Anisotropic Hardy Spaces Research Articles

Anisotropic Hardy spaces associated with ball quasi-Banach function spaces and their applications

Anisotropic Hardy spaces associated with ball quasi-Banach function spaces and their applications

Sharp bilinear decomposition for products of both anisotropic Hardy spaces and their dual spaces with its applications to endpoint boundedness of commutators

Sharp bilinear decomposition for products of both anisotropic Hardy spaces and their dual spaces with its applications to endpoint boundedness of commutators

Anisotropic ball Campanato-type function spaces and their applications

Let A be a general expansive matrix and let X be a ball quasi-Banach function space on $${\mathbb {R}}^n$$ , whose certain power (namely its convexification) supports a Fefferman–Stein vector-valued maximal inequality and the associate space of whose other power supports the boundedness of the powered Hardy–Littlewood maximal operator. The authors first introduce some anisotropic ball Campanato-type function spaces associated with both A and X, prove that these spaces are dual spaces of anisotropic Hardy spaces $$H_X^A({\mathbb {R}}^n)$$ associated with both A and X, and obtain various anisotropic Littlewood–Paley function characterizations of $$H_X^A({\mathbb {R}}^n)$$ . Also, as applications, the authors establish several equivalent characterizations of anisotropic ball Campanato-type function spaces, which, combined with the atomic decomposition of tent spaces associated with both A and X, further induce their Carleson measure characterization. All these results have a wide range of generality and, particularly, even when they are applied to Morrey spaces and Orlicz-slice spaces, some of the obtained results are also new. The novelties of this article are reflected in that, to overcome the essential difficulties caused by the absence of both an explicit expression and the absolute continuity of the quasi-norm $$\Vert \cdot \Vert _X$$ under consideration, the authors embed X under consideration into the anisotropic weighted Lebesgue space with certain special weight and then fully use the known results of this weighted Lebesgue space.

A partial differential equation characterization of anisotropic Hardy spaces

AbstractWe obtain a differential characterization for the anisotropic Hardy space by identifying it with a parabolic Hardy space associated with a general continuous group. This allows to be defined using a parabolic differential equation of Calderón and Torchinsky. We also provide a classification of dilations corresponding to equivalent anisotropic Hardy spaces with respect to linear transformations.

Operator-valued anisotropic Hardy and BMO spaces

Let M be a von Neumann algebra equipped with a normal semifinite faithful trace τ, and A be an expansive dilation on Rn. In this article, the author introduces the operator-valued anisotropic BMO space BMOA(Rn,M) and obtains a predual space of BMOA(Rn,M), which is the operator-valued anisotropic Hardy space HA1(Rn,M). Moreover, as the Lq-spaces analogues of the BMO spaces, the author introduces the operator-valued anisotropic BMO-type space LqMOA(Rn,M)(2<q<∞), and establishes the duality between LqMOA(Rn,M) and the operator-valued anisotropic Hardy space HAp(Rn,M) with p=(q−1)/q. In addition, the author also obtains some interpolation results and the equivalence of HAp(Rn,M) and Lp(Rn,M) with p∈(1,∞). As an application, the boundedness of anisotropic Calderón-Zygmund operators on BMOA(Rn,M) is established. When A:=2In×n, part of results coincides with those of T. Mei [33], where In×n denotes the n×n unit matrix.

The Fourier transform of anisotropic Hardy spaces with variable exponents and their applications

Let $A$ be an expansive dilation on $\mathbb{R}^n$, and $p(\cdot):\mathbb{R}^n\rightarrow(0,\,\infty)$ be a variable exponent function satisfying the globally log-H\"{o}lder continuous condition. Let $\mathcal{H}^{p(\cdot)}_A({\mathbb {R}}^n)$ be the variable anisotropic Hardy space defined via the non-tangential grand maximal function. In this paper, the authors obtain that the Fourier transform of $f\in \mathcal{H}^{p(\cdot)}_A({\mathbb {R}}^n)$ coincides with a continuous function $F$ on $\mathbb{R}^n$ in the sense of tempered distributions. As applications, the authors further conclude a higher order convergence of the continuous function $F$ at the origin and then give a variant of the Hardy-Littlewood inequality in the setting of anisotropic Hardy spaces with variable exponents.

Fourier transform of variable anisotropic Hardy spaces with applications to Hardy-Littlewood inequalities

Fourier transform of variable anisotropic Hardy spaces with applications to Hardy-Littlewood inequalities

Maximal Function Characterizations of Hardy Spaces on Rn with Pointwise Variable Anisotropy

In 2011, Dekel et al. developed highly geometric Hardy spaces Hp(Θ), for the full range 0&lt;p≤1, which were constructed by continuous multi-level ellipsoid covers Θ of Rn with high anisotropy in the sense that the ellipsoids can rapidly change shape from point to point and from level to level. In this article, when the ellipsoids in Θ rapidly change shape from level to level, the authors further obtain some real-variable characterizations of Hp(Θ) in terms of the radial, the non-tangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy spaces of Bownik.

