Intrinsic Square Function Characterizations of Musielak-Orlicz Hardy Spaces

Abstract

In this chapter, for any α ∈ (0, 1] and \(s \in \mathbb{Z}_{+}\), we establish the s-order intrinsic square function characterizations of \(H^{\varphi }(\mathbb{R}^{n})\) by means of the intrinsic Lusin area function S α, s , the intrinsic g-function g α, s or the intrinsic g λ ∗-function g λ, α, s ∗ with the best known range λ ∈ (2 + 2(α + s)∕n, ∞), which are defined via \(\mathrm{Lip}_{\alpha }(\mathbb{R}^{n})\) functions supporting in the unit ball. A φ-Carleson measure characterization of the Musielak-Orlicz Campanato space \(\mathcal{L}_{\varphi,1,s}(\mathbb{R}^{n})\) is also established via the intrinsic function. To obtain these characterizations, we first show that these s-order intrinsic square functions are pointwisely comparable with those similar-looking s-order intrinsic square functions defined via \(\mathrm{Lip}_{\alpha }(\mathbb{R}^{n})\) functions without compact supports.