Weak Hardy Spaces Research Articles

Weak Hardy spaces associated with para-accretive functions and their applications

<p>In this paper, we introduced a new class of weak Hardy spaces, denoted by $ H^{p, \infty}_b $, and provided an analysis of their atomic decomposition. As an application, we established the boundedness of Calderón-Zygmund operators (CZOs) from $ H^p $ to $ H^{p, \infty}_b $ including cases at the critical exponent</p><p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ p = \frac{n}{n+\delta}, $\end{document} </tex-math></disp-formula></p><p>where $ \delta $ represents the regularity index of the distributional kernel. Moreover, the boundedness of CZOs from $ H^{p, \infty} $ to $ H^{p, \infty}_b $ was demonstrated for</p><p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \frac{n}{n+\delta}<p\leq 1. $\end{document} </tex-math></disp-formula></p>

Molecular characterization of weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type with its applications to Littlewood–Paley function characterizations

Abstract Let ( 𝕏 , d , μ ) {(\mathbb{X},d,\mu)} be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and let X ⁢ ( 𝕏 ) {X(\mathbb{X})} be a ball quasi-Banach function space on 𝕏 {\mathbb{X}} . In this article, the authors introduce the weak Hardy space W ⁢ H ~ X ⁢ ( 𝕏 ) {\widetilde{WH}_{X}(\mathbb{X})} associated with X ⁢ ( 𝕏 ) {X(\mathbb{X})} via the Lusin area function. Then the authors characterize W ⁢ H ~ X ⁢ ( 𝕏 ) {\widetilde{WH}_{X}(\mathbb{X})} by the molecule, the grand maximal function, and the Littlewood–Paley g-function and g λ * {g^{*}_{\lambda}} -function. Moreover, all these results have a wide generality. Particularly, the results of this article are also new even when they are applied, respectively, to weighted Lebesgue spaces, Orlicz spaces, and variable Lebesgue spaces, which actually are new even on RD-spaces (that is, spaces of homogeneous type with additional reverse doubling condition). The main novelties of this article exist in that the authors take full advantage of the geometrical properties of 𝕏 {\mathbb{X}} expressed by both the dyadic cubes and the exponential decay of the approximations of the identity to overcome the difficulties caused by the deficiencies of both the explicit expression of the quasi-norm of X ⁢ ( 𝕏 ) {X(\mathbb{X})} and the reverse doubling condition of μ, and that the authors use the tent space on 𝕏 × ℤ {\mathbb{X}\times\mathbb{Z}} to characterize W ⁢ H ~ X ⁢ ( 𝕏 ) {\widetilde{WH}_{X}(\mathbb{X})} by the Littlewood–Paley g λ * {g^{*}_{\lambda}} -function, where the range of λ might be best possible in some cases.

Weak Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Decompositions, Real Interpolation, and Calderón–Zygmund Operators

Let $$({\mathbb {X}},d,\mu )$$ be a space of homogeneous type in the sense of Coifman and 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 use the Aoki–Rolewicz theorem and both the dyadic system and the exponential decay of approximations of the identity on $${\mathbb {X}}$$ , which closely connect with the geometrical properties of $${\mathbb {X}}$$ , to overcome the difficulties caused by the deficiency of both the triangle inequality of $$\Vert \cdot \Vert _{X({\mathbb {X}})}$$ and the reverse doubling assumption of the measure $$\mu $$ under consideration, and also use the relation between the convexification of $$X({\mathbb {X}})$$ and the weak ball quasi-Banach function space $$WX({\mathbb {X}})$$ associated with $$X({\mathbb {X}})$$ to prove that the infinite summation of atoms converges in the space of distributions on $${\mathbb {X}}$$ . Moreover, all these results have a wide range of generality and, particularly, even when they are applied to the weighted Lebesgue space, the Orlicz space, and the variable Lebesgue space, the obtained results are also new and, actually, some of them are new even on RD-spaces (namely, spaces of homogeneous type satisfying the additional reverse doubling condition).

