Anal. Geom. Metr. Spaces 2016; 4:336–362 Research Article Open Access Xiaming Chen, Renjin Jiang, and Dachun Yang* Hardy and Hardy-Sobolev Spaces on Strongly Lipschitz Domains and Some Applications DOI 10.1515/agms-2016-0017 Received November 8, 2016; accepted December 19, 2016 n p Abstract: Let Ω ⊂ R be a strongly Lipschitz domain. In this article, the authors study Hardy spaces, Hr (Ω) Hp Ω H1,p Ω H1,p Ω Ω p 2 n and z ( ), and Hardy-Sobolev spaces, r ( ) and z,0 ( ) on , for ( n+1 , 1]. The authors establish grand maximal function characterizations of these spaces. As applications, the authors obtain some div-curl lemmas in these settings and, when Ω is a bounded Lipschitz domain, the authors prove that the divergence u f f 2 Hp Ω H1,p Ω equation div = for z ( ) is solvable in z,0 ( ) with suitable regularity estimates. Keywords: Hardy space; Hardy-Sobolev space; grand maximal function; div-curl formula; divergence equa- tion MSC: Primary 42B30; Secondary 42B37, 46E35, 42B25, 35F15 1 Introduction n A domain Ω of R (n ≥ 2) is said to be strongly Lipschitz if it is a Lipschitz domain and its boundary ∂Ω is a nite union of parts of rotated graphs of Lipschitz maps and, at most, one of these parts is possibly unbounded; for the denition of Lipschitz domains, we refer the reader to [12]. p n p n The real Hardy space H (R ), p 2 (0, 1], is important since it is a good substitution of L (R ) and has many applications in PDE and analysis; see Stein [26]. Hardy spaces on domains, with applications to the boundary value problems for the Laplace equation, have been studied, for instance, in [2, 6–8, 10, 15, 18– 21, 23, 27, 28]. There are mainly two kinds of Hardy spaces on domains, one is the restriction of Hardy spaces to domains, the other is the collection of elements in Hardy spaces with supports in the considered domains. Let us recall the denition as follows (see, for instance, [3, 8, 21]). n For an open set Ω ⊂ R , let D(Ω) be the collection of all smooth functions compactly supported in Ω and D0 Ω D Ω p 2 n Hardy space Hp Ω ( ) the space of all distributions on ( ). For ( n+1 , 1], the z ( ) is dened as p 0 n p n Hz (Ω) := ff 2 D (R ): f 2 H (R ), suppf ⊂ Ωg p quasi-norm kf k p kf k p n Hardy space H Ω equipped with the Hz (Ω) := H (R ), and the r ( ) as p 0 p n Hr (Ω) := ff 2 D (Ω): 9F 2 H (R ), s. t. FjΩ = f g quasi-norm kf k p kFk p n fF 2 equipped with the Hr (Ω) := infF H (R ), where the inmum is taken over the set p n 0 n 0 p p H (R ): FjΩ = f g. Since D (R ) ⊂ D (Ω), it follows that, every f in Hz (Ω) is actually in Hr (Ω) with kf k p kf k p Hz (Ω) ≥ Hr (Ω). Xiaming Chen: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China, E-mail: [email protected] Renjin Jiang: Center for Applied Mathematics, Tianjin University, Tianjin 300072, China and School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China, E-mail: [email protected] *Corresponding Author: Dachun Yang: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathemat- ics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China, E-mail: [email protected] © 2016 Xiaming Chen et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. Hardy and Hardy-Sobolev Spaces Ë 337 Our rst main result is the grand maximal function characterization of Hardy spaces on domains. To this end, we recall the following two grand maximal functions. x 2 n p 2 n F Ω ϕ 2 D n For any R and ( n+1 , 1], denote by x( ) the collection of all (R ), for which there exists a cube Q such that supp ϕ ⊂ Q, x 2 Q, cQ 2 Ω and 1 kϕk n ` krϕk n L∞(R ) + Q L∞(R ) ≤ jQj , here and hereafter, cQ denotes the center of the cube Q and `Q its side length. For each x 2 Ω, let Gx(Ω) := fϕ 2 Fx(Ω): ϕ = 0 on ∂Ωg. 0 n n For each f 2 D (R ) and any x 2 R , let Mz f(x) := sup jhf , ϕij. ϕ2Fx(Ω) * p* np q p W1,q Ω Ω Let := n−p , := p*−1 and 0 ( ) denote the Sobolev space with zero boundary values on . For each f W1,q Ω x 2 Ω bounded linear functional on 0 ( ), for any , let Mr f(x) := sup jhf , ϕχΩij, ϕ2Gx(Ω) where χΩ denotes the characteristic function of Ω. From the fact that ϕ 2 Gx(Ω), it follows that ϕχΩ 2 W1,q Ω M f 0 ( ), which implies that r is well dened. Then we have the following characterizations concerning Hardy spaces. In what follows, for any q 2 0 1 1 [1, ∞], we denote by q its conjugate index, namely, q + q0 = 1. Theorem 1.1. Let Ω be a strongly Lipschitz domain and p 2 n . ( n+1 , 1] p 0 n p (i) If Ω is bounded, then f 2 Hz (Ω) if and only if f 2 D (R ), supp f ⊂ Ω, Mz f 2 L (Ω) and hf , ϕi = 0 for n each ϕ 2 D with ϕ ≡ on Ω. Moreover, kf k p is equivalent to kMz f k p with the equivalent positive (R ) 1 Hz (Ω) L (Ω) constants independent of f. p 0 n p (ii) If Ω is unbounded, then f 2 Hz (Ω) if and only if f 2 D (R ), supp f ⊂ Ω, Mz f 2 L (Ω). Moreover, kf k p is equivalent to kMz f k p with the equivalent positive constants independent of f. Hz (Ω) L (Ω) In [15, Theorem 5.1], Jonsson et al. gave a result similar to Theorem 1.1. Notice that, in their Theorem 5.1, the considered set has to be unbounded (having the global Markov property) and, in the implication that p p,∞,s p,∞,s Mz f 2 L (Ω) implies f 2 H (Ω), where H (Ω) is the atomic Hardy space on Ω, f is required to belong 1 to Lloc(Ω). To the best of our knowledge, Theorem 1.1 as above is new except the case p = 1 and Ω being unbounded. We have the following characterization for the Hardy space of restricted type, whose proof uses some ideas from the proofs of Theorem 1.1 and Miyachi [21, Theorems 1 and 4]. Theorem 1.2. Let Ω be a strongly Lipschitz domain, p 2 n and q p* 0. Then f 2 Hp Ω if and only if ( n+1 , 1] = ( ) r ( ) 1,q p f is a bounded linear functional on W Ω and Mr f 2 L Ω . Moreover, kf k p is equivalent to kMr f k p 0 ( ) ( ) Hr (Ω) L (Ω) with the equivalent positive constants independent of f . p We wish to emphasize that, whereas the proof of the characterization of Hr (Ω) (Theorem 1.2) is relatively p simpler, the arguments for Hz (Ω) are not trivial, and highly depend on the geometric structure of the under- lying domain. The proof of Theorem 1.1 is composed by three steps: we rst deal with the case Ω being special Lipschitz, then the case Ω being bounded, and nally Ω being unbounded; see Section 2 below. Based on Theorems 1.1 and 1.2, we then establish grand maximal characterizations of vector-valued Hardy spaces; see Corollaries 2.13 and 2.14 below. p n In the same manner as H (R ) for p 2 (0, 1], Hardy-Sobolev spaces play important roles in partial dif- ferential equations and analysis at the borderline cases. We refer the reader to [3, 4, 13, 16, 17, 22, 24, 25] for relevant developments concerning Hardy-Sobolev spaces. Let us begin with recalling the denition of Hardy- Sobolev spaces; see Auscher et al. [3] for p = 1. 338 Ë Xiaming Chen et al. Denition 1.3. Ω ⊂ n p 2 n Hardy-Sobolev space H˙ 1,p Ω Let R be an open set and ( n+1 , 1]. The r ( ) is dened as 1,p 0 p H˙ r (Ω) := ff 2 D (Ω): rf 2 Hr (Ω)g 1,p equipped with the quasi-norm dened by kf k 1,p := krf k p . The Hardy-Sobolev space H˙ z (Ω) is also H˙ r (Ω) Hr (Ω) dened as 1,p 0 n p H˙ z (Ω) := ff 2 D (R ): rf 2 Hz (Ω)g equipped with the quasi-norm kf k 1,p := krf k p . H˙ z (Ω) Hz (Ω) n Above, rf denotes the distributional derivative of f. Precisely, if U ⊂ R is an arbitrary open set, then, 0 0 for any f 2 D (U), rf 2 D (U) is a distributional derivative of f if, for each ϕ 2 D(U) and i 2 f1, ... , ng, it holds true that hf , ∂i ϕi = − h∂i f , ϕi . Our second main result is the following grand maximal characterization of Hardy-Sobolev spaces, which p p 2 n generalizes corresponding results by Auscher et al.