Real-variable characterizations of new anisotropic mixed-norm Hardy spaces

Abstract

Let p ∈ (0, ∞) n and A be a general expansive matrix on R n . In this article, via the non-tangential grand maximal function, the authors first introduce the anisotropic mixednorm Hardy spaces H p A (R n ) associated with A and then establish their radial or non-tangential maximal function characterizations. Moreover, the authors characterize H p A (R n ), respectively, by means of atoms, finite atoms, Lusin area functions, Littlewood-Paley g-functions or g * λfunctions via first establishing an anisotropic Fefferman-Stein vector-valued inequality on the mixed-norm Lebesgue space L p (R n ). In addition, the authors also obtain the duality between H p A (R n ) and the anisotropic mixed-norm Campanato spaces. As applications, the authors establish a criterion on the boundedness of sublinear operators from H p A (R n ) into a quasi-Banach space. Applying this criterion, the authors then obtain the boundedness of anisotropic convolutional δ-type and non-convolutional β-order Calderón-Zygmund operators from H p A (R n ) to itself [or to L p (R n )]. As a corollary, the boundedness of anisotropic convolutional δ-type Calderón-Zygmund operators on the mixed-norm Lebesgue space L p (R n ) with p ∈ (1, ∞) n is also presented.

A Survey on Some Anisotropic Hardy-Type Function Spaces

“…Henceforth, we call Q := {Q k α } k∈Z,α∈E k from Lemma 3.3 dyadic cubes and k the level, denoted by (Q k α ), of the dyadic cube Q k α for any k ∈ Z and α ∈ E k . The following technical lemma is necessary, which is just [51,Lemma 6.14]. In what follows, for any t ∈ R, we denote by t the least integer not less than t. Then there exists a positive constant C such that, for any k, i ∈ Z, {c Q } Q∈Q ⊂ [0, ∞) with Q as in Lemma 3.3, and…”

“…In 2014, based on the work of both Bownik [12] and Ky [60], Li et al [68] introduced the anisotropic Musielak-Orlicz Hardy space, which was a generalization of the anisotropic Hardy space of Bownik [12], the weighted anisotropic Hardy space of Bownik et al [16] as well as the Musielak-Orlicz Hardy space of Ky [60]. Recently, the anisotropic product Musielak-Orlicz Hardy space was studied by Fan et al in [32] and the anisotropic mixed-norm Hardy space by Huang et al in [51].…”

“…We point out that the integrable exponent of the Hardy space H p A (R n ) from[51] is a vector p ∈ (0, ∞) n , whose associated basic function space is the mixed-norm Lebesgue space L p (R n ), which has different orders of integrability in different variables; however, the integrable exponent of the anisotropic variable Hardy space H…”

Dual spaces of mixed-norm martingale Hardy spaces

“…Moreover Musielak-Orlicz-Hardy spaces were studied in Yang et al [35]. The mixed norm classical Hardy spaces have been developed in Cleanthous et al [4] and intensively studied by Huang et al in [12,13,14,15,16].…”

“…Mixed-norm Lebesgue and Hardy spaces were investigated in a great number of papers (e.g. in [2,3,4,9,10,12,13,14,15,16]).…”