Calderón-Zygmund operators with variable kernels acting on weak Musielak-Orlicz Hardy spaces

Abstract. Let $\varphi : \mathbb R^n \times [0,\infty) \rightarrow [0,\infty)$ satisfy that $\varphi(x,\cdot)$ is an Orlicz function for any given $x \in \mathbb R^n$, and $\varphi(\cdot,t)$ is a Muckenhoupt $A_\infty$ weight uniformly in $t \in (0,\infty)$. The weak Musielak-Orlicz Hardy space $WH^\varphi(\mathbb R^n)$ is defined to be the set of all tempered distributions such that their grand maximal functions belong to the weak Musielak-Orlicz space $WL^\varphi(\mathbb R^n)$. In this paper, we discuss the boundedness of the Calderon-Zygmund operator with variable kernel from $WH^\varphi(\mathbb R^n)$ to $WL^\varphi(\mathbb R^n)$. These results are new even for the classical weighted weak Hardy space and probably new for the classical weak Hardy space.

Two boundedness criteria for a class of operators on Musielak–Orlicz Hardy spaces and applications

AbstractLet φ : ℝn × [0, ∞) → [0, ∞) satisfy that φ(x, · ), for any given x ∈ ℝn, is an Orlicz function and φ( · , t) is a Muckenhoupt A∞ weight uniformly in t ∈ (0, ∞). The (weak) Musielak–Orlicz Hardy space Hφ(ℝn) (WHφ(ℝn)) generalizes both the weighted (weak) Hardy space and the (weak) Orlicz Hardy space and hence has wide generality. In this paper, two boundedness criteria for both linear operators and positive sublinear operators from Hφ(ℝn) to Hφ(ℝn) or from Hφ(ℝn) to WHφ(ℝn) are obtained. As applications, we establish the boundedness of Bochner–Riesz means from Hφ(ℝn) to Hφ(ℝn), or from Hφ(ℝn) to WHφ(ℝn) in the critical case. These results are new even when φ(x, t): = Φ(t) for all (x, t) ∈ ℝn × [0, ∞), where Φ is an Orlicz function.

Boundedness of Fractional Integral Operators in Weighted Weak Hardy Spaces on Homogeneous Spaces

In this paper, we shall study the theory of weighted weak Hardy spaces 𝐻𝜔 𝑝,∞ on space of homogeneous type satisfying some reverse doubling condition. More precisely, we will give atomic decomposition characterizations of 𝐻𝜔 𝑝,∞ . Then we use this decomposition to derive the boundedness of fractional integral operators in 𝐻𝜔 𝑝,∞ and prove an 𝐻𝜔 𝑝,∞ interpolation theorem. As an applications, the boundedness of Nagel-Stein’s singular and fractional integral operators in 𝐻𝜔 𝑝,∞ are derived.

Intrinsic Square Function Characterizations of Variable Weak Hardy Spaces

Let $p(\cdot) \colon \mathbb{R}^n \to (0,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, via using the atomic and Littlewood-Paley function characterizations of variable weak Hardy space $W\!H^{p(\cdot)}(\mathbb{R}^n)$, the author establishes its intrinsic square function characterizations including the intrinsic Littlewood-Paley $g$-function, the intrinsic Lusin area function and the intrinsic $g_{\lambda}^{\ast}$-function.

Molecular characterization of anisotropic weak Musielak-Orlicz Hardy spaces and their applications

Let \begin{document}$ A $\end{document} be a real \begin{document}$ n\times n $\end{document} matrix with all its eigenvalues \begin{document}$ \lambda $\end{document} satisfy \begin{document}$ |\lambda|>1 $\end{document} . Let \begin{document}$ \varphi: \mathbb{R}^n\times[0, \, \infty)\to[0, \, \infty) $\end{document} be an anisotropic Musielak-Orlicz function, i.e., \begin{document}$ \varphi(x, \cdot) $\end{document} is an Orlicz function uniformly in \begin{document}$ x\in{\mathbb{R}^n} $\end{document} and \begin{document}$ \varphi(\cdot, \, t) $\end{document} is an anisotropic Muckenhoupt \begin{document}$ \mathcal {A}_\infty({\mathbb{R}^n}) $\end{document} weight uniformly in \begin{document}$ t\in(0, \, \infty) $\end{document} . In this article, the authors introduce the anisotropic weak Musielak-Orlicz Hardy space \begin{document}$ WH^{\varphi}_A(\mathbb{R}^n) $\end{document} via the grand maximal function and establish its molecular characterization which are anisotropic extensions of Liang, Yang and Jiang (Math. Nachr. 289: 634-677, 2016). As an application, the boundedness of anisotropic Calderon-Zygmund operators from \begin{document}$ H_A^\varphi(\mathbb{R}^n) $\end{document} to \begin{document}$ WH_A^\varphi(\mathbb{R}^n) $\end{document} in the critical case is presented.

