Hardy-Orlicz Spaces Research Articles

Nevanlinna and Carleson functions of analytic self‐maps on multiply‐connected domains

AbstractIn the paper we prove that the Nevanlinna counting function and the Carleson function of analytic self‐maps of finitely connected domains with smooth boundary are equivalent. Using this result we obtain the characterization of compact composition operators on Hardy–Orlicz spaces.

Some Estimates for Maximal Bochner—Riesz Means on Musielak—Orlicz Hardy Spaces

Let 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 Musielak–Orlicz Hardy space Hφ(ℝn) is defined to be the space of all tempered distributions whose grand maximal functions belong to the Musielak–Orlicz space Lφ(ℝn). In this paper, the authors establish the boundedness of maximal Bochner–Riesz means T*δ from Hφ(ℝn) to WLφ(ℝn) or Lφ(ℝn). These results are also new even when φ(x, t):= Φ(t) for all (x, t) ∈ ℝn × [0, ∞), where Φ is an Orlicz function.

Atomic characterizations of weak martingale Musielak–Orlicz Hardy spaces and their applications

Let (Ω,F,P) be a probability space, and let φ:Ω×[0,∞)→[0,∞) be a Musielak–Orlicz function. In this article, we establish the atomic characterizations of weak martingale Musielak–Orlicz Hardy spaces WHφs(Ω), WHφM(Ω), WHφS(Ω), WPφ(Ω), and WQφ(Ω). We then use these atomic characterizations to obtain the boundedness of σ-sublinear operators from weak martingale Musielak–Orlicz Hardy spaces to weak Musielak–Orlicz spaces, as well as some martingale inequalities which further clarify the relationships among these weak martingale Musielak–Orlicz Hardy spaces. All these results improve and generalize the corresponding results on weak martingale Orlicz–Hardy spaces. Moreover, we improve all the known results on weak martingale Musielak–Orlicz Hardy spaces. In particular, both the boundedness of σ-sublinear operators and the martingale inequalities, for weak weighted martingale Hardy spaces as well as for weak weighted martingale Orlicz–Hardy spaces, are new.

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.

Real-variable characterizations of Orlicz-slice Hardy spaces

In this paper, the authors first introduce a class of Orlicz-slice spaces which generalize the slice spaces recently studied by Auscher et al. Based on these Orlicz-slice spaces, the authors then 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 Bonami and Feuto as well as the Hardy-amalgam space of de Paul Ablé and Feuto as special cases. Their characterizations via the atom, the molecule, various maximal functions, the Poisson integral and the Littlewood–Paley functions are also obtained. As an application of these characterizations, the authors establish their finite atomic characterizations, which further induce a description of their dual spaces and a criterion on the boundedness of sublinear operators from these Orlicz-slice Hardy spaces into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of [Formula: see text]-type Calderón–Zygmund operators on these Orlicz-slice Hardy spaces. All these results are new even for slice Hardy spaces and, moreover, for Hardy-amalgam spaces, the Littlewood–Paley function characterizations, the dual spaces and the boundedness of [Formula: see text]-type Calderón–Zygmund operators on these Hardy-type spaces are also new.

Multiplication between Hardy spaces and their dual spaces

For any p∈(0,1) and α=1/p−1, let Hp(Rn) and Cα(Rn) be the Hardy and the Campanato spaces on the n-dimensional Euclidean space Rn, respectively. In this article, the authors find suitable Musielak–Orlicz functions Φp, defined by setting, for any x∈Rn and t∈[0,∞),Φp(x,t):={t1+[t(1+|x|)n]1−pwhenn(1/p−1)∉N,t1+[t(1+|x|)n]1−p[log⁡(e+|x|)]pwhenn(1/p−1)∈N, and then establish a bilinear decomposition theorem for multiplications of functions in Hp(Rn) and its dual space Cα(Rn). To be precise, for any f∈Hp(Rn) and g∈Cα(Rn), the authors prove that the product of f and g, viewed as a distribution, can be decomposed into S(f,g)+T(f,g), where S is a bilinear operator bounded from Hp(Rn)×Cα(Rn) to L1(Rn) and T a bilinear operator bounded from Hp(Rn)×Cα(Rn) to the Musielak–Orlicz Hardy space HΦp(Rn) associated with the above Musielak–Orlicz function Φp. Such a bilinear decomposition is sharp when nα∉N, in the sense that any other vector space Y⊂HΦp(Rn) adapted to the above bilinear decomposition should satisfy (L1(Rn)+Y)⁎=(L1(Rn)+HΦp(Rn))⁎. To obtain the sharpness, the authors establish a characterization of the class of pointwise multipliers of Cα(Rn) by means of the dual space of HΦp(Rn), which is of independent interest. As an application, an estimate of the div-curl product involving the space HΦp(Rn) is discussed. This article naturally extends the known sharp bilinear decomposition of H1(Rn)×BMO(Rn).

