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).

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 |