Brett D. Wick, Shengkun Wu
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.
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Fourier multipliers and pseudo-differential operators on Fock-Sobolev spaces
本文研究了$L^2(\mathbb{R}^n)$上有界算子经Bargmann变换导出的Fock空间算子，确定了对应Fourier乘子与伪微分算子的算子，并分析了其在Fock-Sobolev空间上的有界性。
We extend the de Branges-Beurling theorem characterizing the shift-invariant spaces boundedly contained in the Hardy space of square-summable power series to the full Fock space over $\mathbb{C} ^d$. Here, the full Fock space is identified as the \emph{Non-commutative (NC) Hardy Space} of square-summable Taylor series in several non-commuting variables. We then proceed to study lattice operations on NC kernels and operator-valued multipliers between vector-valued Fock spaces. In particular, we demonstrate that the operator-valued Fock space multipliers with common coefficient range space form …
Bounded multiplier algebras arising from Fock representation associated to semigroups
本文引入与左消半群Fock表示相关的乘子代数，证明其为含单位Banach代数，群情形为C*-代数，并刻画了与Hardy代数及非交换Hardy代数的等距同构，最终利用算子空间技术证明其为算子代数。
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…
The Lifting Property for Frame Multipliers and Toeplitz Operators
本文证明正符号框架乘子是coorbit空间间的Banach同构，推广提升定理至Gabor框架和Fock空间，得到调制空间与加权Fock空间的新同构结果。
We study the fractal uncertainty principle in the joint time-frequency representation, and we prove a version for the Short-Time Fourier transform with Gaussian window on the modulation spaces. This can equivalently be formulated in terms of projection operators on the Bargmann-Fock spaces of entire functions. Specifically for signals in $L^2(\mathbb{R}^d)$, we obtain norm estimates of Daubechies' time-frequency localization operator localizing on porous sets. The proof is based on the maximal Nyquist density of such sets, and for multidimensional Cantor iterates we derive explicit upper bound…
We study the range of time-frequency localization operators acting on modulation spaces and prove a lifting theorem. As an application we also characterize the range of Gabor multipliers, and, in the realm of complex analysis, we characterize the range of certain Toeplitz operators on weighted Bargmann-Fock spaces. The main tools are the construction of canonical isomorphisms between modulation spaces of Hilbert-type and a refined version of the spectral invariance of pseudodifferential operators. On the technical level we prove a new class of inequalities for weighted gamma functions.
We study a general metric constrained interpolation problem in a de Branges-Rovnyak space $\mathcal{H}(K_S)$ associated with a contractive multiplier $S$ between two Fock spaces along with its commutative counterpart, a de Branges-Rovnyak space associated with a Schur multiplier on the Drury-Arveson space of the unit ball of $\mathbb{C}^n$.
Let $L$ be a non-negative self-adjoint operator on $L^2(X)$. Assume that operator $L$ generates the analytic semigroup $\{e^{-tL}\}_{t>0}$ whose kernels satisfy the standard Gaussian upper bounds. This paper investigates the sharp weighted inequalities for spectral multipliers $F(L)$ and their commutators with BMO functions. The obtained results in this paper can be considered to be improvements of those in \cite{DSY}[Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers, {\it J. Funct. Anal.}, {\bf 260} (2011), 1106-1131].
Arveson's Gauss-Bonnet-Chern Formula for Hilbert Modules in the Multiplier-Algebra Framework
本文在乘子代数框架下，证明了对于单位球上正则酉不变完全Nevanlinna-Pick空间，Arveson的Gauss-Bonnet-Chern公式无条件成立，并通过有限缺陷子模的尖锐分类解决了Arveson有限缺陷问题，进而应用于向量值再生核Hilbert模的酉等价刚性。
Spectral multipliers in a general Gaussian setting
本文研究一般高斯设定下 Ornstein-Uhlenbeck 算子的谱乘子，证明右半平面 Laplace 变换型函数对应的算子关于不变测度为弱 (1,1) 型，并通过核估计、Mehler 核零点界和增强极大算子定理完成论证。
Let $L$ be a non-negative self adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type. Assume that $L$ generates a holomorphic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ have Gaussian upper bounds but possess no regularity in variables $x$ and $y$. In this article, we study weighted $L^p$-norm inequalities for spectral multipliers of $L$. We show sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate $L^2$ estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers…
This paper concerns harmonic analysis of the Ornstein--Uhlenbeck operator L on the Euclidean space. We examine the method of decomposing a spectral multiplier ϕ(L) into three parts according to the notion of admissibility, which quantifies the doubling behaviour of the underlying Gaussian measure γ. We prove that the above-mentioned admissible decomposition is bounded in L^p(γ) for 1 < p \leq 2 in a certain sense involving the Gaussian conical square function. The proof relates admissibility with E. Nelson's hypercontractivity theorem in a novel way.