Variable Anisotropic Hardy Spaces with Variable Exponents

Abstract Let p(·) : ℝ n → (0, ∞] be a variable exponent function satisfying the globally log-Hölder continuous and let Θ be a continuous multi-level ellipsoid cover of ℝ n introduced by Dekel et al. [12]. In this article, we introduce highly geometric Hardy spaces Hp (·)(Θ) via the radial grand maximal function and then obtain its atomic decomposition, which generalizes that of Hardy spaces Hp (Θ) on ℝ n with pointwise variable anisotropy of Dekel et al. [16] and variable anisotropic Hardy spaces of Liu et al. [24]. As an application, we establish the boundedness of variable anisotropic singular integral operators from Hp (·)(Θ) to Lp (·)(ℝ n ) in general and from Hp (·)(Θ) to itself under the moment condition, which generalizes the previous work of Bownik et al. [6] on Hp (Θ).

Molecular Decomposition of Anisotropic Hardy Spaces With Variable Exponents

Let A be an expansive dilation on ℝn, and p(·): ℝn → (0, ∞) be a variable exponent function satisfying the globally log-Holder continuous condition. Let \(H_A^{p\left(\cdot \right)}\left({{\mathbb{R}^n}} \right)\) be the variable anisotropic Hardy space defined via the non-tangential grand maximal function. In this paper, the authors establish its molecular decomposition, which is still new even in the classical isotropic setting (in the case A:= 2In×n). As applications, the authors obtain the boundedness of anisotropic Calderon-Zygmund operators from \(H_A^{p\left(\cdot \right)}\left({{\mathbb{R}^n}} \right)\) to Lp(·)(ℝn) or from \(H_A^{p\left(\cdot \right)}\left({{\mathbb{R}^n}} \right)\) to itself.

Molecular characterizations of variable anisotropic Hardy spaces with applications to boundedness of Calderón–Zygmund operators

Let $$p(\cdot ):\ \mathbb {R}^n\rightarrow (0,\infty ]$$ be a variable exponent function satisfying the globally log-Holder 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 linear operators on $$H_A^{p(\cdot )}(\mathbb {R}^n)$$ , which is used to prove the boundedness of anisotropic Calderon–Zygmund operators on $$H_A^{p(\cdot )}(\mathbb {R}^n)$$ . In addition, the boundedness of anisotropic Calderon–Zygmund operators from $$H_A^{p(\cdot )}(\mathbb {R}^n)$$ to the variable Lebesgue space $$L^{p(\cdot )}(\mathbb {R}^n)$$ is also presented. All these results are new even in the classical isotropic setting.

The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents

Abstract In this article, the authors establish the characterizations of a class of anisotropic Herz-type Hardy spaces with two variable exponents associated with a non-isotropic dilation on ℝ n {{\mathbb{R}}}^{n} in terms of molecular decompositions. Using the molecular decompositions, the authors obtain the boundedness of the central δ-Calderón-Zygmund operators on the anisotropic Herz-type Hardy space with two variable exponents.

A Multiplier Theorem on Anisotropic Hardy Spaces

AbstractWe present a multiplier theorem on anisotropic Hardy spaces. When m satisfies the anisotropic, pointwise Mihlin condition, we obtain boundedness of the multiplier operator Tm: → , for the range of p that depends on the eccentricities of the dilation A and the level of regularity of a multiplier symbol m. This extends the classical multiplier theorem of Taibleson andWeiss.

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 Hardy-Lorentz spaces

Let p∈ (0,1], q ∈(0,∞] and A be a general expansive matrix on ℝn. Let HAp,q(ℝn) be the anisotropic Hardy-Lorentz spaces associated with A defined via the non-tangential grand maximal function. In this article, the authors characterize HAp,q(ℝn) in terms of the Lusin-area function, the Littlewood-Paley g-function or the Littlewood-Paley gλ*-function via first establishing an anisotropic Fefferman-Stein vector-valued inequality in the Lorentz space Lp,qℝn. All these characterizations are new even for the classical isotropic Hardy-Lorentz spaces on ℝn. Moreover, the range of λ in the gλ*-function characterization of HAp,q(ℝn) coincides with the best known one in the classical Hardy space Hp(ℝn) or in the anisotropic Hardy space HAp(ℝn).

