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Weighted norm inequalities and related topics. (English) Zbl 0578.46046
North-Holland Mathematics Studies, 116 Notas de Matemática (104). Amsterdam - New York - Oxford: North-Holland. X, 604 p. \$ 64.75; Dfl. 175.00 (1985).
The book is a very good exposition of a wide range of topics concentrating around weighted norm inequalities for singular integral operators and Hardy spaces. The authors have presented in a very natural and readable manner an enormously rich material with a lot of interplays making it rather difficult for exposition. A reader is supposed to know basic facts from functional analysis, complex analysis, measure and integration. Each chapter is concluded by a very useful section ”Notes and Further Results” containing reviews of rich additional material.
The organization of the book is given by the contents: Classical Theory of Hardy Spaces (Harmonic functions. Poisson representation. Subharmonic Functions. Riesz factorization theorem. Some classical inequalities. The conjugate function. $$H^ p$$ as a linear space. Canonical factorization theorem. The Helson-Szegö theorem. The dual of $$H^ 1$$. The Corona theorem.). Calderon-Zygmund Theory (The Hardy-Littlewood maximal function and the Calderon-Zygmund decomposition. Norm estimates for the maximal function. The sharp maximal function and the space BMO. Harmonic and subharmonic functions in a half-space. Singular integral operators. Multipliers). Real Variable Theory of Hardy Spaces $$(H^ p$$ spaces for the upper half plane. Maximal function characterization of $$H^ p$$ atomic decomposition. $$H^ p$$ spaces in higher dimensions. Dual spaces. Interpolation of operators between $$H^ p$$ spaces. Estimates for operators acting an $$H^ p$$ spaces.). Weighted norm inequalities. (The condition $$A_ p$$. The reverse Hölder’s inequality and the condition $$A_{\infty}$$. Weighted Norm Inequalities for singular integrals. Two- weights norm inequalities for maximal operators. Factorization and extrapolation. Weights in product spaces). Vector Valued Inequalities. (Operators acting on vector valued functions: some basic facts. A theorem of Marcinkiewicz and Zygmund. Vector valued singular integrals. Applications: Some maximal inequalities. Applications: Some Littlewood- Paley theory. Vector valued inequalities and $$A_ p$$ weights.) Factorization Theorems and Weighted Norm Inequalities. (Factorization through $$weak$$-L$${}^ p$$. Nikishin’s theorem. Factorization through $$L^ p$$. Factorization of operators with values in $$L^ q$$. The dual problem. Weighted norm inequalities and vector valued inequalities. The two weights problem for classical operators. Weights of the form $$| x|^ a$$ for homogeneous operators. Directional Hilbert transforms).
Reviewer: N.M.Zobin

##### MSC:
 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis 46E15 Banach spaces of continuous, differentiable or analytic functions 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 42B25 Maximal functions, Littlewood-Paley theory 42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces