和分析框架并应用于多解析Fock空间，解释了已有文献中的观测现象、解答相关开放问题，推导Toeplitz量子化核的显式刻画，证明Berezin变换的单射性，并给出$\u2702_1$-代数的多种刻画。

The Smirnov classes for the Fock space and complete Pick spaces

For a Hilbert function space $\mathcal H$ the Smirnov class $\mathcal N^+(\mathcal H)$ is defined to be the set of functions expressible as a ratio of bounded multipliers of $\mathcal H$, whose denominator is cyclic for the action of $Mult(\mathcal H)$. It is known that for spaces $\mathcal H$ with complete Nevanlinna-Pick (CNP) kernel, the inclusion $\mathcal H\subset \mathcal N^+(\mathcal H)$ holds. We give a new proof of this fact, which includes the new conclusion that every $h\in\mathcal H$ can be expressed as a ratio $b/a\in\mathcal N^+(\mathcal H)$ with $1/a$ already belonging to $\math…

Toeplitz operators defined by sesquilinear forms: Fock space case

The classical theory of Toeplitz operators in spaces of analytic functions deals usually with symbols that are bounded measurable functions on the domain in question. A further extension of the theory was made for symbols being unbounded functions, measures, and compactly supported distributions, all of them subject to some restrictions. In the context of a reproducing kernel Hilbert space we propose a certain framework for a `maximally possible' extension of the notion of Toeplitz operators for a `maximally wide' class of `highly singular' symbols. Using the language of sesquilinear forms we …

Atomic decompositions for operators in reproducing kernel Hilbert spaces

In this paper, we give some results concerning atomic decompositions for operators on reproducing kernel Hilbert spaces, using frame theory techniques. We provide also generalizations for atomic decompositions of some theorems for reproducing kernel Hilbert spaces. In particular, we obtain atomic decomposition results for operators on Bergman spaces and Fock spaces in a simple manner.

Hankel Bilinear forms on generalized Fock-Sobolev spaces on ${\mathbb C}^n$

We characterize the boundedness of Hankel bilinear forms on a product of generalized Fock-Sobolev spaces on ${\mathbb C}^n$ with respect to the weight $(1+|z|)^ρe^{-\fracα2|z|^{2\ell}}$, for $\ell\ge 1$, $α>0$ and $ρ\in{\mathbb R}$. We obtain a weak decomposition of the Bergman kernel with estimates and a Littlewood-Paley formula, which are key ingredients in the proof of our main results. As an application, we characterize the boundedness, compactness and the membership in the Schatten class of small Hankel operators on these spaces.

Toeplitz算子在带有$A_{\infty}$型权重的加权Fock空间上

Toeplitz operators on weighted Fock spaces with $A_{\infty}$-type weights

本文研究了在带有$A_{\infty}$型权重的加权Fock空间上Toeplitz算子的有界性、紧致性及Schatten $p$-类性质，并探讨了其在Volterra算子和加权复合算子中的应用。

Bosonic kernels

We show that the bosonic Fock representation of a complex Hilbert space admits a purely algebraic kernel calculus; as an illustration, we use it to reproduce the standard integral kernel formulae for metaplectic operators within the complex-wave representation.

$L^p$ bounds for singular integrals and maximal singular integrals with rough kernels

Convolution type Calderón-Zygmund singular integral operators with rough kernels $\pv \Om(x)/|x|^n$ are studied. A condition on $\Om$ implying that the corresponding singular integrals and maximal singular integrals map $L^p \to L^p$ for $1<p<\nf$ is obtained. This condition is shown to be different from the condition $\Om\in H^1(\sn)$.

具有广义核的多线性分数次积分算子

Multilinear Fractional Integral Operators with Generalized Kernels

本文研究核条件弱于Dini条件的多线性分数次积分算子，证明了其在加权Lebesgue空间和变指数Lebesgue空间上的有界性，并首次建立了相应多线性交换子的有界性结果。

On Integral Operators with Operator Valued Kernels

Here Lq-Lp boundedness of integral operator with operator-valued kernels is studied and the main result is applied to convolution operators. Using these results Besov space regularity for Fourier multiplier operator is established.

福克空间上识别托普利兹算子的傅里叶准则

A Fourier Criterion for the Toeplitzness of Operators on Fock Spaces

给出福克空间上识别有界可测符号托普利兹算子的傅里叶准则，通过反对角限制的傅里叶逆变换恢复符号，还用于刻画加权复合算子和广义沃尔泰拉型算子的托普利兹可表示性。

Spectrum of partial integral operators with degenerate kernel

In the paper we consider self-adjoint partial integral operators of Fredholm type $T$ with a degenerate kernel on the space $L_2([a,b]\times[c,d]).$ Essential and discrete spectra of $T$ are described.

