在Rn 上關於哈地空間的理論最早是由Fefferman 和Stein所提出的, 他們所提出的結果提供了許多涉及卷積算子的精確估計的應用。Hardy 空間最重要的應用之一是當p ∈ (0, 1] 時，它們是Lebesgue 空間的良好替代品。這些在Rn 上Hardy空間理論也對於各種分析領域和偏微分方程扮演著重要角色，但是要檢查一個tempered distribution f 是否屬於Hp 是不容易的；然而Coifman and Latter 給了它原子的刻畫就解決了這個問題， Coifman 針對一維的狀況作了原子的定義，而Latter 則將它的結果做了推廣到多維的原子定義。之後，Jonsson, Sj¨ogren 和Wallin 三人則針對特別的閉子集上研究哈地空間的性質，而Miyachi 則是討論開集上的哈地空間。本文則針對特定開集做有關原子分解及極大函數刻畫。

The theory of Hardy spaces over Rn was originated by Fefferman and Stein , which was generalized several years ago to the case of proper subsets of Rn. The theory of Hardy spaces on the Euclidean space Rn plays an important role in various fields of analysis and partial differential equations; Their work resulted in many applications involving sharp estimates for convolution operators. It is not immediately apparent how much of a role the differential structure of Rn plays in obtaining these results. One of the most important applications of Hardy spaces is that they are good substitutes of Lebesgue spaces when p ∈ (0, 1]. However, it is not so easy to check whether a tempered distribution f belongs to Hp. An explicit representation theorem for functions in Hp, p ≤ 1, is given by Coifman and Latter, by means of a purely real variable constructure. The pioneering work of generalization was done by Jonsson, Sj¨ogren and Wallin for the case of suitable closed subsets and by Miyachi for the case of open subsets. In this article we study Hardy spaces over certain open subsets Ω ⊂ Rn. We first define the Hardy space on Ω by means of atoms, and then give different maximal function characterizations.

[1] M. Bownik, Anisotropic Hardy spaces and wavelets, Mem. Amer. Math. Soc. 164 (2003), no. 781,
1-122.
[2] M. Bownik, Anisotropic Triebel-Lizorkin spaces with doubling measures, J. Geom. Anal. 17 (2007),
387-424.
[3] A. P. Calderon, An atomic decomposition of distributions in parabolic Hp spaces, Adv. Math. 25
(1977), 216-225.
[4] A. P. Calderon and A. Torchinsky, Parabolic maximal functions associated with a distribution, Adv.
Math. 16 (1975), 1-64.
[5] A. P. Calderon and A. Torchinsky, Parabolic maximal functions associated with a distribution II,
Adv. Math. 24 (1977), 101-171.
[6] D.-C. Chang, G. Dafni, and E. M. Stein, Hardy spaces, BMO, and boundary value problems for
the Laplacian on a smooth domain in Rn, Trans. Amer. Math. Soc. 351 (1999), 1605-1661.
[7] D.-C. Chang, S. G. Krantz, and E. M. Stein, Hp theory on a smooth domain in Rn and elliptic
boundary value problems, J. Funct. Anal. 114 (1993), 286-347.
[8] R. R. Coifman, A real variable characterization of Hp, Studia Math. 51 (1974), 269-274.
[9] R. R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes, Compensated compactness and Hardy
spaces, J. Math. Pures Appl. 72 (1993), 247-286.
[10] C. Feerman and E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), 137-193.
[11] G. B. Folland and E. M. Stein, Hp Spaces on Homogeneous Group, Princeton University Press,
Princeton, N. J., 1982.
[12] A. Jonsson, P. Sjogren, and H. Wallin, Hardy and Lipschitz spaces on subsets of Rn, Studia Math.
80 (1984), 141-166.
[13] R. H. Latter, A characterization of Hp(Rn) in terms of atoms, Studia Math. 62 (1978), 93-101.
[14] C.-C. Lin and K. Stempak, Atomic Hp spaces and their duals on open subsets of Rd, Forum Math.
27 (2015), 2129-2156.
[15] A. Miyachi, Hp spaces over open subsets of Rn, Studia Math. 95 (1990), 205-228.
[16] S. Muller, Hardy space methods for nonlinear partial dierential equations, Tatra Mt. Math. Publ.
4 (1994), 159-168.
[17] S. Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Muller,
Comm. Partial Dierential Equations 19 (1994), 277-319.
[18] E. M. Stein Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals,