We study Hardy spaces associated with a general multidimensional Bessel operator $\mathbb{B}_ν$. This operator depends on a multiparameter of type $ν$ that is usually restricted to a product of half-lines. Here we deal with the Bessel operator in the general context, with no restrictions on the type parameter. We define the Hardy space $H^1$ for $\mathbb{B}_ν$ in terms of the maximal operator of the semigroup of operators $\exp(-t\mathbb{B}_ν)$. Then we prove that, in general, $H^1$ admits an atomic decomposition of local type.
We characterize the Hardy space $H^1$ in the rational Dunkl setting associated with the reflection group $\mathbb Z_2^n$ by means of Riesz transforms. As a corollary we obtain a Riesz transform characterization of $H^1$ for product of Bessel operators in $(0,\infty)^n$.
We analyse Morrey spaces, generalised Morrey spaces and Campanato spaces on homogeneous groups. The boundedness of the Hardy-Littlewood maximal operator, Bessel-Riesz operators, generalised Bessel-Riesz operators and generalised fractional integral operators in generalised Morrey spaces on homogeneous groups is shown. Moreover, we prove the boundedness of the modified version of the generalised fractional integral operator and Olsen type inequalities in Campanato spaces and generalised Morrey spaces on homogeneous groups, respectively. Our results extend results known in the isotropic Euclidea…
We provide the weak factorization of the Hardy spaces $H^{p}(\mathbb{R}_+, dm_λ)$ in the Bessel setting, for $p\in \left(\frac{2λ+ 1}{2λ+ 2}, 1\right]$. As a corollary we obtain a characterization of the boundedness of the commutator $[b, R_{Δ_λ}]$ from $L^{q}(\mathbb{R}_+, dm_λ)$ to $L^{r}(\mathbb{R}_+, dm_λ)$ when $b\in \textrm{Lip}_α(\mathbb{R}_+, dm_λ)$ provided that $α= \frac{1}{q} - \frac{1}{r}$. The results are a slight generalization and modification of the work of Duong, Li, Yang, and the second named author, which in turn are based on modifications and adaptations of work by Uchiyama…
The Hardy-Morrey spaces related to Laplace-Bessel differential equations are introduced in terms of maximal functions. The atomic decomposition theory which has the same cancellation properties of the $H^{p}_{Δ_ν}(\mathbb{R}^{n}_{+})$ Hardy spaces is established.
Consider the Bessel operator with a potential on L^2((0,infty), x^a dx), namely Lf(x) = -f"(x) - a/x f'(x) + V(x)f(x). We assume that a>0 and V\in L^1_{loc}((0,infty), x^a dx) is a non-negative function. By definition, a function f\in L^1((0,infty), x^a dx) belongs to the Hardy space H^1(L) if sup_{t>0} |e^{-tL} f| \in L^1((0,infty), x^a dx). Under certain assumptions on V we characterize the space H^1(L) in terms of atomic decompositions of local type. In the second part we prove that this characterization can be applied to L for a \in (0,1) with no additional assumptions on the potential V.
A new proof is given of the atomic decomposition of Hardy spaces Hp, in the classical setting of Rn. The new method can be used to establish atomic decomposition of maximal Hardy spaces in general setting and non classical settings.
This paper is devoted to Hardy type inequalities with remainders for compactly supported smooth functions on open sets in the Euclidean space. We establish new inequalities with weight functions depending on the distance function to the boundary of the domain. One-dimensional $L_1$ and $L_p$ inequalities and their multidimensional analogues are proved. We consider spatial inequalities in open convex domains with the finite inner radius. Constants in these inequalities depend on the roots of parametric Lamb equation for the Bessel function and turn out to be sharp in some particular cases.
We constructively determine the Pietsch measure of the 2-summing multiplication operator \[\mathcal{M}_u:\ell^{\infty} \rightarrow H^p, \quad (φ_I) \mapsto \sum φ_Ix_Ih_I. \] Our construction of the Pietsch measure for the multiplication operator $\mathcal{M}_u$ involves the Haar coefficients of $u$ and its atomic decomposition.
We show that any weakly separated Bessel system of model spaces in the Hardy space on the unit disc is a Riesz system and we highlight some applications to interpolating sequences of matrices. This will be done without using the recent solution of the Feichtinger conjecture, whose natural generalization to multi-dimensional model sub-spaces of $\mathrm{H}^2$ turns out to be false.
