REAL-VARIABLE CHARACTERIZATIONS OF VARIABLE HARDY SPACES ON LIPSCHITZ DOMAINS OF ℝ n

Let Ω be a proper open subset of Rn and p(·) : Ω → (0, ∞) be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the author introduces the “geometrical” variable Hardy spaces H p(·) r (Ω) and H p(·) z (Ω) on Ω, 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 Rn. Moreover, the author further introduces the “geometrical” variable local Hardy spaces h p(·) r (Ω), and then establishes the atomic characterization of h p(·) r (Ω) when Ω is a bounded Lipschitz domain of Rn.