A \(d\)-Ahlfors-regular quasi-metric space \((X,\rho, \mu)\) consists of a set \(X\), an equivalence class of quasi-metrics \(\rho\), and a nonnegative measure \(\mu\) such that \[ c \, r^d \leq \mu \big( B_\rho (x,r) \big) \leq C \, r^d, \qquad x\in X, \quad r>0, \] \(0<c<C< \infty\) (or restricted to some \(r\)), where \(B_\rho (x,r)\) are balls centered at \(x\in X\) of radius \(r\) with respect to \(\rho\). Here \(0<d<\infty\).
The book deals mainly with the theory of Hardy spaces \(H^p (X)\), extending the classical theory of the Hardy spaces \(H^d (\mathbb R^d)\), restricted to \(\frac{d}{d+1} <p<\infty\), preferably \(\frac{d}{d+1} <p \leq 1\). After the general set-up of the “Geometry of quasi-metric spaces” (Chapter 2) and “Analysis of spaces of homogeneous type” (Chapter 3), the Hardy spaces \(H^p (X)\) are introduced in Chapter 4, “Maximal theory of Hardy spaces”, in terms of distributions on \((X, \rho, \mu)\) and related grand maximal functions following the nowadays classical Fefferman-Stein approach in \(\mathbb R^d\). Chapter 5, “Atomic theory of Hardy spaces”, introduces related atomic Hardy spaces. They coincide with the already introduced Hardy spaces in Chapter 4. This theory is extended in Chapter 6, “Molecular and ionic theory of Hardy spaces”, to characterizations in terms of molecules and ions (being atoms where the first moments must be small, but not necessarily zero). This is complemented in Chapter 7, “Further results”, by some special assertions, including duality. Chapter 8, “Boundedness of linear operators defined on \(H^p (X)\)”, applies this theory to several types of linear operators, including fractional integral operators and operators of Calderón-Zygmund type. Chapter 9, “Besov and Triebel-Lizorkin spaces on Ahlfors-regular quasi-metric spaces”, extends this theory to the indicated spaces. The subtitle of the book “A sharp theory” refers to the intention of the authors to develop the theory of Hardy \(H^p (X)\) within natural restrictions for \(p\), \(p_0 <p \leq 1\), where \(p_0\) reflects the underlying geometry. This generalizes not only the natural restriction \(p_0 = \frac{d}{d+1}\) in \(\mathbb R^d\), but improves decisively some results already available in the literature in a remarkable and natural way.