Variable Anisotropic Hardy Spaces and Their Applications

Let $p(\cdot) \colon \mathbb{R}^n \to (0,\infty]$ be a variable exponent function satisfying the globally log-Hölder continuous condition and $A$ a general expansive matrix on $\mathbb{R}^n$. In this article, the authors first introduce the variable anisotropic Hardy space $H_A^{p(\cdot)}(\mathbb{R}^n)$ associated with $A$, via the non-tangential grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of $H_A^{p(\cdot)}(\mathbb{R}^n)$, respectively, by means of atoms, finite atoms, the Lusin area function, the Littlewood-Paley $g$-function or $g_{\lambda}^{\ast}$-function. As applications, the authors first establish a criterion on the boundedness of sublinear operators from $H^{p(\cdot)}_A(\mathbb{R}^n)$ into a quasi-Banach space. Then, applying this criterion, the authors show that the maximal operators of the Bochner-Riesz and the Weierstrass means are bounded from $H^{p(\cdot)}_A(\mathbb{R}^n)$ to $L^{p(\cdot)}(\mathbb{R}^n)$ and, as consequences, some almost everywhere and norm convergences of these Bochner-Riesz and Weierstrass means are also obtained. These results on the Bochner-Riesz and the Weierstrass means are new even in the isotropic case.

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.

Molecular characterization of anisotropic Musielak–Orlicz Hardy spaces and their applications

Let A be an expansive dilation on ℝn and φ : ℝn × [0, ∞) → [0, ∞) an anisotropic Musielak–Orlicz function. Let HAφ(ℝn) be the anisotropic Hardy space of Musielak–Orlicz type defined via the grand maximal function. In this article, the authors establish its molecular characterization via the atomic characterization of HAφ(ℝn). The molecules introduced in this article have the vanishing moments up to order s and the range of s in the isotropic case (namely, A := 2In×n) coincides with the range of well-known classical molecules and, moreover, even for the isotropic Hardy space Hp(ℝn) with p ∈ (0, 1] (in this case, A := 2In×n, φ(x, t) := tp for all x ∈ ℝn and t ∈ [0, ∞)), this molecular characterization is also new. As an application, the authors obtain the boundedness of anisotropic Calderon–Zygmund operators from HAφ(ℝn) to Lφ(ℝn) or from HAφ(ℝn) to itself.

LITTLEWOOD-PALEY CHARACTERIZATIONS OF ANISOTROPIC HARDY SPACES OF MUSIELAK-ORLICZ TYPE

Let $\varphi : \mathbb{R}^n\times [0,\,\infty)\to[0,\infty)$ be a Musielak-Orlicz function and $A$ an expansive dilation. Let $H^\varphi_A({\mathbb {R}}^n)$ be the anisotropic Hardy space of Musielak-Orlicz type defined via the grand maximal function. Its atomic characterization and some other maximal function characterizations of $H^\varphi_A({\mathbb {R}^n})$, in terms of the radial, the non-tangential and the tangential maximal functions, are known. In this article, the authors further obtain their characterizations in terms of the Lusin-area function, the $g$-function or the $g^\ast_\lambda$-function via first establishing an anisotropic Peetre's inequality of Musielak-Orlicz type. Moreover, the range of $\lambda$ in the $g^\ast_\lambda$-function characterization of $H^\varphi_A({\mathbb {R}}^n)$ coincides with the known best conclusions in the case when $H^\varphi_A({\mathbb {R}}^n)$ is the classical Hardy space $H^p({\mathbb{R}}^n)$ or the anisotropic Hardy space $H^p_A({\mathbb {R}}^n)$ or their weighted variants, where $p\in(0,\,1]$.

Anisotropic Hardy Spaces of Musielak-Orlicz Type with Applications to Boundedness of Sublinear Operators

Let φ : ℝn × [0, ∞)→[0, ∞) be a Musielak-Orlicz function and A an expansive dilation. In this paper, the authors introduce the anisotropic Hardy space of Musielak-Orlicz type, H A φ(ℝn), via the grand maximal function. The authors then obtain some real-variable characterizations of H A φ(ℝn) in terms of the radial, the nontangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy space H A p(ℝn) with p ∈ (0,1] and are new even for its weighted variant. Finally, the authors characterize these spaces by anisotropic atomic decompositions. The authors also obtain the finite atomic decomposition characterization of H A φ(ℝn), and, as an application, the authors prove that, for a given admissible triplet (φ, q, s), if T is a sublinear operator and maps all (φ, q, s)-atoms with q < ∞ (or all continuous (φ, q, s)-atoms with q = ∞) into uniformly bounded elements of some quasi-Banach spaces ℬ, then T uniquely extends to a bounded sublinear operator from H A φ(ℝn) to ℬ. These results are new even for anisotropic Orlicz-Hardy spaces on ℝn.