Atomic and Littlewood–Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

Abstract

Let a := (a 1 , .

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

“…By this, the monotone convergence property of increasing sequences on L p (R n ) (see Lemma 3.10 below) and the obtained boundedness of the anisotropic Hardy-Littlewood maximal operator on L p (R n ) again, we then prove Theorem 3.15. Moreover, using the obtained radial maximal function characterizations of H p A (R n ) and Lemma 3.3, we also show that, for any given [27,Lemma 3.15], we show that some estimates related to L p (R n ) norms for some series of functions can be reduced into dealing with the L r (R n ) norms of the corresponding functions (see Lemma 4.5 below), which plays a key role in the proof of Theorem 4.7 and is also of independent interest. Then, using this key lemma, the obtained vector-valued inequality and some arguments similar to those used in the proof of [ [27,Theorem 5.9] to the present setting of anisotropic mixed-norm Hardy spaces.…”

“…Moreover, due to the notable work of Calderón and Torchinsky [10] on parabolic Hardy spaces and also in order to meet the requirements arising in the development of harmonic analysis and partial differential equations, there has been an increasing interest in extending classical function spaces from Euclidean spaces to some more general underlying spaces; see, for instance, [6,15,27,55,61]. In particular, as a generalization of both the isotropic Hardy space and the parabolic Hardy space of Calderón and Torchinsky [10], in 2003, Bownik [6] first introduced the anisotropic Hardy space H p A (R n ) with p ∈ (0, ∞), where A is a general expansive matrix on R n (see Definition 2.1 below).…”

“…To complete the real-variable theory of this Hardy space H p a (R n ), Huang et al [27] established several equivalent real-variable characterizations of H p a (R n ), respectively, in terms of atoms, finite atoms, Lusin area functions, Littlewood-Paley g-functions or g * λ -functions and also obtained the boundedness of anisotropic Calderón-Zygmund operators from H p a (R n ) to itself [or to L p (R n )]. In [28], via the atomic and the finite atomic characterizations from [27], Huang et al further proved that the dual space of H p a (R n ), with p ∈ (0, 1] n , is the anisotropic mixed-norm Campanato space. For more progresses about the real-variable theory of this sort of anisotropic function spaces, we refer the reader to [15,32,58].…”

“…Based on the aforementioned mixed-norm Lebesgue space L p (R n ) in [5] and also motivated by the works of Cleanthous et al [15] as well as Hart et al [25], in this article, we introduce a new kind of anisotropic mixed-norm Hardy spaces H p A (R n ) with p ∈ (0, ∞) n and some general expansive matrix A, via borrowing some ideas from the real-variable theories of the anisotropic Hardy space H p A (R n ) in [6] and the anisotropic mixed-norm Hardy space H p a (R n ) in [15,27,28]. To be precise, the anisotropic mixed-norm Hardy space H p A (R n ) is first introduced via the nontangential grand maximal function and then characterized by means of radial or non-tangential maximal functions.…”

“…Obviously, L p(·) (R n ) and L p (R n ) cannot cover each other, so do the variable anisotropic Hardy space H p(·) A (R n ) of [40] and the Hardy space H p A (R n ) of the present article. In addition, the real-variable theory of mixed-norm Hardy space H p a (R n ) associated with a vector a ∈ [1, ∞) n was established in [15,27,28]. Observe that the space H p a (R n ), introduced by Cleanthous et al [15], is included in the Hardy space H p A (R n ) of the present article as a special case in the sense of equivalent quasi-norms (see Proposition 6.6 below).…”