Real‐variable characterizations of local Hardy spaces on spaces of homogeneous type. Issue 5 (1st April 2021)

• Record Type: Journal Article
• Authors: He, Ziyi; Yang, Dachun; Yuan, Wen
• Is Part Of: Mathematische Nachrichten. Volume 294:Issue 5(2021)
• Page Start: 900
• Page End: 955
• Publication Date: 2021-04-01
• Subjects: atom -- Littlewood–Paley function -- local Hardy space -- maximal function -- space of homogeneous type; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201900320
• Languages: English
• ISSNs: 0025-584X

Abstract

Abstract: Let ( X, d, μ ) be a space of homogeneous type, with upper dimension μ, in the sense of R. R. Coifman and G. Weiss. Let η be the Hölder regularity index of wavelets constructed by P. Auscher and T. Hytönen. In this article, the authors introduce the local Hardy space h ∗, p ( X ) via local grand maximal functions and also characterize h ∗, p ( X ) via local radial maximal functions, local non‐tangential maximal functions, local atoms and local Littlewood–Paley functions. Furthermore, the authors establish the relationship between the global and the local Hardy spaces. Finally, the authors also obtain the finite atomic characterizations of h ∗, p ( X ) . As an application, the authors give the dual spaces of h ∗, p ( X ) when p ∈ ( ω / ( ω + η ), 1 ), which further completes the result of G. Dafni and H. Yue on the dual space of h ∗, 1 ( X ) . This article also answers the question of R. R. Coifman and G. Weiss on the nonnecessity of any additional geometric assumptions except the doubling condition for the radial maximal function characterization of H cw 1 ( X ) when μ ( X ) < ∞ .