Factorization theory and \(A_p\) weights.

*(English)*Zbl 0558.42012In a series of papers, culminating with the paper under review, the author has developed a new and powerful approach to the theory of weighted norm inequality with \(A_ p\) weights [see Proc. Am. Math. Soc. 83, 673–679 (1981; Zbl 0477.42011); Lect. Notes Math. 908, 86–101 (1982; Zbl 0191.42019)]. The central idea in this approach is the use of the theory of factorization of operators developed by B. Maurey.

The results obtained are nothing less than spectacular. For example, the factorization theorem of P. W. Jones [Ann. Math. (2) 111, 511–530 (1980; Zbl 0493.42030)] which was previously available through complicated “hard analysis” is reduced to a piece of basic abstract nonsense. It was pointed out by P. Jones (loc. cit.) that his factorization theorem could be used to interpolate weighted \(A_ p\) norm inequalities in the expected way. A remarkable result proved in this paper is that \(A_ p\) norm inequalities need not be interpolated since they can be extrapolated. For example, a particular case of the extrapolation theorem of Rubio de Francia states that if for a fixed \(p\), the operator \(T\) is bounded in \(L^ p(w)\) for all \(w\in A_ p\), then \(T\) is bounded in \(L^ q(w)\) for every \(w\in A_ p\), \(1<q<\infty\).

The results obtained are nothing less than spectacular. For example, the factorization theorem of P. W. Jones [Ann. Math. (2) 111, 511–530 (1980; Zbl 0493.42030)] which was previously available through complicated “hard analysis” is reduced to a piece of basic abstract nonsense. It was pointed out by P. Jones (loc. cit.) that his factorization theorem could be used to interpolate weighted \(A_ p\) norm inequalities in the expected way. A remarkable result proved in this paper is that \(A_ p\) norm inequalities need not be interpolated since they can be extrapolated. For example, a particular case of the extrapolation theorem of Rubio de Francia states that if for a fixed \(p\), the operator \(T\) is bounded in \(L^ p(w)\) for all \(w\in A_ p\), then \(T\) is bounded in \(L^ q(w)\) for every \(w\in A_ p\), \(1<q<\infty\).

Reviewer: M. Milman