Grand Lebesgue 空间中的齐次核积分算子

Homogeneous kernel integral operators in Grand Lebesgue Spaces

本文研究作用于两个 Grand Lebesgue 空间之间、带齐次核的多重线性积分算子，给出其算子范数的估计与精确计算。

Singular integrals with product kernels associated with mixed homogeneities and Hardy spaces

This paper is motivated by Phong and Stein's paper on non-standard singular integrals with mixed homogeneities. Our purpose is to study these new non-standard convolution singular integrals and establish the boundedness of these singular integrals on the Hardy spaces.

Hardy核诱导的积分算子的加权不等式

Weighted inequality of integral operators induced by Hardy kernels

针对双倍权，本文得到单位圆盘上Hardy核诱导积分算子单权不等式成立的充要条件，证实了Guo和Wang在该情形下的猜想。

多解析Fock核的水平Fourier变换

Horizontal Fourier transform of the polyanalytic Fock kernel

本文研究多解析Fock空间中与水平Weyl平移可交换的算子代数，通过计算水平Fourier变换将空间分解为Hermite函数乘积，并证明其等距同构于向量函数空间，算子代数同构于矩阵函数代数。

Fock空间上复对称加权复合算子的刻画

A Characterization of Complex Symmetric Weighted Composition Operators on the Fock Space

该研究未预先指定共轭算子，借助再生核与生成函数刻画Fock空间上有界复对称加权复合算子，所得共轭算子多为反线性高斯积分算子，部分为加权复合共轭算子。

Mixed multidimensional integral operators with piecewise constant kernels and their representations

We consider the algebra of mixed multidimensional integral operators. In particular, Fredholm integral operators of the first and second kind belongs to this algebra. For the piecewise constant kernels we provide an explicit representation of the algebra as a product of simple matrix algebras. This representation allows us to compute the inverse operators (or to solve the corresponding integral equations) and to find the spectrum explicitly. Moreover, explicit traces and determinants are also constructed. So, roughly speaking, the analysis of integral operators is reduced to the analysis of ma…

Generalized Integral Operators and Schwartz Kernel Theorem

In connection with the classical Schwartz kernel theorem, we show that in the framework of Colombeau generalized functions a large class of linear mappings admit integral kernels. To do this, we need to introduce news spaces of generalized functions with slow growth and the corresponding adapted linear mappings. Finally, we show that in some sense Schwartz' result is contained in our main theorem.

Multiple Weighted Estimates for Multilinear Commutators of Multilinear Singular Integrals with Generalized Kernels

In this paper, the weighted Lp boundedness of multilinear commutators and iterated commutators of multilinear singular integral operators with generalized kernels is established, where the weight is multiple weight. Our results are generalizations of the corresponding results for multilinear singular integral operators with standard kernels and Dini kernels.

Optimality of function spaces for kernel integral operators

We explore boundedness properties of kernel integral operators acting on rearrangement-invariant (r.i.) spaces. In particular, for a given r.i. space $X$ we characterize its optimal range partner, that is, the smallest r.i. space $Y$ such that the operator is bounded from $X$ to $Y$. We apply the general results to Lorentz spaces to illustrate their strength.

Positive fixed points of Hammerstein's integral operators with degenerate kernel

In this paper we study positive fixed points of Hammerstein integral operators with degenerate kernel in the cone of C[0, 1]. Problem on a number of positive fixed points of the Hammerstein integral operator leads to the study positive roots of polynomials with real coefficients. Consider a model on a Cayley tree with nearest-neighbor interactions and with the set [0, 1] of spin values. The uniqueness translational-invariant Gibbs measures for the given model is proved.

Singular Integrals with Flag Kernels on Homogeneous Groups: I

Let $\mathcal K$ be a flag kernel on a homogeneous nilpotent Lie group $G$. We prove that operators $T$ of the form $T(f)= f*\mathcal K$ form an algebra under composition, and that such operators are bounded on $L^{p}(G)$ for $1<p<\infty$.

L²上带有柯西核的本质半交换奇异积分算子

Essentially semi-commuting singular integral operators with Cauchy kernel on $L^{2}$

该研究刻画了L²上两个柯西奇异积分算子的半交换子为紧算子的条件，通过Hankel算子等工具研究柯西奇异积分算子，得到其本质正规的充要条件。

Fractional type multilinear commutators generated by fractional integral with rough variable kernel and local Campanato functions on generalized vanishing local Morrey spaces

In this paper, we consider the boundedness of fractional type multilinear commutators generated by fractional integral with rough variable kernel and local Campanato functions on both generalized local (central) Morrey spaces and generalized vanishing local Morrey spaces, under generic size conditions which are satisfied by most of the operators in harmonic analysis, respectively.

