A Hardy space analysis of the Báez-Duarte criterion for the RH

Three Classification Results In The Theory Of Weighted Hardy Spaces On The Ball

We present a natural family of Hilbert function spaces on the d-dimensional complex unit ball and classify which of them satisfy that subsets of the ball yield isometrically isomorphic subspaces if and only if there is an analytic automorphism of the ball taking one to the other. We also characterize pairs of weighted Hardy spaces on the unit disk which are isomorphic via a composition operator by a simple criterion on their respective sequences of weights.

支持ζ函数无零点半平面的Hardy空间逼近

A Hardy space approximation supporting zero-free half-planes for the $ζ$-function

基于Bagchi的Báez-Duarte准则，证明若E属于G_k在H^2(C_α)中的闭线性张成，则ζ在C_α中无零点，并数值分析序列s_n。

An atomic decomposition characterization of flag Hardy spaces $H^p_F({\rr}^{n}\times{\rr}^m)$ with applications

In this paper, we give an atomic decomposition characterization of flag Hardy spaces $H^p_F({\rr}^n\times {\rr}^m)$ for $0<p\le 1$, which were introduced in \cite{hl1}. A remarkable feature of atoms of such flag Hardy spaces is that these atoms have only partial cancellation conditions. As an application, we prove a boundedness criterion for operators on flag Hardy spaces.

Criterion of the boundedness of singular integrals on spaces of homogeneous type

It was well known that geometric considerations enter in a decisive way in many questions of harmonic analysis. The main purpose of this paper is to provide the criterion of the boundedness for singular integrals on the Hardy spaces and as well as on its dual, particularly on $\bmo$ for spaces of homogeneous type $(X, d,μ)$ in the sense of Coifman and Weiss, where the quasi-metric $d$ may have no regularity and the measure $μ$ satisfies only the doubling property. We make no additional geometric assumptions on the quasi-metric or the doubling measure and thus, the results of this paper extend …

Real-Variable Characterizations of Orlicz-Slice Hardy Spaces

In this article, the authors first introduce a class of Orlicz-slice spaces which generalize the slice spaces recently studied by P. Auscher et al. Based on these Orlicz-slice spaces, the authors introduce a new kind of Hardy type spaces, the Orlicz-slice Hardy spaces, via the radial maximal functions. This new scale of Orlicz-slice Hardy spaces contains the variant of the Orlicz-Hardy space of A. Bonami and J. Feuto as well as the Hardy-amalgam space of Z. V. de P. Ablé and J. Feuto as special cases. Their characterizations via the atom, the molecule, various maximal functions, the Poisson in…

Universal Toeplitz operators on the Hardy space over the polydisk

The Invariant Subspace Problem (ISP) for Hilbert spaces asks if every bounded linear operator has a non-trivial closed invariant subspace. Due to the existence of universal operators (in the sense of Rota), the ISP may be solved by describing the invariant subspaces of these operators alone. We characterize all anaytic Toeplitz operators $T_ϕ$ on the Hardy space $H^2(\mathbb{D}^n)$ over the polydisk $\mathbb{D}^n$ for $n>1$ whose adjoints satisfy the Caradus criterion for universality, that is, when $T_ϕ^*$ is surjective and has infinite dimensional kernel. In particular if $ϕ$ in a non-consta…

Molecular Characterizations of Variable Anisotropic Hardy Spaces with Applications to Boundedness of Calderón-Zygmund Operators

Let $p(\cdot):\ \mathbb{R}^n\to(0,\infty]$ be a variable exponent function satisfying the globally log-Hölder continuous condition and $A$ a general expansive matrix on $\mathbb{R}^n$. Let $H_A^{p(\cdot)}(\mathbb{R}^n)$ be the variable anisotropic Hardy space associated with $A$ defined via the non-tangential grand maximal function. In this article, via the known atomic characterization of $H_A^{p(\cdot)}(\mathbb{R}^n)$, the author establishes its molecular characterization with the known best possible decay of molecules. As an application, the author obtains a criterion on the boundedness of …

Compact embeddings of model subspaces of the Hardy space

We study embeddings of model (star-invariant) subspaces $K^p_Θ$ of the Hardy space $H^p$, associated with an inner function $Θ$. We obtain a criterion for the compactness of the embedding of $K^p_Θ$ into $L^p(μ)$ analogous to the Volberg--Treil theorem on bounded embeddings and answer a question posed by Cima and Matheson. The proof is based on Bernstein inequalities for functions in $K^p_Θ$. Also we study measures $μ$ such that the embedding operator belongs to a Schatten--von Neumann ideal.

