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Weighted inequalities in Lorentz and Orlicz spaces. (English) Zbl 0751.46021
Singapore etc.: World Scientific Publishing Co. Pte. Ltd.. xii, 233 p. (1991).
Inequalities play a crucial role in mathematical analysis and its applications. Now, we have a new book written with a great skill by two specialists, devoted to exposition of recent and almost recent results in a part of this domain. The main idea is concentrated around the notion of a maximal function (m.f.) $$Mf(x)=\sup_ B {1 \over | B|} \int_ B | f(y)| dy$$, where $$B$$ is any ball with centre at $$x\in \mathbb{R}^ n$$, and a number of its generalizations as the generalized m.f., one-sided m.f., fractional order m.f., etc. These notions, originated by G. H. Hardy in 1930 and continued by B. Muckenhoupt in 1972, are involved in a number of inequalities, mostly with weights, in Orlicz classes, Zygmund classes and Lorentz spaces. The results presented in the book were obtained mostly in the last 10 years, by the authors of the book and by such mathematicians as M. Gabidzashvili, I. Genebashvili, A. Gogatishvili, P. Ortega Salvador, L. Pick, E. T. Sawyer and others. It is very fortunate that there appeared a book containing important recent results which are published often in journals of easily found libreries.
The contents of the book are:
1. Integral operators in nonweighted Orlicz classes,
2. Maximal functions and potentials in weighted Orlicz classes,
3. Singular integrals in weighted Orlicz classes,
4. Integral operators in weighted Zygmund classes,
5. Fractional maximal function in weighted Lorentz spaces,
6. Potentials and Riesz transforms in weighted Lorentz spaces.
I think this book will serve well as a recent monograph as well as a reference book.

##### MSC:
 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis 42B25 Maximal functions, Littlewood-Paley theory 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems