Real-variable characterizations of Orlicz-slice Hardy spaces

Abstract

In this article, the authors first introduce a class of Orlicz-slice spaces which generalize the slice spaces recently studied by P. Auscher et al. Based on these Orlicz-slice spaces, the authors 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 A. Bonami and J. Feuto as well as the Hardy-amalgam space of Z. V. de P. Ablé and J. 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 δ-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 δ-type Calderón-Zygmund operators on these Hardy-type spaces are also new. )(R n ), where Φ(t) := t log(e+t) for any t ∈ [0, ∞) is an Orlicz function, and applied these Hardy-type spaces to study the linear decomposition of the product of the Hardy space H 1 (R n ) and its dual space BMO (R n ) as well as the local Hardy space h 1 (R n ) and its dual space bmo (R n ). Moreover, very recently, Cao et al. [12] applied h Φ * (R n ) to study the bilinear decomposition of the product of the local Hardy space h 1 (R n ) and its dual space bmo (R n ). Recall that both the Hardy type spaces H Φ * (R n ) and h Φ * (R n ) were defined in [8] via the (local) radial maximal functions, while h Φ * (R n ) in [12] was defined via the local grand maximal function. Moreover, no other real-variable characterizations of both the Hardy type spaces H Φ * (R n ) and h Φ * (R n ) are known so far. On the other hand, recently, to study the classification of weak solutions in the natural classes for the boundary value problems of a t-independent elliptic system in the upper plane, Auscher and Mourgoglou [6] introduced the slice spaces E

Applications of Hardy Spaces Associated with Ball Quasi-Banach Function Spaces

“…To this end, we employ some estimates from [60]. Indeed, the following estimates were obtained in [60] (see [60, pp. 54-56, the estimates of II 1 , II 2 and II 3 ] for the details): for any x ∈ (4Q j ) ∁ ,…”

“…It was proved in [60, Lemma 2.28] that X is a ball quasi-Banach function space; however, it might not be a quasi-Banach function space (see, for instance, [59, Remark 7.4(i)]). Moreover, the assumption (2.2) holds true for the space X, any s ∈ (0, 1] and any θ ∈ (0, min{s, p − Φ , r}) (see [60,Lemma 4.3]); similarly, (2.3) also holds true for the space X, any s ∈ (0, 1] and any θ ∈ (0, min{s, 2p − Φ /s, r}). To state the following assumption on X, we need the notion of the associate space.…”

Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces II: Littlewood–Paley Characterizations and Real Interpolation

“…On another hand, observe that (E p Φ ) t (R n ) when p = t = 1 goes back to the amalgam space (L Φ , ℓ 1 )(R n ) introduced by Bonami and Feuto [11], where Φ(t) := t log(e+t) for any t ∈ [0, ∞), and the Hardy space H Φ * (R n ) associated with the amalgam space (L Φ , ℓ 1 )(R n ) was applied by Bonami and Feuto [11] to study the linear decomposition of the product of the Hardy space H 1 (R n ) and its dual space BMO (R n ). Another main motivation to introduce (HE q Φ ) t (R n ) in [81] exists in that it is a natural generalization of H Φ * (R n ) in [11]. In the last part of this section, we focus on the weak Orlicz-slice Hardy space (W HE q Φ ) t (R n ) built on the Orlicz-slice space (E q Φ ) t (R n ).…”

“…Then (E q Φ ) t (R n ) satisfies Assumption 2.7 (see [81,Lemma 4.3]) and Assumption 2.9 (see [80,Proposition 7.40]). Furthermore, from [81,Lemmas 4.4], we deduce that, for any given r 0 ∈ (0, min{1, p − Φ }) and p 0 ∈ (max{1, p + Φ }, ∞), (3.8) with X := (E q Φ ) t (R n ) holds true. Thus, all the assumptions of main theorems in Sections 3 and 4 are satisfied.…”

Riesz transforms, Cauchy–Riemann systems, and Hardy-amalgam spaces

“…In this paper we study Hardy spaces H p,q (R d ), 0 < p, q < ∞, modeled over amalgam spaces (L p , q )(R d ). These spaces have been investigated recently by Ablé and Feuto ( [1], [2] and [3]; see also [34]).…”