Power-weighted Hardy type integral inequalities with additional terms on finite intervals
N.I. Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal University（喀山联邦大学 N.I. 罗巴切夫斯基数学力学研究所）；Institute of Software Development and Engineering, Innopolis University（因诺波利斯大学软件研发工程学院）
This paper defines local weighted Hardy spaces with variable exponent. Local Hardy spaces permit atomic decomposition, which is one of the main themes in this paper. A consequence is that the atomic decomposition is obtained for the functions in the Lebesgue spaces with exponentially decaying exponent. As an application, we obtain the boundedness of singular integral operators, the Littlewood--Paley characterization and wavelet decomposition.
Let $Δ$ be the Dunkl Laplacian on $\mathbb R^N$ associated with a normalized root system $R$ and a multiplicity function $k(α)\geq 0$. We say that a function $f$ belongs to the Hardy space $H^1_Δ$ if the nontangential maximal function $\mathcal M_H f(\mathbf x)=\sup_{\| \mathbf x-\mathbf y\|<t} |\exp(t^2Δ)f(\mathbf x)|$ belongs to $L^1(w(\mathbf x)\, d\mathbf x)$, where $w(\mathbf x)=\prod_{α\in R} |\langle α,\mathbf x\rangle|^{k(α)}$. We prove that $H^1_Δ$ coincides with the space $H^1_{\rm atom}(\mathbb R^N, \| \mathbf x-\mathbf y\|, w(\mathbf x)d\mathbf x)$ understood as the atomic Hardy sp…
Let $Δ$ and $L=Δ-\|\mathbf x\|^2$ be the Dunkl Laplacian and the Dunkl harmonic oscillator respectively. We define the Hardy space $\mathcal H^1$ associated with the Dunkl harmonic oscillator by means of the nontangential maximal function with respect to the semigroup $e^{tL}$. We prove that the space $\mathcal H^1$ admits characterizations by relevant Riesz transforms and atomic decompositions. The atoms which occur in the atomic decompositions are of local type.
We consider a nonnegative self-adjoint operator $L$ on $L^2(X)$, where $X\subseteq \mathbb{R}^d$. Under certain assumptions, we prove atomic characterizations of the Hardy space $$H^1(L) = ł\{f\in L^1(X) \ : \ \ {\|}\sup_{t>0} \ |\exp(-tL)f \ | \ {\|}_{L^1(X)}<\infty\ \}.$$ We state simple conditions, such that $H^1(L)$ is characterized by atoms being either the classical atoms on $X\subseteq \mathbb{R}^d$ or local atoms of the form $|Q|^{-1}χ_Q$, where $Q\subseteq X$ is a cube (or cuboid). One of our main motivation is to study multidimensional operators related to orthogonal expansions. We p…
Consider the multidimensional Bessel operator $$B f(x) = -\sum_{j=1}^N \left(\partial_j^2 f(x) +\frac{α_j}{x_j} \partial_j f(x)\right), \quad x\in(0,\infty)^N. $$ Let $d = \sum_{j=1}^N \max(1,α_j+1)$ be the homogeneous dimension of the space $(0,\infty)^N$ equipped with the measure $x_1^{α_1}... x_N^{α_N} dx_1...dx_N$. In the general case $α_1,...,α_N >-1$ we prove multiplier theorems for spectral multipliers $m(B)$ on $L^{1,\infty}$ and the Hardy space $H^1$. We assume that $m$ satisfies the classical Hörmander condition $$\sup_{t>0} \left||η(\cdot) m(t\cdot)\right||_{W^{2,β}(\mathbb{R})}<\in…
For $f \in H^p(δ^2)$, $0<p\leq 2$, with Haar expansion $f=\sum f_{I \times J}h_{I\times J}$ we constructively determine the Pietsch measure of the $2$-summing multiplication operator \[\mathcal{M}_f:\ell^{\infty} \rightarrow H^p(δ^2), \quad (φ_{I\times J}) \mapsto \sum φ_{I\times J}f_{I \times J}h_{I \times J}. \] Our method yields a constructive proof of Pisier's decomposition of $f \in H^p(δ^2)$ \[|f|=|x|^{1-θ}|y|^θ\quad\quad \text{ and }\quad\quad \|x\|_{X_0}^{1-θ}\|y\|^θ_{H^2(δ^2)}\leq C\|f\|_{H^p(δ^2)}, \] where $X_0$ is Pisier's extrapolation lattice associated to $H^p(δ^2)$ and $H^2(δ^2…
The main purpose of this paper is to give an estimate for the Fourier Laguerre transform on Hardy spaces in the setting of Laguerre hypergroup. The atomic and molecular characterization is investigated which allows us to prove a version of Hormander's multiplier theorem on H^p (0<p\leq1).
