The Hardy space \(H^1\) on non-homogeneous metric spaces (Q2898401)

According to \textit{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 \textit{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 \(r>0\), NEWLINE\[NEWLINE \mu(B(x,r)) \leq \lambda(x,r) \leq C \lambda(x, r/2), NEWLINE\]NEWLINE where \( B(x,r) = \{ y \in X : d(x,y) < r \}\).NEWLINENEWLINEObviously, 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 NEWLINE\[NEWLINE \mu( \{ y \in {\mathbb R}^n: | x-y | <r \}) \leq C r^a. NEWLINE\]NEWLINE By taking \(\lambda(x,r) = Cr^a\), we see that \(({\mathbb R}^n, | \cdot |, \mu)\) is also upper doubling measure space.NEWLINENEWLINEA metric space \((X,d)\) is said to be \textit{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 \(N_0\).NEWLINENEWLINELet \((X,d,\mu)\) 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 \(L^1(\mu)\).