Suppose $A$ is a positive real linear transformation on a finite dimensional complex inner product space $V$. The reproducing kernel for the Fock space of square integrable holomorphic functions on $V$ relative to the Gaussian measure $dμ_A(z)=\frac {\sqrt {\det A}} {π^n}e^{-{\rm Re}< Az,z>} dz$ is described in terms of the holomorphic--antiholomorphic decomposition of the linear operator $A$. Moreover, if $A$ commutes with a conjugation on $V$, then a restriction mapping to the real vectors in $V$ is polarized to obtain a Segal--Bargmann transform, which we also study in the Gaussian-measure …
Let $L$ be a non-negative self adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type. Assume that $L$ generates a holomorphic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy generalized $m$-th order Gaussian estimates. In this article, we study singular and dyadically supported spectral multipliers for abstract self-adjoint operators. We show that in this setting sharp spectral multiplier results follow from Plancherel or Stein-Tomas type estimates. These results are applicable to spectral multipliers for large classes of operators including $m$-th order elliptic di…
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。
We study a class of spectral multipliers ϕ(L) for the Ornstein--Uhlenbeck operator L arising from the Gaussian measure on R^n and find a sufficient condition for integrability of ϕ(L)f in terms of the admissible conical square function and a maximal function.
We characterize the non-commutative Aleksandrov--Clark measures and the minimal realization formulas of contractive and, in particular, isometric non-commutative rational multipliers of the Fock space. Here, the full Fock space over $\mathbb{C} ^d$ is defined as the Hilbert space of square--summable power series in several non-commuting formal variables, and we interpret this space as the non-commutative and multi-variable analogue of the Hardy space of square--summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov--Clark measur…
In this article we construct a generalized Gaussian process coming from Coxeter groups of type B. It is given by creation and annihilation operators on an $(α,q)$-Fock space, which satisfy the commutation relation $$ b_{α,q}(x)b_{α,q}^\ast(y)-qb_{α,q}^\ast(y)b_{α,q}(x)=\langle x, y\rangle I+α\langle \overline{x}, y \rangle q^{2N}, $$ where $x,y$ are elements of a complex Hilbert space with a self-adjoint involution $x\mapsto\bar{x}$ and $N$ is the number operator with respect to the grading on the $(α,q)$-Fock space. We give an estimate of the norms of creation operators. We show that the dist…
Let $ν_{α,q}$ be the probability and orthogonality measure for the $q$-Meixner-Pollaczek orthogonal polynomials, which has appeared in \cite{BEH15} as the distribution of the $(α,q)$-Gaussian process (the Gaussian process of type B) over the $(α,q)$-Fock space (the Fock space of type B). The main purpose of this paper is to find the radial Bargmann representation of $ν_{α,q}$. Our main results cover not only the representation of $q$-Gaussian distribution by \cite{LM95}, but also of $q^2$-Gaussian and symmetric free Meixner distributions on $\mathbb R$. In addition, non-trivial commutation rel…
The full Fock space over $\mathbb C ^d$ can be identified with the free Hardy space, $H^2 (\mathbb B ^d _\mathbb N)$ - the unique non-commutative reproducing kernel Hilbert space corresponding to a non-commutative Szegö kernel on the non-commutative, multi-variable open unit ball $\mathbb B ^d _\mathbb N := \bigsqcup _{n=1} ^\infty \left( \mathbb C^{n\times n} \otimes \mathbb C ^d \right) _1$. Elements of this space are free or non-commutative functions on $\mathbb B ^d _\mathbb N$. Under this identification, the full Fock space is the canonical non-commutative and several-variable analogue of…
We study the boundedness of Toeplitz operators on Segal-Bargmann spaces in various contexts. Using Gutzmer's formula as the main tool we identify symbols for which the Toeplitz operators correspond to Fourier multipliers on the underlying groups. The spaces considered include Fock spaces, Hermite and twisted Bergman spaces and Segal-Bargmann spaces associated to Riemannian symmetric spaces of compact type.