New molecular characterizations of anisotropic Musielak–Orlicz Hardy spaces and their applications

Let φ:Rn×[0,∞)→[0,∞) be an anisotropic growth function and A a general expansive matrix on Rn. Let HAφ(Rn) be the anisotropic Musielak–Orlicz Hardy space associated with A defined via the non-tangential grand maximal function. In this article, the authors establish a new molecular characterization of HAφ(Rn) via its known atomic characterizations, which positively answer a question mentioned by Li et al. (2016) [34]. As an application, the authors obtain a criterion on the boundedness of linear operators on HAφ(Rn), which further implies the boundedness of integral anisotropic Calderón–Zygmund operators on HAφ(Rn). In addition, the boundedness of these operators from HAφ(Rn) to the Musielak–Orlicz space Lφ(Rn) is also presented.

Small Bergman-Orlicz and Hardy-Orlicz spaces, and their composition operators

We show that the weighted Bergman-Orlicz space $A_{\alpha}^{\psi}$ coincides with some weighted Banach space of holomorphic functions if and only if the Orlicz function $\psi$ satisfies the so-called $\Delta^{2}$--condition. In addition we prove that this condition characterizes those $A_{\alpha}^{\psi}$ on which every composition operator is bounded or order bounded into the Orlicz space $L_{\alpha}^{\psi}$. This provides us with estimates of the norm and the essential norm of composition operators on such spaces. We also prove that when $\psi$ satisfies the $\Delta^{2}$--condition, a composition operator is compact on $A_{\alpha}^{\psi}$ if and only if it is order bounded into the so-called Morse-Transue space $M_{\alpha}^{\psi}$. Our results stand in the unit ball of $\mathbb{C}^{N}$.

Atomic and maximal function characterizations of Musielak–Orlicz–Hardy spaces associated to non-negative self-adjoint operators on spaces of homogeneous type

Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a non-negative self-adjoint operator on $L^2(\mathcal{X})$ whose heat kernels satisfy the Gaussian upper bound estimates. Assume that the growth function $\varphi:\ \mathcal{X}\times[0,\infty) \to[0,\infty)$ satisfies that $\varphi(x,\cdot)$ is an Orlicz function and $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights). Let $H_{\varphi,\,L}(\mathcal{X})$ be the Musielak-Orlicz-Hardy space defined via the Lusin area function associated with the heat semigroup of $L$. In this article, the authors characterize the space $H_{\varphi,\,L}(\mathcal{X})$ by means of atoms, non-tangential and radial maximal functions associated with $L$. In particular, when $\mu(\mathcal{X})<\infty$, the local non-tangential and radial maximal function characterizations of $H_{\varphi,\,L}(\mathcal{X})$ are obtained. As applications, the authors obtain various maximal function and the atomic characterizations of the "geometric" Musielak-Orlicz-Hardy spaces $H_{\varphi,\,r}(\Omega)$ and $H_{\varphi,\,z}(\Omega)$ on the strongly Lipschitz domain $\Omega$ in $\mathbb{R}^n$ associated with second-order self-adjoint elliptic operators with the Dirichlet and the Neumann boundary conditions; even when $\varphi(x,t):=t$ for any $x\in\mathbb{R}^n$ and $t\in[0,\infty)$, the equivalent characterizations of $H_{\varphi,\,z}(\Omega)$ given in this article improve the known results via removing the assumption that $\Omega$ is unbounded.

