Michał Strzelecki
Let $H$ be the Hardy operator and $I$ the identity operator acting on functions on the real half-line. We find optimal bounds for the operator $H - I$ in the setting of power weights and the cases of positive decreasing functions, positive functions, and general functions. As a byproduct, we obtain some results about the optimal relations between the norms of $H$ and its dual.
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Polarized Hardy--Stein identity
本文提出 Bregman 余散度及相关积分形式，证明了向量函数与两个实函数典范配对情形下的极化 Hardy–Stein 恒等式。
We study functorial properties of the spaces $R(X)$, introduced in [Studia Math. 197 (2010), 69-79] as a central tool in the analysis of the Hardy operator minus the identity on decreasing functions. In particular, we provide conditions on a minimal Lorentz space $Λ_φ$ so that the equation $R(X)=Λ_φ$ has a solution within the category of rearrangement invariant (r.i.) spaces. Moreover, we show that if $R(X)=Λ_φ$, then we can always take $X$ to be the minimal r.i. Banach range space for the Hardy operator defined in $Λ_φ$.
The purpose of this paper is to introduce a new class of singular integral operators in the Dunkl setting involving both the Euclidean metric and the Dunkl metric. Then we provide the $T1$ theorem, the criterion for the boundedness on $L^2$ for these operators. Applying this singular integral operator theory, we establish the Littlewood-Paley theory and the Dunkl-Hardy spaces. As applications, the boundedness of singular integral operators, particularly, the Dunkl-Rieze transforms, on the Dunkl-Hardy spaces is given. The $L^2$ theory and the singular integral operator theory play crucial roles…
The investigation of the asymptotic behavior of trigonometric series near the origin is a prominent topic in mathematical analysis. For trigonometric series in one variable, this problem was exhaustively studied by various authors in a series of publications dating back to the work of G. H. Hardy, 1928. Trigonometric series in several variables have got less attention. The aim of the work is to partially fill this gap by finding the asymptotics of trigonometric series in several variables with the terms, having a form of `one minus the cosine' up to a decreasing power-like factor: \[ \sum_{z\i…
Hardy spaces of discrete holomorphic functions on the upper half-lattice
该研究在正方形格点离散全纯性框架下，发展上半格离散全纯函数的Hardy空间\(H^p\)理论，证明相关公式与恒等式，描述再生核等，比较离散与经典理论，还证明对偶性等结果。
An Anisotropic Balian-Low Phenomenon: Geometric Obstructions to Wavelet Frames
本文研究扩张膨胀矩阵相关的各向异性Hardy空间中小波框架的解析稳定性，通过建立重建误差的确定性算子范数下界，揭示各向异性Balian-Low现象，其障碍源于几何不相容指数。
A new class of integrals involving the confluent hypergeometric function ${}_1F_{1}(a;c;z)$ and the Riemann $Ξ$-function is considered. It generalizes a class containing some integrals of S. Ramanujan, G.H. Hardy and W.L. Ferrar and gives as by-products, transformation formulas of the form $F(z,α)=F(iz,β)$, where $αβ=1$. As particular examples, we derive an extended version of the general theta transformation formula and generalizations of certain formulas of Ferrar and Hardy. A one-variable generalization of a well-known identity of Ramanujan is also given. We conclude with a generalization o…
