Weak Generalized Herz–Hardy Spaces

Abstract

AbstractThe main target of this chapter is to establish a complete real-variable theory of weak generalized Herz–Hardy spaces. Note that the classical weak Hardy space \(WH^p(\mathbb {R}^{n})\), with p ∈ (0, 1], plays an important role in the study of the boundedness of operators in harmonic analysis. Indeed, in 1986, to find the biggest function space X such that Calderón–Zygmund operators are bounded from X to the weak Lebesgue space \(WL^1(\mathbb {R}^{n})\), Fefferman and Soria (Studia Math. 85:1–16, 1986) introduced the weak Hardy space \(WH^1(\mathbb {R}^{n})\). In addition, let δ ∈ (0, 1] and T be a convolutional δ-type Calderón–Zygmund operator. It is well known that, for any given p ∈ (n∕(n + δ), 1], T is bounded on the classical Hardy space \(H^p(\mathbb {R}^{n})\) (see [4]). However, T is not bounded on \(H^{n/(n+\delta )}(\mathbb {R}^{n})\), which is called the critical case or the endpoint case .