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The Hardy space \(H^1\) with non-doubling measures and their applications. (English) Zbl 1316.42002

Lecture Notes in Mathematics 2084. Cham: Springer (ISBN 978-3-319-00824-0/pbk; 978-3-319-00825-7/ebook). xiii, 653 p. (2013).
During the last century Harmonic Analysis was developed not only on euclidean spaces with the Lebesgue measure, but with respect to weights or even measures satisfying the doubling condition where the same tools can be applied. Also Calderón-Zygmund theory, singular integrals, Littlewood-Paley theory as well as the theory of Hardy spaces and BMO were extended to the so-called homogeneous spaces, that is metric spaces \((X,d,\mu)\) with measures satisfying the doubling condition \(\mu(B(x,2r))\leq C \mu(B(x,r))\) for \(x\in X\) and \(r>0\). Most of the results in the classical theory could be shifted to this new setting and a number of applications were achieved. Only in the last decades the use of weaker assumptions on the measure were shown to be enough to be able to develop Hardy spaces and Calderón-Zygmund operators in this abstract setting. The book under review is devoted to the study of the Hardy spaces defined with respect to non-doubling measures and different applications to boundedness of classical operators in this new setting. The authors consider two possible extensions of Hardy spaces and Calderón-Zygmund theory to the setting of non-doubling measures. The first one is to keep the underlying space as \(\mathbb R^D\) but replacing the Lebesgue measure by a measure \(\mu\) satisfying the “polynomial growth condition” given by \(\mu(B(x,r))\leq C_0r^n \) for some \(0<n\leq D\) and the second one is the setting of metric spaces, where they consider the notion introduced by T. Hytönen [Publ. Mat., Barc. 54, No. 2, 485–504 (2010; Zbl 1246.30087)] called non-homogeneous spaces (which includes both homogeneous and with polynomial growth). The monograph contains two main blocks, the first one consisting of the analysis of spaces defined on \(\mathbb R^D\) with non-doubling measures \(\mu\), which is divided into six chapters, and the second one devoted to spaces defined on non-homogeneous metric spaces, which is divided into two chapters. We now explain a bit the contents of the first part of the book, where most of the results are taken from the work by X. Tolsa [Math. Ann. 319, No. 1, 89–149 (2001; Zbl 0974.42014)] and his collection of papers on the topic. The first chapter is of introductory character and includes, in the setting of measures of polynomial growth, some topics concerning covering lemmas, the notion of doubling cube, Lebesgue’s differentiation theorem and Calderón-Zygmund decomposition. In chapter 2 the authors study the approximation of the identity in this setting. They first introduce the notion of the coefficient \(\delta(Q,R)\) associated to a pair of cubes \(Q\) and \(R\) given by \[ \delta(Q,R)=\max\Big\{ \int_{Q_R\setminus Q}\frac{d\mu}{|x-z_Q|^n}, \int_{R_Q\setminus R}\frac{d\mu(x)}{|x-z_R|^n}\Big\}, \] where \(R_Q\) stands for the smallest cube concentric with \(R\) containing \(Q\) and \(R\) and \(a_R\) is the center of \(R\). Using this notion they study cubes of different generations (which correspond to dyadic cubes in the classical situation) and they manage to construct the functions which originate the kernels of the approximation of the identity to be used in the setting. The purpose of Chapter 3 is to introduce the spaces \(H^1(\mu)\) and \(BMO\)-type spaces, establishing the John-Nirenberg inequality, equivalent formulations to \(RBMO(\mu)\) and \(H^1(\mu)\) and the duality result. The basic definition is the space \(RBMO(\mu)\) consisting of locally integrable functions satisfying two conditions: \[ \sup_{Q}\frac{1}{\mu(\eta Q)}\int_Q |f- m_{\tilde Q}f|d\mu<\infty \,\, \text{and}\,\, |m_Q(f)-m_R(f)|\leq C(1+\delta(Q,R)) \] for any two doubling cubes \(Q\subset R\), where \(\tilde Q\) stands for the smallest cube of the form \(2^kQ\) for some \(k\in \mathbb Z_+\) satisfying \(\mu(2Q)\leq 2^{D+1}\mu(Q)\). In chapter five the authors first establish some weighted estimates for the local sharp maximal functions and several interpolation results and then analyze boundedness of singular integrals on \(L^p(\mu)\) and \(H^1(\mu)\), as well as the boundedness of maximal singular operators and commutators. The last chapter of this first part contains the versions of Littlewood-Paley and maximal operators adapted to the approximations of the identity generated in the context of measures with polynomial growth.
The second part of the book follows a similar program. The aim is to do an analogous development but now in the setting of non-homogeneous spaces. The authors consider \((X,d,\nu)\) such that \(d\) is a metric on \(X\) and \(\nu\) is a Borel measure satisfying the upper doubling condition \(\nu(B(x,r))\leq \lambda(x,r)\), where \(\lambda:X\times (0,\infty)\to (0,\infty)\) is non-decreasing in the second variable and \(\lambda(x,r)\leq C \lambda(x,r/2)\), and also geometrically doubling, that is that any ball \(B(x,r)\) can be covered by at most \(N_0\) balls of radius \(r/2\). In this context the notion of \(RBMO(X,\nu)\) and the corresponding \(H^1(X,\nu)\) together with their equivalent formulations are presented. Finally, the boundedness of the Calderón-Zygmund operators in the unweighted and weighted situation, or linear and multilinear commutators acting on these spaces are provided in the two chapters included in this second part.

MSC:

42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
42B30 \(H^p\)-spaces
42B35 Function spaces arising in harmonic analysis
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
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