$K$-frames and atomic systems for an operator $K$ in Hilbert spaces were introduced by Gavruta \cite{12} and further studied by Xio, Zhu and Gavruta \cite{21}. In this paper, we have introduced the notion of an approximative atomic system for an operator $K$ in Banach spaces and obtained interesting results. A complete characterization of family of approximative local atoms of subspace of Banach space has been obtained. Also, a necessary and sufficient condition for the existence of an approximative atomic system for an operator $K$ is given. Finally, explicit methods are given for the constru…
Maximal and atomic Hardy spaces Hp and HAp , are considered in the setting of a doubling metric measure space in the presence of a non-negative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. It is shown that Hp = HAp with equivalent norms.
We consider a homogeneous tree endowed with a nondoubling flow measure $μ$ of exponential growth and a probabilistic Laplacian $\mathcal{L}$ self-adjoint with respect to $μ$. We prove that the maximal characterization in terms of the heat and the Poisson semigroup of $\mathcal{L}$ and the Riesz transform characterization of the atomic Hardy space introduced in a previous work fail.
Institute Modeling and Decision Sciences (MODS Institute)（建模与决策科学研究所）；Center for Basic Sciences, School of Engineering and Architecture, Pontifical Bolivarian University（波利瓦尔天主教大学工程与建筑学院基础科学中心）；Department of Mathematics, Faculty of Education and Sciences, University of Sucre（苏克雷大学教育与科学学院数学系）
The celebrated Fefferman's theorems on the general form of linear functionals on the Hardy space $H^1$ over the circle group is generalized to the case of an arbitrary compact Abelian group with totally ordered dual. Several corollaries that can be applied to multiple lacunary Fourier series and atomic decomposition on the two dimensional torus are obtained.
Let $({\mathcal X},d,μ)$ be a metric measure space of homogeneous type in the sense of R. R. Coifman and G. Weiss and $H^1_{\rm at}({\mathcal X})$ be the atomic Hardy space. Via orthonormal bases of regular wavelets and spline functions recently constructed by P. Auscher and T. Hytönen, together with obtaining some crucial lower bounds for regular wavelets, the authors give an unconditional basis of $H^1_{\rm at}({\mathcal X})$ and several equivalent characterizations of $H^1_{\rm at}({\mathcal X})$ in terms of wavelets, which are proved useful.
We consider an infinite homogeneous tree V endowed with the usual metric d defined on graphs and a weighted measure μ. The metric measure space V,d,μ) is nondoubling and of exponential growth, hence the classical theory of Hardy spaces does not apply in this setting. We construct an atomic Hardy space H^1 on (V,d,μ) and investigate some of its properties, focusing in particular on real interpolation properties and on boundedness of singular integrals on H^1.
Let $(Ω,\mathcal{F},\mathbb{P})$ be a probability space and $φ:\ Ω\times[0,\infty)\to [0,\infty)$ be a Musielak--Orlicz function. In this article, the authors establish the atomic characterizations of weak martingale Musielak--Orlicz Hardy spaces $WH_φ^s(Ω)$, $WH_φ^M(Ω)$, $WH_φ^S(Ω)$, $WP_φ(Ω)$ and $WQ_φ(Ω)$. Using these atomic characterizations, the authors then obtain the boundedness of sublinear operators from weak martingale Musielak--Orlicz Hardy spaces to weak Musielak--Orlicz spaces, and some martingale inequalities which further clarify the relationships among $WH_φ^s(Ω)$, $WH_φ^M(Ω)$,…
In this paper we study Hardy spaces associated with non-negative self-adjoint operators and develop their vector-valued theory. The complex interpolation scales of vector-valued tent spaces and Hardy spaces are extended to the endpoint p=1. The holomorphic functional calculus of L is also shown to be bounded on the associated Hardy space H^1_L(X). These results, along with the atomic decomposition for the aforementioned space, rely on boundedness of certain integral operators on the tent space T^1(X).
