Hytönen, Tuomas; Yang, Dachun; Yang, Dongyong The Hardy space \(H^1\) on non-homogeneous metric spaces. (English) Zbl 1250.42076 Math. Proc. Camb. Philos. Soc. 153, No. 1, 9-31 (2012). According to T. Hytönen [Publ. Mat., Barc. 54, No. 2, 485–504 (2010; Zbl 1246.30087)], a metric measure space \((X,d,\mu)\) is said to be upper doubling if \(\mu\) is a Borel measure on \(X\) and there exist a dominating function \(\lambda (x, r)\) and a positive constant \(C\) such that for each \(x \in X, r \to \lambda(x,r)\) is non-decreasing and, for all \(x \in X\) and \(r>0\), \[ \mu(B(x,r)) \leq \lambda(x,r) \leq C \lambda(x, r/2), \] where \( B(x,r) = \{ y \in X : d(x,y) < r \}\).Obviously, a space of homogeneous type is a special case of upper doubling spaces, where one can take the dominating function \(\lambda(x,r) = \mu(B(x,r))\). Moreover, let \(\mu\) be a non-negative Radon measure on \({\mathbb R}^n\) which only satisfies the polynomial growth condition \[ \mu( \{ y \in {\mathbb R}^n: | x-y | <r \}) \leq C r^a. \] By taking \(\lambda(x,r) = Cr^a\), we see that \(({\mathbb R}^n, | \cdot |, \mu)\) is also upper doubling measure space.A metric space \((X,d)\) is said to be geometrically doubling if there exists some \(N_0 \in {\mathbb N}\) such that for any ball \(B(x,r) \subset X\), there exists a finite ball covering \(\{ B(x_i, r/2)\}_i\) of \(B(x,r)\) such that the cardinality of this covering is at most \(N_0\).Let \((X,d,\mu)\) be a metric space satisfying the upper doubling condition and the geometrical doubling condition. T. Hytönen introduced the regularized BMO space RBMO\((\mu)\). The authors introduce the atomic Hardy space \(H^1(\mu)\) and show that the dual space of \(H^1(\mu)\) is RBMO\((\mu)\). As an application they obtain the boundedness of Calderón–Zygmund operators from \(H^1(\mu)\) to \(L^1(\mu)\). Reviewer: Yasuo Komori (Numazu) Cited in 1 ReviewCited in 61 Documents MSC: 42B30 \(H^p\)-spaces 42B35 Function spaces arising in harmonic analysis Keywords:Hardy space; RBMO; non-homogeneous space; upper doubling; Calderón–Zygmund operator Citations:Zbl 1246.30087 PDFBibTeX XMLCite \textit{T. Hytönen} et al., Math. Proc. Camb. Philos. Soc. 153, No. 1, 9--31 (2012; Zbl 1250.42076) Full Text: DOI arXiv References: [1] DOI: 10.1090/S0002-9939-98-04317-2 · Zbl 0897.28008 [2] Yosida, Functional Analysis (1995) · Zbl 0830.46001 [3] DOI: 10.1007/BF02393237 · Zbl 1060.30031 [4] DOI: 10.1006/aima.2001.2011 · Zbl 1015.42010 [5] DOI: 10.1007/s10114-011-9118-7 · Zbl 1266.42061 [6] DOI: 10.1007/PL00004432 [7] Journé, Calderón–Zygmund operators, pseudodifferential operators and the Cauchy integral of Calderón (1983) [8] DOI: 10.1007/BF02546524 · Zbl 0097.28501 [9] DOI: 10.1007/BF02392690 · Zbl 1065.42014 [10] Hytönen, Publ. Mat. 54 pp 485– (2010) · Zbl 1246.30087 [11] DOI: 10.1090/S0002-9939-08-09365-9 · Zbl 1273.42021 [12] DOI: 10.1017/S030500410400800X · Zbl 1063.42012 [13] DOI: 10.1090/S0002-9939-98-04201-4 · Zbl 0897.28007 [14] DOI: 10.1007/978-1-4613-0131-8 · Zbl 0985.46008 [15] DOI: 10.1090/S0002-9904-1977-14325-5 · Zbl 0358.30023 [16] Coifman, Analyse harmonique non-commutative sur certains espaces homogènes (1971) · Zbl 0224.43006 [17] DOI: 10.1090/S0002-9939-05-07781-6 · Zbl 1113.42008 [18] DOI: 10.1090/S0002-9947-02-03131-8 · Zbl 1021.42010 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.