The Characterizations of Anisotropic Mixed-Norm Hardy Spaces on $\mathbb{R}^n$ by Atoms and Molecules

Let $\vec{p}\in(0,\,\infty)^n$, $A$ be an expansive dilation on $\mathbb{R}^n$,and $H^{\vec{p}}_A({\mathbb {R}}^n)$ be the anisotropic mixed-norm Hardy space defined via the non-tangential grand maximal function studied by \cite{hlyy20}. In this paper, the authors establish new atomic and molecular decompositions of $H^{\vec{p}}_A({\mathbb {R}}^n)$. As an application, the authors obtain a boundedness criterion for a class of linear operators from $H^{\vec{p}}_{A}(\mathbb{R}^n)$ to $H^{\vec{p}}_{A}(\mathbb{R}^n)$. Part of results are still new even in the classical isotropic setting (in the cas…

离散图上拟线性算子的Hardy权空间

The space of Hardy-weights for quasilinear operators on discrete graphs

本文在一般加权图上研究p-Schrödinger算子的Hardy不等式，通过广义容量给出Hardy权空间的Maz'ya型刻画，并利用临界性理论探讨极小化元存在性及其与Cheeger型估计的联系。

The Periodic Dilation Completeness Problem: Cyclic vectors in the Hardy space over the infinite-dimensional polydisk

The classical completeness problem raised by Beurling and independently by Wintner asks for which $ψ\in L^2(0,1)$, the dilation system $\{ψ(kx):k=1,2,\cdots\}$ is complete in $L^2(0,1)$, where $ψ$ is identified with its extension to an odd $2$-periodic function on $\mathbb{R}$. This difficult problem is nowadays commonly called as the Periodic Dilation Completeness Problem (PDCP). By Beurling's idea and an application of the Bohr transform, the PDCP is translated as an equivalent problem of characterizing cyclic vectors in the Hardy space $\mathbf{H}_\infty^2$ over the infinite-dimensional pol…

矩阵加权 Hardy 空间的极大函数与原子刻画及其在 Calderón--Zygmund 算子有界性中的应用

Maximal Function and Atomic Characterizations of Matrix-Weighted Hardy Spaces with Their Applications to Boundedness of Calderón--Zygmund Operators

本文通过矩阵加权 grand 非切向极大函数引入矩阵加权 Hardy 空间 $H^p_W$，利用多种极大函数与原子对其进行刻画，并应用有限原子刻画证明了 Calderón--Zygmund 算子在该空间上的有界性。

Real-Variable Characterizations of New Anisotropic Mixed-Norm Hardy Spaces

Let $\vec{p}\in(0,\infty)^n$ and $A$ be a general expansive matrix on $\mathbb{R}^n$. In this article, via the non-tangential grand maximal function, the authors first introduce the anisotropic mixed-norm Hardy spaces $H_A^{\vec{p}}(\mathbb{R}^n)$ associated with $A$ and then establish their radial or non-tangential maximal function characterizations. Moreover, the authors characterize $H_A^{\vec{p}}(\mathbb{R}^n)$, respectively, by means of atoms, finite atoms, Lusin area functions, Littlewood-Paley $g$-functions or $g_λ^\ast$-functions via first establishing an anisotropic Fefferman-Stein ve…

Boundedness criterion for sublinear operators and commutators on generalized mixed Morrey spaces

In this paper, the author studies the boundedness for a large class of sublinear operator $T_α, α\in[0,n)$ generated by Calder{ó}n-Zygmund operators ($α=0$) and generated by fractional integral operator ($α>0$) on generalized mixed Morrey spaces $M^φ_{\vec{q}}(\Bbb R^n)$. Moreover, the boundeness for the commutators of $T_α, α\in[0,n)$ on generalized mixed Morrey spaces $M^φ_{\vec{q}}(\Bbb R^n)$ is also studied. As applications, we obtain the boundedness for Hardy-Littlewood maximal operator, Calderón-Zygmund singular integral operators, fractional integral operator, fractional maximal operato…

哈代型空间的平方函数刻画

Square function characterization of Hardy-type spaces

研究在特定假设下将哈代空间的刻画推广到哈代型空间\(X_A\)，并得出复合算子\(C_\varphi: \mathcal{B}^\omega \to X_A\)有界性和紧性的精确准则，推广了一维相关结果。

乘积Hardy空间在同型空间上的研究：离散乘积Calderón型再生公式、原子刻画及乘积Calderón-Zygmund算子

Product Hardy Spaces on Spaces of Homogeneous Type: Discrete Product Calderón-Type Reproducing Formula, Atomic Characterization, and Product Calderón--Zygmund Operators

本文研究了乘积Hardy空间的构造与性质，通过离散乘积Calderón型再生公式建立了原子刻画，并探讨了乘积Calderón-Zygmund算子的有界性。

Local Hardy Spaces of Musielak-Orlicz Type and Their Applications

Let $ϕ: \mathbb{R}^n\times[0,\fz)\rightarrow[0,\fz)$ be a function such that $ϕ(x,\cdot)$ is an Orlicz function and $ϕ(\cdot,t)\in A^{\mathop\mathrm{loc}}_{\infty}(\mathbb{R}^n)$ (the class of local weights introduced by V. S. Rychkov). In this paper, the authors introduce a local Hardy space $h_ϕ(\mathbb{R}^n)$ of Musielak-Orlicz type by the local grand maximal function, and a local $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{bmo}_ϕ(\mathbb{R}^n)$ which is further proved to be the dual space of $h_ϕ(\mathbb{R}^n)$. As an application, the authors prove that the class of pointwise multipli…

Dirichlet级数的Hardy-Hilbert空间上的复合-微分算子

Composition-differentiation operators on Hardy-Hilbert space of Dirichlet series

本文研究了Dirichlet级数Hardy-Hilbert空间上的复合-微分算子，建立了基于平均计数函数衰减的紧性判据，给出了零特征符号下的有界性充分条件，并推导了特定符号下的范数与逼近数估计及谱刻画。

Anisotropic Hardy-Lorentz Spaces and Their Applications

Let $p\in(0,1]$, $q\in(0,\infty]$ and $A$ be a general expansive matrix on $\mathbb{R}^n$. The authors introduce the anisotropic Hardy-Lorentz space $H^{p,q}_A(\mathbb{R}^n)$ associated with $A$ via the non-tangential grand maximal function and then establish its various real-variable characterizations in terms of the atomic or the molecular decompositions, the radial or the non-tangential maximal functions, or the finite atomic decompositions. All these characterizations except the $\infty$-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on $\mathbb{R}^n$…

Hardy空间上的广义Hilbert算子

Generalized Hilbert operators on Hardy spaces

该数学研究刻画了Hardy空间上广义Hilbert算子的有界性条件，否定了相关猜想，明确了乘子空间的包含关系与结构。

Boundedness of Fractional Integrals on Special John--Nirenberg--Campanato and Hardy-Type Spaces via Congruent Cubes

Let $p\in[1,\infty]$, $q\in[1,\infty)$, $s\in\mathbb{Z}_+:=\mathbb{N}\cup\{0\}$, and $α\in\mathbb{R}$. In this article, the authors first find a reasonable version $\widetilde{I}_β$ of the (generalized) fractional integral $I_β$ on the special John--Nirenberg--Campanato space via congruent cubes, $JN_{(p,q,s)_α}^{\mathrm{con}}(\mathbb{R}^n)$, which coincides with the Campanato space $\mathcal{C}_{α,q,s}(\mathbb{R}^n)$ when $p=\infty$. To this end, the authors introduce the vanishing moments up to order $s$ of $I_β$. Then the authors prove that $\widetilde{I}_β$ is bounded from $JN_{(p,q,s)_α}^…

关于算子的弱类型$(p,a)$准则及其应用

A weak type $(p,a)$ criterion for operators, and applications

本文研究了在齐型空间上算子的弱类型$(p,a)$准则，通过分析Riesz势和Riesz变换型算子的弱类型性质，以及谱乘子的$L^p-L^q$有界性，揭示了其在Hardy空间与BMO空间中的有界性。