##
**Weight theory for integral transforms on spaces of homogeneous type.**
*(English)*
Zbl 0955.42001

Pitman Monographs and Surveys in Pure and Applied Mathematics. 92. Harlow: Longman. xii, 410 p. (1998).

The book is devoted to the the description of the theory of the weighted inequalities for a wide class of integral transforms (integral operators) on spaces of homogeneous type. This class includes: the singular integral operators, the multi-dimensional Riesz potentials and one-dimensional fractional integrals (Riemann-Liouville transform), maximal and fractional maximal functions, Fourier operators, etc. The main results in the book are given for the Lorentz and Orlicz spaces (in particular, for \(L_p\)-spaces). Except these classes in the book are considered more general Lorentz-Orlicz spaces and also weighted BMO-spaces, weighted Campanato-Morrey classes, Zygmund classes. From this point of view this new book may be considered as a continuation and extension of the book by V. Kokilashvili and M. Krbec: “Weighted inequalities in Lorentz and Orlicz spaces” (1991; Zbl 0751.46021). But it should be noted that large part of the present book is based on new results (mainly of the authors).

Contents: Preface, 1. Basic ingredients, 2. Maximal functions in Lorentz spaces, 3. Two weight criteria for integral transforms with positive kernels, 4. Solution of two weight problems for fractional maximal functions, 5. Maximal functions and singular integrals in \(L^{p,\Phi}\), 6. Maximal functions and Fourier operators in Orlicz classes, 7. Inequalities for singular integrals, 8. Modular inequalities for fractional maximal functions, 9. Two weight inequalities for singular integrals on homogeneous groups, Bibliography, Index.

The main aim of the book is the systematical and full solution of the weighted problems in terms of the necessary and sufficient conditions on the weight and an application to classical operators in Euclidean spaces. The main attention in the book is given to weighted inequalities for maximal functions (fractional maximal functions, maximal singular integrals). These results for different mentioned spaces may be found in the Chapters 2, 4, 5, and 6. In Chapter 3 the weighted problem is considered for the wide class of integral transforms which have a positive kernel and this chapter is a core for the investigations in the book and for applications in other chapters. In Chapters 7, 8 are given some new results for classical operators (singular integrals or fractional integrals) in spaces of homogeneous type. Finally, Chapter 9 is devoted to weighted inequalities in the case of homogeneous group (inequalities for singular integral operators, inequalities for Hardy and Hardy type operators) while Chapter 1 contains some preliminary results about spaces of homogeneous type, weights (\(A_p\)-weights and some generalizations) and homogeneous groups. The book should be very interesting for specialists in the field of real and complex analysis and operator theory. The wide and useful bibliography contains more than 250 papers or books, more than 100 of them were published in the recent 10 years.

Contents: Preface, 1. Basic ingredients, 2. Maximal functions in Lorentz spaces, 3. Two weight criteria for integral transforms with positive kernels, 4. Solution of two weight problems for fractional maximal functions, 5. Maximal functions and singular integrals in \(L^{p,\Phi}\), 6. Maximal functions and Fourier operators in Orlicz classes, 7. Inequalities for singular integrals, 8. Modular inequalities for fractional maximal functions, 9. Two weight inequalities for singular integrals on homogeneous groups, Bibliography, Index.

The main aim of the book is the systematical and full solution of the weighted problems in terms of the necessary and sufficient conditions on the weight and an application to classical operators in Euclidean spaces. The main attention in the book is given to weighted inequalities for maximal functions (fractional maximal functions, maximal singular integrals). These results for different mentioned spaces may be found in the Chapters 2, 4, 5, and 6. In Chapter 3 the weighted problem is considered for the wide class of integral transforms which have a positive kernel and this chapter is a core for the investigations in the book and for applications in other chapters. In Chapters 7, 8 are given some new results for classical operators (singular integrals or fractional integrals) in spaces of homogeneous type. Finally, Chapter 9 is devoted to weighted inequalities in the case of homogeneous group (inequalities for singular integral operators, inequalities for Hardy and Hardy type operators) while Chapter 1 contains some preliminary results about spaces of homogeneous type, weights (\(A_p\)-weights and some generalizations) and homogeneous groups. The book should be very interesting for specialists in the field of real and complex analysis and operator theory. The wide and useful bibliography contains more than 250 papers or books, more than 100 of them were published in the recent 10 years.

Reviewer: N.K.Karapetyants (Rostov-na-Donu)

### MSC:

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

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

42B25 | Maximal functions, Littlewood-Paley theory |

43A85 | Harmonic analysis on homogeneous spaces |

26A33 | Fractional derivatives and integrals |

46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |