Real-variable Characterizations of Orlicz-Hardy Spaces on Strongly Lipschitz Domains of ℝⁿ

2011/07/17 by Yang, Dachun, Yang, Sibei
#35J25 #42B20 #42B25 #42B37 #47B38 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Primary: 42B30 #Secondary: 42B35

paper · doi:10.48550/arxiv.1107.3267

Abstract

Let Ω be a strongly Lipschitz domain of ℝⁿ, whose complement in ℝⁿ is unbounded. Let L be a second order divergence form elliptic operator on L² (Ω) with the Dirichlet boundary condition, and the heat semigroup generated by L have the Gaussian property (G_diam(Ω)) with the regularity of their kernels measured by μ∈(0,1], where diam(Ω) denotes the diameter of Ω. Let Φ be a continuous, strictly increasing, subadditive and positive function on (0,∞) of upper type 1 and of strictly critical lower type p_Φ∈(n/(n+μ),1]. In this paper, the authors introduce the Orlicz-Hardy space H_{Φ, r}(Ω) by restricting arbitrary elements of the Orlicz-Hardy space H_Φ(ℝⁿ) to \boz and establish its atomic decomposition by means of the Lusin area function associated with \e^-tL\_t≥0. Applying this, the authors obtain two equivalent characterizations of H_{Φ, r}(\boz) in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by L.

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