New martingale inequalities and applications to Fourier analysis

Let (Ω,F,P) be a probability space and φ:Ω×[0,∞)→[0,∞) be a Musielak–Orlicz function. In this article, the authors prove that the Doob maximal operator is bounded on the Musielak–Orlicz space Lφ(Ω). Using this and extrapolation method, the authors then establish a Fefferman–Stein vector-valued Doob maximal inequality on Lφ(Ω). As applications, the authors obtain the dual version of the Doob maximal inequality and the Stein inequality for Lφ(Ω), which are new even in weighted Orlicz spaces. The authors then establish the atomic characterizations of martingale Musielak–Orlicz Hardy spaces Hφs(Ω), Pφ(Ω), Qφ(Ω), HφS(Ω) and HφM(Ω). From these atomic characterizations, the authors further deduce some martingale inequalities between different martingale Musielak–Orlicz Hardy spaces, which essentially improve the corresponding results in Orlicz space case and are also new even in weighted Orlicz spaces. By establishing the Davis decomposition on HφS(Ω) and HφM(Ω), the authors obtain the Burkholder–Davis–Gundy inequality associated with Musielak–Orlicz functions. Finally, using the previous martingale inequalities, the authors prove that the maximal Fejér operator is bounded from Hφ[0,1) to Lφ[0,1), which further implies some convergence results of the Fejér means; these results are new even for the weighted Hardy spaces.

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.

CLASSICAL PROPERTIES OF COMPOSITION OPERATORS ON HARDY–ORLICZ SPACES ON PLANAR DOMAINS

In this paper we study composition operators on Hardy–Orlicz spaces on multiply connected domains whose boundaries consist of finitely many disjoint analytic Jordan curves. We obtain a characterization of order-bounded composition operators. We also investigate weak compactness and the Dunford–Pettis property of these operators.

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 $p\in (0,1]$.

Bilinear decompositions of products of local Hardy and Lipschitz or BMO spaces through wavelets

Let [Formula: see text] and [Formula: see text] be the local Hardy space in the sense of D. Goldberg. In this paper, the authors establish two bilinear decompositions of the product spaces of [Formula: see text] and their dual spaces. More precisely, the authors prove that [Formula: see text] and, for any [Formula: see text], [Formula: see text], where [Formula: see text] denotes the local BMO space, [Formula: see text], for any [Formula: see text] and [Formula: see text], the inhomogeneous Lipschitz space and [Formula: see text] a variant of the local Orlicz–Hardy space related to the Orlicz function [Formula: see text] for any [Formula: see text] which was introduced by Bonami and Feuto. As an application, the authors establish a div-curl lemma at the endpoint case.

Intrinsic Structures of Certain Musielak–Orlicz Hardy Spaces

For any $$p\in (0,\,1]$$ , let $$H^{\Phi _p}(\mathbb {R}^n)$$ be the Musielak–Orlicz Hardy space associated with the Musielak–Orlicz growth function $$\Phi _p$$ , defined by setting, for any $$x\in \mathbb {R}^n$$ and $$t\in [0,\,\infty )$$ , $$\begin{aligned}&\Phi _{p}(x,\,t)\\&\quad := {\left\{ \begin{array}{ll} \displaystyle \frac{t}{\log {(e+t)}+[t(1+|x|)^n]^{1-p}}&{} \quad \text {when} n(1/p-1)\notin \mathbb N \cup \displaystyle \frac{t}{\log (e+t)+[t(1+|x|)^n]^{1-p}[\log (e+|x|)]^p}&{} \quad \text {when} n(1/p-1)\in \mathbb N\cup \{0\}, \end{array}\right. } \end{aligned}$$ which is the sharp target space of the bilinear decomposition of the product of the Hardy space $$H^p(\mathbb {R}^n)$$ and its dual. Moreover, $$H^{\Phi _1}(\mathbb {R}^n)$$ is the prototype appearing in the real-variable theory of general Musielak–Orlicz Hardy spaces. In this article, the authors find a new structure of the space $$H^{\Phi _p}(\mathbb {R}^n)$$ by showing that, for any $$p\in (0,\,1]$$ , $$H^{\Phi _p}(\mathbb {R}^n)=H^{\phi _0}(\mathbb {R}^n) +H_{W_p}^p({{{\mathbb {R}}}^n})$$ and, for any $$p\in (0,\,1)$$ , $$H^{\Phi _p}(\mathbb {R}^n)=H^{1}(\mathbb {R}^n) +H_{W_p}^p({{{\mathbb {R}}}^n})$$ , where $$H^1(\mathbb {R}^n)$$ denotes the classical real Hardy space, $$H^{\phi _0}({{{\mathbb {R}}}^n})$$ the Orlicz–Hardy space associated with the Orlicz function $$\phi _0(t):=t/\log (e+t)$$ for any $$t\in [0,\infty )$$ , and $$H_{W_p}^p(\mathbb {R}^n)$$ the weighted Hardy space associated with certain weight function $$W_p(x)$$ that is comparable to $$\Phi _p(x,1)$$ for any $$x\in \mathbb {R}^n$$ . As an application, the authors further establish an interpolation theorem of quasilinear operators based on this new structure.

Atomic decomposition and weak factorization in generalized Hardy spaces of closed forms

We give an atomic decomposition of closed forms on Rn, the coefficients of which belong to some Hardy space of Musielak–Orlicz type. These spaces are natural generalizations of weighted Hardy–Orlicz spaces, when the Orlicz function depends on the space variable. One of them, called Hlog, appears naturally when considering products of functions in the Hardy space H1 and in BMO. As a main consequence of the atomic decomposition, we obtain a weak factorization of closed forms whose coefficients are in Hlog. Namely, a closed form in Hlog is the infinite sum of the wedge product between an exact form in the Hardy space H1 and an exact form in BMO. The converse result, which generalizes the classical div–curl lemma, is a consequence of [4]. As a corollary, we prove that the real-valued Hlog space can be weakly factorized.

Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

Let $$\vec A: = \left( {{A_1},{A_2}} \right)$$ be a pair of expansive dilations and φ: ℝ n ×ℝ m ×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ via the anisotropic Lusin-area function and establish its atomic characterization, the $$\vec g$$ -function characterization, the $$\vec g_\lambda ^*$$ -function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet ( $$\left( {\varphi ,q,\vec s} \right)$$ ), if T is a sublinear operator and maps all ( $$\left( {\varphi ,q,\vec s} \right)$$ )-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ to L φ (R n × R m ) and from $$H_{\vec A}^\varphi \left( {{\mathbb{R}^n} \times {\mathbb{R}^m}} \right)$$ to itself, whose kernels are adapted to the action of $$\vec A$$ . The results of this article essentially extend the existing results for weighted product Hardy spaces on ℝ n × ℝ m and are new even for classical product Orlicz-Hardy spaces.

Hardy spaces for ball quasi-Banach function spaces

This article unifies the theory for Hardy spaces built on Banach lattices on R-n satisfying certain weak conditions on indicator functions of balls. The authors introduce a new family of function spaces, named the ball quasi-Banach function spaces, to define Hardy type spaces. The ones in this article extend classical Hardy spaces and include various known function spaces, for example, Hardy-Lorentz spaces, Hardy-Herz spaces, Hardy-Orlicz spaces, Hardy-Morrey spaces, Musielak-Orlicz-Hardy spaces, variable Hardy spaces and variable Hardy-Morrey spaces. Among them, Hardy-Herz spaces are shown to naturally arise in the context of any function spaces above. The example of Hardy-Morrey spaces shows that the absolute continuity of the quasi-norm is not necessary, which is used to guarantee the density of the set of functions having compact supports in Hardy spaces for ball quasi-Banach function spaces, but the decomposition result on these Hardy-type spaces never requires this absolute continuity of the quasi-norm. Moreover, via assuming that the powered Hardy-Littlewood maximal operator satisfies certain Fefferman-Stein vector-valued maximal inequality as well as it is bounded on the associate space, the atomic characterizations of Hardy type spaces are obtained. Although the results are based on the rather abstract theory of function spaces, they improve and extend the results for Orlicz spaces and Musielak-Orlicz spaces. Moreover, local Hardy type spaces and Hardy type spaces associated with operators in this setting are also studied.

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.

Intrinsic atomic and molecular decompositions of Hardy–Musielak–Orlicz spaces

We introduce the Hardy type space for Musielak–Orlicz spaces. It includes several existing Hardy type spaces such as the Hardy–Orlicz spaces and the Hardy spaces with variable exponents. Furthermore, we develop an atomic decomposition such that the size condition just relies on the norms of Musielak–Orlicz spaces. This gives us a nature extension of the molecular decompositions to the Hardy type space for Musielak–Orlicz spaces.