Weighted Hardy spaces.

*(English)*Zbl 0676.42021
Lecture Notes in Mathematics, 1381. Berlin etc.: Springer-Verlag. iv, 193 p. DM 30.00 (1989).

The authors give the detailed version of their announcement in Bull. Am. Math. Soc., New Ser. 3, 1053-1056 (1980; Zbl 0452.43004) several results developed since then, and some applications.

In Chapter 1 they introduce the notion of weighted measures in the general context of homogeneous spaces, including the theory of \(A_ p\) weights. Chapter 2 deals with the P. Jones decomposition of these weights, including the control of the doubling condition. In Chapter 3 they treat the properties of the sharp maximal functions and the local sharp maximal functions. Chapter 4 deals with the functions defined on the upper-half space \({\mathbb{R}}_+^{n+1}\), including the nontangential maximal functions and the area function. Then in Chapter 5 they discuss the extensions of a tempered distribution on \({\mathbb{R}}^ n\) to the upper- half space by means of convolutions with the dilates of Schwartz functions. In Chapter 6 they introduce the weighted Hardy spaces. They construct in Chapter 7 a dense class of functions for these spaces of distributions. In Chapter 8 they construct the atomic decomposition for these spaces and in Chapter 9 they describe an extension of the Fefferman \(H^ 1\) duality result. They construct the dual to the Hardy spaces in Chapter 10, and deal with the continuity of various singular integral and multiplier operators on these spaces in Chapter 11. Finally in Chapter 12 they show how the complex method of interpolation applies in these context.

In Chapter 1 they introduce the notion of weighted measures in the general context of homogeneous spaces, including the theory of \(A_ p\) weights. Chapter 2 deals with the P. Jones decomposition of these weights, including the control of the doubling condition. In Chapter 3 they treat the properties of the sharp maximal functions and the local sharp maximal functions. Chapter 4 deals with the functions defined on the upper-half space \({\mathbb{R}}_+^{n+1}\), including the nontangential maximal functions and the area function. Then in Chapter 5 they discuss the extensions of a tempered distribution on \({\mathbb{R}}^ n\) to the upper- half space by means of convolutions with the dilates of Schwartz functions. In Chapter 6 they introduce the weighted Hardy spaces. They construct in Chapter 7 a dense class of functions for these spaces of distributions. In Chapter 8 they construct the atomic decomposition for these spaces and in Chapter 9 they describe an extension of the Fefferman \(H^ 1\) duality result. They construct the dual to the Hardy spaces in Chapter 10, and deal with the continuity of various singular integral and multiplier operators on these spaces in Chapter 11. Finally in Chapter 12 they show how the complex method of interpolation applies in these context.

Reviewer: K.Yabuta

##### MSC:

42B30 | \(H^p\)-spaces |

42B25 | Maximal functions, Littlewood-Paley theory |

42-02 | Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces |

42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |

42B15 | Multipliers for harmonic analysis in several variables |