In this paper, we will define $a$-deformed Laguerre operators $L_{a,α}$ and $a$-deformed Laguerre holomorphic semigroups on $L^2\left(\left(0,\infty\right),dμ_{a,α}\right)$. Then we give a spherical harmonic expansion, which reduces to the Bochner-type identity when taking the boundary value $z=\frac{πi}2$, of the $(k,a)$-generalized Laguerre semigroup introduced by S. Ben Saïd, T. Kobayashi and B. Ørsted. And then we prove a Hardy inequality for fractional powers of the $a$-deformed Dunkl harmonic oscillator $\triangle_{k,a}:=\left|x\right|^{2-a}\triangle_k-\left|x\right|^a$ using this expans…
Product Hardy spaces meet ball quasi-Banach function spaces
本文通过 Littlewood-Paley-Stein 理论引入了与球拟 Banach 函数空间关联的乘积 Hardy 空间，建立了基于离散 Calderón 恒等式的分解定理，并利用 Rubio de Francia 外推定理证明了双参数奇异积分算子的有界性。
Real-Variable Theory of Hardy--Lorentz Spaces on Quasi-Ultrametric Spaces of Homogeneous Type with Reverse-Doubling Property
本文基于超RD空间构造逼近单位，推导了Calderón复现公式，并建立了Hardy-Lorentz空间的小波特征，证明了与Campanato-Lorentz空间的对偶定理及真实插值定理。
Let $(\mathbb{X},d,μ)$ be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and $X(\mathbb{X})$ a ball quasi-Banach function space on $\mathbb{X}$. In this article, the authors introduce the weak Hardy space $WH_X(\mathbb{X})$ associated with $X(\mathbb{X})$ via the grand maximal function, and characterize $WH_X(\mathbb{X})$ by other maximal functions and atoms. The authors then apply these characterizations to obtain the real interpolation and the boundedness of Calderón--Zygmund operators in the critical case. The main novelties of this article exist in that the authors…
Compactness and Essential Norms of Weighted Bilinear Hardy Operators on General Intervals
本文研究一般区间上加权双线性 Hardy 算子的紧性，证明其紧性等价于有界性与端点尾部拟范数消失，并给出本质范数、有限秩距离与非紧性测度的精确恒等式，以及显式紧性判据和逼近数收敛结果。
The Littlewood-Paley formula and mean counting function for vertical limits of Dirichlet series
本文证明了Hardy空间$\mathscr{H}^p$中Dirichlet级数的Littlewood-Paley公式，并建立了垂直极限函数的均值计数函数存在性，推广了先前结果。
Almost periodicity and boundary values of Dirichlet series
该研究利用概周期性建立狄利克雷级数哈代空间的Hardy–Stein恒等式与Littlewood–Paley公式对应形式，证明相关极限不可交换，揭示两类边值的性质，并探究其零点集与Blaschke乘积。
Let $X$ be a space of homogeneous type and let $L$ be a sectorial operator with bounded holomorphic functional calculus on $L^2(X)$. We assume that the semigroup $\{e^{-tL}\}_{t>0}$ satisfies Davies-Gaffney estimates. In this paper, we introduce a new type of paraproduct operators that is constructed via certain approximations of the identity associated to $L$. We show various boundedness properties on $L^p(X)$ and the recently developed Hardy and BMO spaces $H^p_L(X)$ and $BMO_L(X)$. In generalization of standard paraproducts constructed via convolution operators, we show $L^2(X)$ off-diagona…
Spectral enclosures and stability for non-self-adjoint discrete Schroedinger operators on the half-line
本文研究半直线上带复Robin边界和复值势的非自伴离散薛定谔算子，建立了可求和势的最优谱包络、谱稳定性的一般小性条件，并证明了一个可生成最优离散Hardy不等式的一般恒等式。
For a bounded function $φ$ on the unit circle $\mathbb T$, let $T_φ$ be the associated Toeplitz operator on the Hardy space $H^2$. Assume that the kernel $$K_2(φ):=\{f\in H^2:\,T_φf=0\}$$ is nontrivial. Given a unit-norm function $f$ in $K_2(φ)$, we ask whether an identity of the form $|f|^2=\frac12\left(|f_1|^2+|f_2|^2\right)$ may hold a.e. on $\mathbb T$ for some $f_1,f_2\in K_2(φ)$, both of norm $1$ and such that $|f_1|\ne|f_2|$ on a set of positive measure. We then show that such a decomposition is possible if and only if either $f$ or $\overline{zφf}$ has a nontrivial inner factor. The pr…
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