Atomic characterizations of variable Hardy spaces on domains and their applications
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Tusi Mathematical Research Group
ORIGINAL PAPER
Atomic characterizations of variable Hardy spaces on domains and their applications Xiong Liu1 Received: 17 May 2020 / Accepted: 27 October 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract Let 𝛺 be a proper open subset of ℝn and p(⋅) ∶ 𝛺 → (0, ∞) a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the author introduces the variable Hardy space H p(⋅) (𝛺) on 𝛺 by the radial maximal function and then characterize the space H p(⋅) (𝛺) via grand maximal functions and atoms. Moreover, the author introduces the variable BMO space BMOp(⋅) (𝛺) and the variable Hölder space 𝛬p(⋅), q, d (𝛺) on 𝛺 . As applications of atomic characterizations of H p(⋅) (𝛺) , the author shows that 𝛬p(⋅), q, d (𝛺) is the dual space of H p(⋅) (𝛺) . In particular, when 𝛺 is a bounded Lipschitz domain p(⋅) in ℝn , the author further obtains H p(⋅) (𝛺) = Hr (𝛺) , BMOp(⋅) (𝛺) = BMOp(⋅) (𝛺) z p(⋅), q, 0 (𝛺) with equivalent (quasi-)norm. Here the variand 𝛬p(⋅), q, 0 (𝛺) = 𝛬z p(⋅) able Hardy space Hr (𝛺) is defined via restricting arbitrary elements of (𝛺) ∶= {f ∈ BMOp(⋅) (ℝn ) ∶ supp (f ) ⊂ 𝛺} and H p(⋅) (ℝn ) to 𝛺 , BMOp(⋅) z p(⋅), q, d p(⋅), q, d n 𝛬z (𝛺) ∶= {f ∈ 𝛬 (ℝ ) ∶ supp (f ) ⊂ 𝛺} , where H p(⋅) (ℝn ) , BMOp(⋅) (ℝn ) p(⋅), q, d n (ℝ ) , respectively, denote the variable Hardy space, the variable BMO and 𝛬 space and the variable Hölder space on ℝn , and 𝛺 denotes the closure of 𝛺 in ℝn . The above results extend the main results in Miyachi (Studia Math 95:205–228, 1990) to the case of variable exponents. Keywords Variable Hardy space · Atom · Maximal function · Variable BMO space · Duality · Domain Mathematics Subject Classification 42B30 · 42B25 · 46A20 · 42B35 · 46E30
Communicated by Juan Seoane Sepúlveda. * Xiong Liu [email protected] 1
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, People’s Republic of China Vol.:(0123456789)
X. Liu
1 Introduction The real-variable theory of Hardy spaces on the n-dimensional Euclidean space ℝn was initiated by Stein and Weiss [28] and then systematically developed by Fefferman and Stein [13]. As the development of the real-variable theory of Hardy spaces, these spaces play important roles in various fields of analysis and partial differential equations (see, for instance, [13, 21, 26, 28]). Moreover, since Hardy spaces on domains of ℝn have significant applications in the regularity theory of elliptic boundary value problems, the research of the real-variable theory of Hardy spaces (see, for instance, [1, 4, 5, 18, 20, 29]) and regularity for the inhomogeneous Dirichlet and Neumann problems (see, for instance, [2, 3, 6, 12]) on domains has inspired great interests. In particular, Miyachi [20] introduced the Hardy space H p (𝛺) in the open subset 𝛺 of ℝn by means of maximal functions and obtained its atomic characterization. Furthermore, the duality theory of Hardy space H p (𝛺) was also studied in p p [20]. Moreover, Chang et al. [6] introduced the Hardy spa
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