关于积分核算子的最优端点

On optimal endpoints for integral kernel operators

研究单变量函数上积分核算子行为，在重排不变拟范数赋范的巴拿赫空间类中找最优端点空间候选者，刻画非增重排具体逐点估计的存在性并构建理论，以此为一些积分核算子提出最优端点空间。

Integral operators with infinitely smooth bi-Carleman kernels of Mercer type

With the aim of applications to solving general integral equations, we introduce and study in this paper a special class of bi-Carleman kernels on $\mathbb{R}\times\mathbb{R}$, called $K^\infty$ kernels of Mercer type, whose property of being infinitely smooth is stable under passage to certain left and right multiples of their associated integral operators. An expansion theorem in absolutely and uniformly convergent bilinear series concerning kernels of this class is proved extending to a general non-Hermitian setting both Mercer's and Kadota's Expansion Theorems for positive definite kernels…

Integral representations of unbounded operators by arbitrarily smooth Carleman kernels

In this paper, we give a characterization of all closed linear operators in a separable Hilbert space which are unitarily equivalent to an integral operator in $L_2(R)$ with bounded and arbitrarily smooth Carleman kernel on $R^2$. In addition, we give an explicit construction of corresponding unitary operators.

Kernels of Integral Equations Can Be Boundedly Infinitely Differentiable on $\mathbb{R}^2$

In this paper, we reduce the general linear integral equation of the third kind in $L^2(Y,μ)$, with largely arbitrary kernel and coefficient, to an equivalent integral equation either of the second kind or of the first kind in $L^2(\mathbb{R})$, with the kernel being the linear pencil of bounded infinitely differentiable bi-Carleman kernels expandable in absolutely and uniformly convergent bilinear series. The reduction is done by using unitary equivalence transformations.

Vanishing generalized Morrey spaces and commutators of Marcinkiewicz integrals with rough kernel associated with Schrödinger operator

Let L=-Δ+V(x) be a Schrodinger operator, where Δ is the Laplacian on while nonnegative potential V(x) belonging to the reverse Holder class. We establish the boundedness of the commutators of Marcinkiewicz integrals with rough kernel associated with Schrodinger operator on vanishing generalized Morrey spaces.

Integral representations of closed operators as bi-Carleman operators with arbitrarily smooth kernels

In this paper, we characterize all closed linear operators in a separable Hilbert space which are unitarily equivalent to an integral bi-Carleman operator in $L_2(R)$ with bounded and arbitrarily smooth kernel on $R^2$. In addition, we give an explicit construction of corresponding unitary operators. The main result is a qualitative sharpening of an earlier result of [5].

Splitting of Volterra Integral Operators with Degenerate Kernels

Volterra integral operators with non-sign-definite degenerate kernels $A(x,t)= \sum_{k=0}^n A_k(x,t)$, $A_k(x,t)= a_k (x) t^k$, are studied acting from one weighted $L_2$ space on $(0,+\infty)$ to another. Imposing an integral doubling condition on one of the weights, it is shown that the operator with the kernel $A(x,t)$ is bounded if and only $n+1$ operators with kernels $A_k(x,t)$ are all bounded. We apply this result to describe spaces of pointwise multipliers in weighted Sobolev spaces on $(0,+\infty)$.

Generalized local Morrey spaces and fractional integral operators with rough kernel

Let $M_{Ω,\a}$ and $I_{Ω,\a}$ be the fractional maximal and integral operators with rough kernels, where $0 < \a < n$. In this paper, we shall study the continuity properties of $M_{Ω,\a}$ and $I_{Ω,\a}$ on the generalized local Morrey spaces $LM_{p,φ}^{x_0}$. The boundedness of their commutators with local Campanato functions is also obtained.

Continuity properties of integral kernels associated with Schroedinger operators on manifolds

For Schroedinger operators (including those with magnetic fields) with singular (locally integrable) scalar potentials on manifolds of bounded geometry, we study continuity properties of some related integral kernels: the heat kernel, the Green function, and also kernels of some other functions of the operator. In particular, we show the joint continuity of the heat kernel and the continuity of the Green function outside the diagonal. The proof makes intensive use of the Lippmann-Schwinger equation.

广义 $q$-Fock 空间与结构恒等式

Generalized $q$-Fock spaces and structural identities

本文利用 $q$-calculus 研究了一族在 Hardy 空间与 Fock 空间间插值的再生核 Hilbert 空间，通过经典算子及其 $q$-calculus 对应算子对其进行刻画，并借此研究了经典后向移位算子与 q-Jackson 导数间的交织算子。

特征值问题中具有可分核的积分算子到矩阵的变换

Transformation from integral operator with separable kernel to matrix in eigenvalue problem

本文针对可分核积分算子的特征值问题，建立其特征值、特征函数与矩阵特征对的关联，证明第二类Fredholm积分方程可转化为矩阵特征计算问题，并用算例验证结论。

Solution to the Volterra integral equations of the first kind with piecewise continuous kernels in class of Sobolev-Schwartz distributions

Sufficient conditions for existence and uniqueness of the solution of the Volterra integral equations of the first kind with piecewise continuous kernels are derived in framework of Sobolev-Schwartz distribution theory. The asymptotic approximation of the parametric family of generalized solutions is constructed. The method for the solution's regular part refinement is proposed using the successive approximations method.