Parametric Marcinkiewicz integrals with rough kernels acting on weak Musielak–Orlicz Hardy spaces

Let φ : R n × [ 0 , ∞ ) → [ 0 , ∞ ) satisfy that φ ( x , ⋅ ) , for any given x ∈ R n , is an Orlicz function and that φ ( ⋅ , t ) is a Muckenhoupt A ∞ weight uniformly in t ∈ ( 0 , ∞ ) . The weak Musielak–Orlicz Hardy space WH φ ( R n ) is defined to be the set of all tempered distributions such that their grand maximal functions belong to the weak Musielak–Orlicz space WL φ ( R n ) . For parameter ρ ∈ ( 0 , ∞ ) and measurable function f on R n , the parametric Marcinkiewicz integral μ Ω ρ related to the Littlewood–Paley g -function is defined by setting, for all x ∈ R n , μ Ω ρ ( f ) ( x ) : = ( ∫ 0 ∞ | ∫ | x − y | ≤ t Ω ( x − y ) | x − y | n − ρ f ( y ) d y | 2 d t t 2 ρ + 1 ) 1 / 2 , where Ω is homogeneous of degree zero satisfying the cancellation condition. In this article, we discuss the boundedness of the parametric Marcinkiewicz integral μ Ω ρ with rough kernel from weak Musielak–Orlicz Hardy space WH φ ( R n ) to weak Musielak–Orlicz space WL φ ( R n ) . These results are new even for the classical weighted weak Hardy space of Quek and Yang, and probably new for the classical weak Hardy space of Fefferman and Soria.

Variable weak Hardy spaces WH L p(·)(ℝ n ) associated with operators satisfying Davies–Gaffney estimates

Abstract Let p ⁢ ( ⋅ ) : ℝ n → [ 0 , 1 ] {p(\,\cdot\,)\colon\mathbb{R}^{n}\to[0,1]} be a variable exponent function satisfying the globally log-Hölder continuous condition, and L a one-to-one operator of type ω in L 2 ⁢ ( ℝ n ) {L^{2}({\mathbb{R}}^{n})} , with ω ∈ [ 0 , π / 2 ) {\omega\in[0,\pi/2)} , which has a bounded holomorphic functional calculus and satisfies the Davies–Gaffney estimates. In this article, we introduce the variable weak Hardy space WH L p ⁢ ( ⋅ ) ⁢ ( ℝ n ) {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} , associated with L via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent space WT p ⁢ ( ⋅ ) ⁢ ( ℝ + n + 1 ) {\mathrm{WT}^{p(\,\cdot\,)}(\mathbb{R}_{+}^{n+1})} , which is also obtained in this article. In particular, when L is non-negative and self-adjoint, we obtain the atomic characterization of WH L p ⁢ ( ⋅ ) ⁢ ( ℝ n ) {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} . As an application of the molecular characterization, when L is the second-order divergence form elliptic operator with complex bounded measurable coefficients, we prove that the associated Riesz transform ∇ ⁡ L - 1 / 2 {\nabla L^{-1/2}} is bounded from WH L p ⁢ ( ⋅ ) ⁢ ( ℝ n ) {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} to the variable weak Hardy space WH p ⁢ ( ⋅ ) ⁢ ( ℝ n ) {\mathrm{WH}^{p(\,\cdot\,)}(\mathbb{R}^{n})} . Moreover, when L is non-negative and self-adjoint with the kernels of { e - t ⁢ L } t > 0 {\{e^{-tL}\}_{t>0}} satisfying the Gaussian upper bound estimates, the atomic characterization of WH L p ⁢ ( ⋅ ) ⁢ ( ℝ n ) {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} is further used to characterize this space via non-tangential maximal functions.

Extension of Pettis integration: Pettis operators and their integrals

In this note, the authors discuss the concepts of a Pettis operator, by which they mean a weak $$^*$$ –weakly continuous linear operator F from a dual Banach space to an $$L_1$$ -space, and of its Pettis integral, understood simply as the dual operator $$F^*$$ of F. Applications to radial limits in weak Hardy spaces of vector-valued harmonic and holomorphic functions are provided.