We obtain a general Marcinkiewicz-type multiplier theorem for mixed systems of strongly commuting operators $L=(L_1,...,L_d);$ where some of the operators in $L$ have only a holomorphic functional calculus, while others have additionally a Marcinkiewicz-type functional calculus. Moreover, we prove that specific Laplace transform type multipliers of the pair $(\mathcal{L},A)$ are of certain weak type $(1,1).$ Here $\mathcal{L}$ is the Ornstein-Uhlenbeck operator while $A$ is a non-negative operator having Gaussian bounds for its heat kernel. Our results include the Riesz transforms $A(\mathcal{…
Let $X$ be a space of homogeneous type and let $L$ be a nonnegative self-adjoint operator on $L^2(X)$ which satisfies a Gaussian estimate on its heat kernel. In this paper we prove a Hömander type spectral multiplier theorem for $L$ on the Besov and Triebel--Lizorkin spaces associated to $L$. Our work not only recovers the boundedness of the spectral multipliers on $L^p$ spaces and Hardy spaces associated to $L$, but also is the first one which proves the boundedness of a general spectral theorem on Besov and Triebel--Lizorkin spaces.
A rational function belongs to the Hardy space, $H^2$, of square-summable power series if and only if it is bounded in the complex unit disk. Any such rational function is necessarily analytic in a disk of radius greater than one. The inner-outer factorization of a rational function, $\mathfrak{r} \in H^2$ is particularly simple: The inner factor of $\mathfrak{r}$ is a (finite) Blaschke product and (hence) both the inner and outer factors are again rational. We extend these and other basic facts on rational functions in $H^2$ to the full Fock space over $\mathbb{C}^d$, identified as the \emph{…
We consider operators acting on a UMD Banach lattice $X$ that have the same algebraic structure as the position and momentum operators associated with the harmonic oscillator $-\frac12Δ+ \frac12|x|^{2} $ acting on $L^{2}(\mathbb{R}^{d})$. More precisely, we consider abstract harmonic oscillators of the form $\frac12 \sum _{j=1} ^{d}(A_{j}^{2}+B_{j}^{2})$ for tuples of operators $A=(A_{j})_{j=1} ^{d}$ and $B=(B_{k})_{k=1} ^{d}$, where $iA_j$ and $iB_k$ are assumed to generate $C_{0}$ groups and to satisfy the canonical commutator relations. We prove functional calculus results for these abstrac…
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。
Let $X$ be a space of homogeneous type and let $L$ be an injective, non-negative, self-adjoint operator on $L^2(X)$ such that the semigroup generated by $-L$ fulfills Davies-Gaffney estimates of arbitrary order. We prove that the operator $F(L)$, initially defined on $H^1_L(X)\cap L^2(X)$, acts as a bounded linear operator on the Hardy space $H^1_L(X)$ associated with $L$ whenever $F$ is a bounded, sufficiently smooth function. Based on this result, together with interpolation, we establish Hörmander type spectral multiplier theorems on Lebesgue spaces for non-negative, self-adjoint operators …
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。
Sharp Gaussian Mixed-Norm Theory for Fock Spaces
本文发展高斯混合范数Fock空间的结构理论，证明最优圆周均值估计，刻画嵌入、原子分解、Carleson测度及对偶，并给出定量重中心定理。
The Smirnov class for the classical Hardy space is the set of ratios of bounded analytic functions on the open complex unit disk with outer denominators. This definition extends naturally to the commutative and non-commutative multi-variable settings of the Drury-Arveson space and the full Fock space over $\mathbb C ^d$. Identifying the Fock space with the free multi-variable Hardy space of non-commutative or free holomorphic functions on the non-commutative open unit ball, we prove that any closed, densely-defined operator affiliated to the right free multiplier algebra of the full Fock space…
The Fock space consists of all entire functions which are square integrable with respect to Gauss measure. The Toeplitz algebra is the C*-algebra generated by the Toeplitz operator with bounded symbol on the Fock space. In this paper, we study the Toeplitz algebra on the Fock space.
Let L be a non-negative, self-adjoint operator on L^2(Ω), where (Ω, d μ) is a space of homogeneous type. Assume that the semigroup {T_t}_{t>0} generated by -L satisfies Gaussian bounds, or more generally Davies-Gaffney estimates. We say that f belongs to the Hardy space H^1_L if the square function S_h f(x)=(\iint_{Γ(x)} |t^2 L e^{-t^2 L} f(y)|^2 \frac{dμ(y)}{μ(B_d(x,t))} \frac{dt}{t})^{1/2} belongs to L^1(Ω, dμ), where Γ(x)={(y,t) \in Ω\times (0,\infty): d(x,y)<t}. We prove spectral multiplier theorems for L on H^1_L.
