LITTLEWOOD–PALEY 𝑔*_𝜆-FUNCTION CHARACTERIZATIONS OF MUSIELAK–ORLICZ HARDY SPACES ON SPACES OF HOMOGENEOUS TYPE

Abstract

Let (𝒳 , 𝑑, 𝜇) be a space of homogeneous type, in the sense of Coifman and Weiss, and 𝜙 : 𝒳 x [0,\infty) -> [0, \infty) satisfy that, for almost every 𝑥 \in 𝒳, 𝜙(𝑥,.) is an Orlicz function and that 𝜙(., 𝑡) is a Muckenhoupt weight uniformly in 𝑡 \in [0,\infty). In this article, by using the aperture estimate of Littlewood–Paley auxiliary functions on the Musielak–Orlicz space 𝐿^𝜙(𝒳), we obtain the Littlewood–Paley 𝑔*_𝜆-function characterization of Musielak–Orlicz Hardy space 𝐻^𝜙(𝒳). Particularly, the range of 𝜆 coincides with the best-known one.

Keywords

• space of homogeneous type
• musielak–orlicz hardy space
• littlewood–paley auxiliary function
• 𝑔*_𝜆-function