[L. Gavruta, Frames for Operators, Appl. comput. Harmon. Anal. 32(2012), 139-144] introduced a special kind of frames, named $K$-frames, where $K$ is an operator, in Hilbert spaces, is significant in frame theory and has many applications. In this paper, first of all, we have introduced the notion of approximative $K$-atomic decomposition in Banach spaces. We gave two characterizations regarding the existence of approximative $K$-atomic decompositions in Banach spaces. Also some results on the existence of approximative $K$-atomic decompositions are obtained. We discuss several methods to cons…
Factorization and Atomic Decomposition in Hardy-Orlicz Spaces on the Upper Half-Plane with Applications to Hankel Operators
Unité de Recherche et d’Expertise Numérique, Université Virtuelle de Côte d’Ivoire（科特迪瓦虚拟大学数字研究与专家单位）；Laboratoire de Mathématiques et Applications, UFR Mathématiques − - Informatique, Université Félix Houphouët-Boigny Abidjan-Cocody（费利克斯·乌弗埃-博瓦尼大学阿比让-科科迪校区数学与信息科学学院数学与应用实验室）；Department of Mathematics, University of Ghana（加纳大学数学系）
We introduce a decreasing one-parameter family $\mathfrak{X}^γ(M)$, $γ>0$, of Banach subspaces of the Hardy-Goldberg space $\mathfrak{h}^1(M)$ on certain nondoubling Riemannian manifolds with bounded geometry and we investigate their properties. In particular, we prove that $\mathfrak{X}^{1/2}(M)$ agrees with the space of all functions in $\mathfrak{h}^1(M)$ whose Riesz transform is in $L^1(M)$, and we obtain the surprising result that this space does not admit an atomic decomposition.
This paper gives a systematic study of operator-valued local Hardy spaces. These spaces are localizations of the Hardy spaces defined by Tao Mei, and share many properties with Mei's Hardy spaces. We prove the ${\rm h}_1$-$\rm bmo$ duality, as well as the ${\rm h}_p$-${\rm h}_q$ duality for any conjugate pair $(p,q)$ when $1<p< \infty$. We show that ${\rm h}_1(\mathbb{R}^d, \mathcal M)$ and ${\rm bmo}(\mathbb{R}^d, \mathcal M)$ are also good endpoints of $L_p(L_\infty(\mathbb{R}^d) \overline{\otimes} \mathcal M)$ for interpolation. We obtain the local version of Calderón-Zygmund theory, and th…
In this paper we introduce Hardy-Lorentz spaces with variable exponents associated to dilation in ${\Bbb R}^n$. We establish maximal characterizations and atomic decompositions for our variable exponent anisotropic Hardy-Lorentz spaces.
基于作者最近与Jan Maas和Jan van Neerven的工作，本文建立了两个范数（一个使用极大函数，另一个使用平方函数）的等价性，这两个范数用于定义具有高斯测度的$\R^{n}$上的Hardy空间，并适用于Ornstein-Uhlenbeck半群.与Mauceri和Meda较早提出的原子高斯Hardy空间不同，本文研究的$h^{1}(\R^{n};dγ)$空间是里斯变换从$h^{1}(\R^{n};dγ)$到$L^{1}(\R^{n};dγ)$有界的.这给出了一个高斯模拟的开创性工作的费曼和斯坦的情况下，勒贝格测度和通常的拉普拉斯算子。
We prove a radial maximal function characterisation of the local atomic Hardy space h^1(M) on a Riemannian manifold M with positive injectivity radius and Ricci curvature bounded from below. As a consequence, we show that an integrable function belongs to h^1(M) if and only if either its local heat maximal function or its local Poisson maximal function are integrable. A key ingredient is a decomposition of Hölder cut-offs in terms of an appropriate class of approximations of the identity, which we obtain on arbitrary Ahlfors-regular metric measure spaces and generalises a previous result of A.…
The rational Dunkl operators are commuting differential-reflection operators on the Euclidean space $R^d$ associated with a root system, that contain some non-local refection terms, and the associated Hardy space is defined by means of the Riesz transforms with respect to the Dunkl operators. The aim of the paper is to prove that its dual can be realized by a class of functions on $R^d$, denoted by $BMC_κ$ in the text, that consists of the underlying functions of a certain type of weighted Carleson measures. Our method is "purely analytic" and does not depend on the atomic decomposition. As a …
In this article, we give a short proof of Hardy's inequality for Hermite expansions of functions in the classical Hardy spaces $H^p({\mathbb R^n})$, by using an atomic decomposition of the Hardy spaces associated with the Hermite operators. When the space dimension is $1$, we obtain a new estimate of Hardy's inequality for Hermite expansions in $H^p({\mathbb R})$ for the range $0<p<1.$
For a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function $k\geq 0$ let $\mathbf N=N+\sum_{α\in R} k(α)$. Let $L=-Δ+V$, $V\geq 0$, be the Dunkl--Schrödinger operator on $\mathbb R^N$. Assume that there exists $q >\max(1,\frac{\mathbf{N}}{2})$ such that $V$ belongs to the reverse Hölder class $\text{RH}^q(dw)$. We prove the Fefferman--Phong inequality for $L$. As an application, we conclude that the Hardy space $H^1_{L}$, which is originally defined by means of the maximal function associated with the semigroup $e^{tL}$, admits an atomic decomposition with local atoms in the…