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…
Optimal stability of regularized spectral differentiation in Sobolev spaces
该研究针对带噪数据下Sobolev空间函数的稳定谱微分问题，引入容许傅里叶乘子，证明正则化微分算子达极小极大最优稳定性速率，扩展了相关结果并验证了多种乘子的表现。
It has long been known that the differential operator $D$ represents a typical examples of unbounded operators in many Banach spaces including the classical Fock spaces, the Fock--Sobolev spaces, and the generalized Fock spaces where the weight decays faster than the Gaussian weight. In this note we identify Fock type spaces where the operator admits some basic spectral structures including compactness and membership in the Schatten $\mathcal{S}_p$ classes. We also showed that its nontrivial spectrum while acting on such spaces is precisely the closed unit disk $\D$ in the complex plane.
Sampling properties of the zeroes of the Gaussian entire function
本文研究Fock空间上高斯整函数零点的采样性质，通过放宽密度与分离条件并估计邻近零点数量，证明了使用d+o(d)个点的随机采样不等式及采样常数的次多项式增长，并恢复了零点几乎必然为唯一性集的结果。
We show that for all q in the interval (-1,1), the Fock representation of the q-commutation relations can be unitarily embedded into the Fock representation of the extended Cuntz algebra. In particular, this implies that the C*-algebra generated by the Fock representation of the q-commutation relations is exact. An immediate consequence is that the q-Gaussian von Neumann algebra is weakly exact for all q in the interval (-1,1).
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.…
Rational Cuntz states peak on the free disk algebra
本文利用 Fock 空间非交换有理乘子的实现理论分析 Cuntz 代数上的态，证明由等距 NC 有理乘子的非交换 Clark 测度产生的一类 Cuntz 态是自由圆盘代数的峰值态。
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.
We consider the $\dbar-$ equation in $\C^1$ in classes of functions with Gaussian decay at infinity. We prove that if the right-hand side of the equation is majorated by $\exp(-q|z|^2)$, with some positive $q$, together with derivatives up to some order, and is orthogonal, as a distribution, to all analytical polynomials, then there exists a solution with decays, together with derivatives, as $\exp(-q'|z|^2)$, for any $q'<q/e$. This result carries over to the $\dbar$-equation in classes of distributions, again, with Gaussian decay at infinity, in some precisely defined sense. The properties of…
Sampling and Interpolation in Gaussian Mixed-Norm Fock Spaces
本文为高斯混合范数Fock空间建立了采样与插值的尖锐判据，以经典密度α/π为临界值，并给出有界线性右逆，推广了经典Hilbert Fock空间理论。
Representer Theorem in Complex Reproducing Kernel Hilbert Spaces with Applications to Fock and Hardy Spaces and Superoscillations
本文探讨复值再现核希尔伯特空间中的代表定理，应用于福克空间、哈代空间及超振荡问题，揭示超振荡与机器学习核方法的联系。
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…
We study various properties of composition operators acting between generalized Fock spaces $\mathcal{F}_φ^p$ and $\mathcal{F}_φ^q$ with weight functions $φ$ grow faster than the classical Gaussian weight function $\frac{1}{2}|z|^2$ and satisfy some mild smoothness conditions. We have shown that if $p\neq q,$ then the composition operator $C_ψ: \mathcal{F}_φ^p \to \mathcal{F}_φ^q $ is bounded if and only if it is compact. This result shows a significance difference with the analogous result for the case when $C_ψ$ acts between the classical Fock spaces or generalized Fock spaces where the weig…
We review the recent results on the Jacobi field of a (real-valued) Lévy process defined on a Riemannian manifold. In the case where the Lévy process is neither Gaussian, nor Poisson, the corresponding Jacobi field acts in an extended Fock space. We also give a unitary equivalent representation of the Jacobi field in a usual Fock space. This representation is inspired by a result by Accardi, Franz, and Skeide (2002).
A complex-analytic proof of square-restricted stable phase retrieval in Fock space
该研究通过复分析方法，结合加权导数范数、分部积分等技术，在Lean中借助大语言模型完成验证，证明了一维Fock空间中高斯点处平方限制形式的局部稳定相位检索的强制性不等式。
Thermal States on Mittag-Leffler Fock Space of the Slitted Plane
本文在狭缝平面的 Mittag-Leffler Fock 空间上构造并研究数态和热态，推广了量子理论基本算符与物理态，为更一般量子态理论奠定基础。
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