New real-variable characterizations of anisotropic weak Hardy spaces of Musielak-Orlicz type

A real $n\times n$ matrix $A$ is called an expansive dilation if all of its eigenvalues $\lambda $ satisfy $|\lambda |\!\!>\!\!1$. Let $\varphi : \mathbb {R}^n\times [0,\infty )\to [0,\infty )$ be a Musielak-Orlicz function. The aim of this article is to find an appropriate general space which includes the weak Hardy space of Fefferman and Soria, the weighted weak Hardy space of Quek and Yang}, the anisotropic weak Hardy space of Ding and Lan, the Musielak-Orlicz Hardy space of Ky and the anisotropic Hardy space of Musielak-Orlicz type of Li, Yang and Yuan. For this reason, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ with $m\in \mathbb {N}$ and obtain some new real-variable characterizations of $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ in terms of the radial, the non-tangential and the tangential maximal functions via a new monotone convergence theorem adapted to the weak anisotropic Musielak-Orlicz space $L^{\varphi , \infty }({\mathbb {R}}^n)$. These maximal function characterizations generalize the known results on the anisotropic weak Hardy space $H^{p, \infty }_A({\mathbb {R}}^n)$ with $p\in (0, 1]$ and are new even for their weighted variants or weak Orlicz-Hardy variants. As an application, the authors show the boundedness of a class of multilinear operators formed by the anisotropic Calderon-Zygmund operators from product weighted Lebesgue space to $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ with $\varphi (x,t):=t^p\omega (x)$ and $\omega \in \mathbb {A}_1(A)$, which is a weighted and non-isotropic extension of Grafakos.

LITTLEWOOD–PALEY CHARACTERIZATIONS OF ANISOTROPIC WEAK MUSIELAK–ORLICZ HARDY SPACES

Let $A$ be an expansive dilation on $\mathbb{R}^{n}$ and $\unicode[STIX]{x1D711}:\mathbb{R}^{n}\times [0,\infty )\rightarrow [0,\infty )$ an anisotropic growth function. In this article, the authors introduce the anisotropic weak Musielak–Orlicz Hardy space $\mathit{WH}_{A}^{\unicode[STIX]{x1D711}}(\mathbb{R}^{n})$ via the nontangential grand maximal function and then obtain its Littlewood–Paley characterizations in terms of the anisotropic Lusin-area function, $g$-function or $g_{\unicode[STIX]{x1D706}}^{\ast }$-function, respectively. All these characterizations for anisotropic weak Hardy spaces $\mathit{WH}_{A}^{p}(\mathbb{R}^{n})$ (namely, $\unicode[STIX]{x1D711}(x,t):=t^{p}$ for all $t\in [0,\infty )$ and $x\in \mathbb{R}^{n}$ with $p\in (0,1]$) are new. Moreover, the range of $\unicode[STIX]{x1D706}$ in the anisotropic $g_{\unicode[STIX]{x1D706}}^{\ast }$-function characterization of $\mathit{WH}_{A}^{\unicode[STIX]{x1D711}}(\mathbb{R}^{n})$ coincides with the best known range of the $g_{\unicode[STIX]{x1D706}}^{\ast }$-function characterization of classical Hardy space $H^{p}(\mathbb{R}^{n})$ or its weighted variants, where $p\in (0,1]$.

Interpolation Between $$H^{p(\cdot )}({\mathbb {R}}^n)$$ H p ( · ) ( R n ) and $$L^\infty ({\mathbb {R}}^n)$$ L ∞ ( R n ) : Real Method

Let $$p(\cdot ):\ {\mathbb {R}}^n\rightarrow (0,\infty )$$ be a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into “good” and “bad” parts and then prove the following real interpolation theorem between the variable Hardy space $$H^{p(\cdot )}({\mathbb {R}}^n)$$ and the space $$L^{\infty }({\mathbb {R}}^n)$$ : $$(H^{p(\cdot )}(\mathbb R^n),L^{\infty }({\mathbb {R}}^n))_{\theta ,\infty }= WH^{p(\cdot )/(1-\theta )}({\mathbb {R}}^n),\quad \mathrm{where}~\theta \in (0,1), \mathrm{and}$$ $$WH^{p(\cdot )/(1-\theta )}({\mathbb {R}}^n)$$ denotes the variable weak Hardy space. As an application, the variable weak Hardy space $$WH^{p(\cdot )}({\mathbb {R}}^n)$$ with $$p_-:=\mathop {\text {ess inf}}\limits _{x\in {{{\mathbb {R}}}^n}}p(x)\in (1,\infty )$$ is proved to coincide with the variable Lebesgue space $$WL^{p(\cdot )}({\mathbb {R}}^n)$$ .

