Real-variable Characterizations of Variable Hardy Spaces on Lipschitz Domains of Rn

Bull. Korean Math. Soc. Published online January 8, 2021

Xiong Liu
School of Mathematics and Statistics, Lanzhou University

Abstract : Let Ω be a proper open subset of R^n and p(·) : Ω → (0, ∞) be a variable exponent function satisfying the globally log-Hölder continuous condition. In this paper, the author first introduces the variable Hardy spaces H^(p(·))_ r (Ω) and H^(p(·))_ z (Ω) on Ω, respectively, by restricting arbitrary elements of H^(p(·))(R^n ) to Ω, and restricting elements of H^(p(·))(R^n ) which are zero outside to Ω, where H^(p(·))(R^n ) denotes the variable Hardy space on R^n and denotes the closure of Ω in R^n , and then obtains the grand maximal function characterizations of H^(p(·))_r (Ω) and H^(p(·))_z (Ω) when Ω is a strongly Lipschitz domain of R^n . Moreover, the author further introduces the variable local Hardy spaces h^(p(·))_r(Ω) and h^(p(·))_z (Ω) in a similar way, and characterizations in terms of atoms or duality theory of h^(p(·))_r (Ω) and h^(p(·))_z (Ω), when Ω is a bounded Lipschitz domain of R^n , are also established. These results are also new even when q ∈ ((1, 2) ∪ (2, ∞]) and p(·) := p for all p ∈ ( n/(n+1) , 1], where for the case that q = 2 are given in [7, 9].