The Hardy–Lorentz spaces H^p,q(Rⁿ)

Abstract

Abstract. We deal with the Hardy-Lorentz spaces H p,q (R n ) where 0 < p ≤ 1, 0 < q ≤ ∞. We discuss the atomic decomposition of the elements in these spaces, their interpolation properties, and the behavior of singular integrals and other operators acting on them.

Anisotropic Hardy-Lorentz spaces and their applications

“…Moreover, it is known that the weak Hardy spaces H p,∞ (R n ) are special cases of the Hardy-Lorentz spaces H p,q (R n ) which, when p = 1 and q ∈ (1, ∞), were introduced and investigated by Parilov [56]. In 2007, Abu-Shammala and Torchinsky [1] studied the Hardy-Lorentz spaces H p,q (R n ) for the full range p ∈ (0, 1] and q ∈ (0, ∞], and obtained their ∞-atomic characterization, real interpolation properties over parameter q, and the boundedness of singular integrals and some other operators on these spaces. In 2010, Almeida and Caetano [2] studied the generalized Hardy spaces which include the classical Hardy-Lorentz spaces H p,q (R n ) investigated in [1] as special cases.…”

“…As the series of works (see, for example, [29,31,3,45,56,1,2]) reveal, the Hardy-Lorentz spaces (as well the weak Hardy spaces) serve as a more subtle research object than the usual Hardy spaces when considering the boundedness of singular integrals, especially, in some endpoint cases, due to the fact that these function spaces own finer structures. However, the real-variable theory of these spaces is still not complete.…”

“…In 2007, Abu-Shammala and Torchinsky [1] studied the Hardy-Lorentz spaces H p,q (R n ) for the full range p ∈ (0, 1] and q ∈ (0, ∞], and obtained their ∞-atomic characterization, real interpolation properties over parameter q, and the boundedness of singular integrals and some other operators on these spaces. In 2010, Almeida and Caetano [2] studied the generalized Hardy spaces which include the classical Hardy-Lorentz spaces H p,q (R n ) investigated in [1] as special cases. To be more precise, Almeida and Caetano [2] obtained some maximal characterizations of these generalized Hardy spaces, some real interpolation results with function parameter and, as applications, they studied the behavior of some classical operators in this generalized setting.…”

“…All these results except the ∞-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on R n . As applications, we first prove that the space H p,q A (R n ) is an intermediate space between H p 1 ,q 1 A (R n ) and H p 2 ,q 2 A (R n ) with 0 < p 1 < p < p 2 < ∞ and q 1 , q, q 2 ∈ (0, ∞], and also between H p,q 1 A (R n ) and H p,q 2 A (R n ) with p ∈ (0, ∞) and 0 < q 1 < q < q 2 ≤ ∞ in the real method of interpolation. We then establish a criterion on the boundedness of sublinear operators from H p,q A (R n ) into a quasi-Banach space.…”

“…Then we introduce the anisotropic Hardy-Lorentz spaces H p,q A (R n ), with p ∈ (0, 1] and q ∈ (0, ∞], via the non-tangential maximal function (see Definition 2.5 below). These anisotropic Hardy-Lorentz spaces include the classical Hardy spaces of Fefferman and Stein ([30]), the classical Hardy-Lorentz spaces of Abu-Shammala and Torchinsky ( [1]), the anisotropic Hardy spaces of Bownik ([9]) and the anisotropic weak Hardy spaces of Ding and Lan ( [24]) as special cases. Some basic properties of H p,q A (R n ) are also obtained in this section (see Propositions 2.7 and 2.8 below).…”