Anisotropic weak Hardy spaces of Musielak-Orlicz type and their applications

Anisotropy is a common attribute of the nature, which shows different characterizations in different directions of all or part of the physical or chemical properties of an object. The anisotropic property, in mathematics, can be expressed by a fairly general discrete group of dilations {Ak: k ∈ Z}, where A is a real n × n matrix with all its eigenvalues λ satisfy |λ| > 1. The aim of this article is to study a general class of anisotropic function spaces, some properties and applications of these spaces. Let φ: Rn×[0,∞) → [0,∞) be an anisotropic p-growth function with p ∈ (0, 1]. The purpose of this article is to find an appropriate general space which includes weak Hardy space of Fefferman and Soria, weighted weak Hardy space of Quek and Yang, and anisotropic weak Hardy space of Ding and Lan. For this reason, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type HAφ,∞(Rn) and obtain its atomic characterization. As applications, we further obtain an interpolation theorem adapted to HAφ,∞(Rn) and the boundedness of the anisotropic Calderon-Zygmund operator from HAφ,∞(Rn) to LAφ,∞(Rn). It is worth mentioning that the superposition principle adapted to the weak Musielak-Orlicz function space, which is an extension of a result of E. M. Stein, M. Taibleson and G. Weiss, plays an important role in the proofs of the atomic decomposition of HAφ,∞(Rn) and the interpolation theorem.

Variable weak Hardy spaces and their applications

Let p(⋅):Rn→(0,∞) be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors first introduce the variable weak Hardy space on Rn, WHp(⋅)(Rn), via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of WHp(⋅)(Rn), respectively, by means of atoms, molecules, the Lusin area function, the Littlewood–Paley g-function or gλ⁎-function. As an application, the authors establish the boundedness of convolutional δ-type and non-convolutional γ-order Calderón–Zygmund operators from Hp(⋅)(Rn) to WHp(⋅)(Rn) including the critical case when p−=n/(n+δ) or when p−=n/(n+γ), where p−:=essinfx∈Rnp(x).

Estimates of some integral operators with bounded variable kernels on the hardy and weak hardy spaces over ℝ n

In this paper, we first introduce \({L^{{\sigma _1}}}{\left( {\log L} \right)^{{\sigma _2}}}\) conditions satisfied by the variable kernels Ω(x, z) for 0 ≤ σ1 ≤ 1 and σ2 ≥ 0. Under these new smoothness conditions, we will prove the boundedness properties of singular integral operators TΩ, fractional integrals TΩ,α and parametric Marcinkiewicz integrals μΩρ with variable kernels on the Hardy spaces Hp(Rn) and weak Hardy spaces WHp(Rn). Moreover, by using the interpolation arguments, we can get some corresponding results for the above integral operators with variable kernels on Hardy–Lorentz spaces Hp,q(Rn) for all p < q < ∞.

Weak Musielak–Orlicz Hardy spaces and applications

Let satisfy that , for any given , is an Orlicz function and is a Muckenhoupt weight uniformly in . In this article, the authors introduce the weak Musielak–Orlicz Hardy space via the grand maximal function and then obtain its vertical or its non–tangential maximal function characterizations. The authors also establish other real‐variable characterizations of , respectively, in terms of the atom, the molecule, the Lusin area function, the Littlewood–Paley g‐function or ‐function. All these characterizations for weighted weak Hardy spaces (namely, and with and ) are new and part of these characterizations even for weak Hardy spaces (namely, and with ) are also new. As an application, the boundedness of Calderón–Zygmund operators from to in the critical case is presented.

Weighted Weak Local Hardy Spaces Associated with Schrödinger Operators

We characterize the weighted weak local Hardy spacesWhρp(ω)related to the critical radius functionρand weightsω∈A∞ρ,∞(Rn)which locally behave as Muckenhoupt’s weights and actually include them, by the atomic decomposition. As an application, we show that localized Riesz transforms are bounded on the weighted weak local Hardy spaces.

Inner functions with derivatives in the weak Hardy space

It is proved that exponential Blaschke products are the inner functions whose derivative is in the weak Hardy space. As a consequence, it is shown that exponential Blaschke products are Frostman shift invariant. Exponential Blaschke products are described in terms of their logarithmic means and also in terms of the behavior of the derivatives of functions in the corresponding model space.