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Weights, extrapolation and the theory of Rubio de Francia. (English) Zbl 1234.46003

Operator Theory: Advances and Applications 215. Basel: Birkhäuser (ISBN 978-3-0348-0071-6/hbk; 978-3-0348-0072-3/ebook). xiii, 280 p. (2011).
This book presents some recent aspects of weighted norm inequalities, in particular, it deals with generalizations and applications of the extrapolation theorem of Rubio de Francia. It is well-written and its contents are well-organized. Chapter 1 is a nice review for classical and modern weighted norm inequalities. The reviewer thinks it is useful to read this chapter not only for researchers and students who are working on weighted norm inequalities but also for mathematicians working in other areas of harmonic analysis. This book consists of two parts: Part I: One-Weight Extrapolation and Part II: Two-Weight Factorization and Extrapolation.
Chapter 2 discusses the Rubio de Francia extrapolation theorem: if the inequality \[ \int | Tf(x) |^p w(x) dx \leq C \int | f(x) |^p w(x) dx \] holds for \(p\) equal to some \(p_0\) and all \(w \in A_{p_0}\), with \(C\) depending only on the \(A_{p_0}\) constant of \(w\), then for \(1<p<\infty\), this norm inequality holds for all \(w\in A_p\). The authors give a new proof for this theorem and discuss how their proof allows a number of powerful generalizations. An outline of their proof is as follows. First introduce two versions of the iteration algorithm of Rubio de Francia. \[ Rh (x) := \sum_{k=0}^{\infty} \frac{M^k h(x)}{2^k \| M \|^k_{L^p(w) \to L^p(w)}}, \] where \(M^k = M \circ \cdots \circ M\) is \(k\) iterations of the Hardy–Littlewood maximal operator \(M\) and \[ R'h (x) := \sum_{k=0}^{\infty} \frac{{M'}^k h(x)}{2^k \| M' \|^k_{L^{p'}(w) \to L^{p'}(w)}}, \] where \(M' h = M(hw)/w\). Then \(| h(x) | \leq Rh(x), | h(x) | \leq R'h(x)\), \(\| Rh \|_{L^p(w)} \leq 2 \| h \|_{L^p(w)}\), \(\| R'h \|_{L^{p'}(w)} \leq 2 \| h \|_{L^{p'}(w)}\), and \(Rh, R'h \cdot w \in A_1\). With the help of these properties the proof is now straightforward. Beyond its simplicity, an important feature of their proof is that it makes clear exactly what the essential ingredients are. They emphasize that they do not assume that \(T\) is linear or even sublinear. Therefore they restate the extrapolation theorem as follows: given a suitably chosen family of pairs of non-negative functions \((f,g)\), if for some \(p_0\) and all \(w \in A_{p_0}\), \(\| f \|_{L^{p_0}(w)} \leq \| g \|_{L^{p_0}(w)}\), then for all \(p\) and \(w \in A_p\), \(\| f \|_{L^{p}(w)} \leq \| g \|_{L^{p}(w)}\). In the following chapters various generalizations of the extrapolation theorem are proved in this manner, and several weighted estimates can be proved by applying them to pairs of the form \((| Tf |, | f | )\). In particular, the authors give new and simpler proofs of known results and prove new theorems.
Chapter 3 introduces a generalization of the Hardy–Littlewood maximal operator and the Muckenhoupt \(A_p\) classes. As an application a vector-valued extension of the Coifman–Fefferman inequality is proved.
Chapter 4 deals with Banach function spaces, and as an application, an extrapolation theorem on variable Lebesgue spaces \(L^{p(\cdot)}\) is proved.
Part II discusses two-weight norm inequalities. The authors state that the study of two-weight norm inequalities are considerably more difficult than in the one-weight case, since the two-weight \(A_p\) condition: \[ \sup_Q \Big( \frac{1}{| Q |} \int_Q u(x)dx \Big) \Big( \frac{1}{| Q |} \int_Q v(x)^{-p'/p} dx \Big)^{p/p'} <\infty \] is not sufficient for the strong \((p,p)\) type inequality for the Hardy–Littlewood maximal operator.
They say that their approach is to find appropriate “\(A_p\) bump conditions” such as \[ \sup_Q \Big( \frac{1}{| Q |} \int_Q u(x)^r dx \Big)^{1/r} \Big( \frac{1}{| Q |} \int_Q v(x)^{-rp'/p} dx \Big)^{1/rp'} <\infty, \] and more generally \[ \sup_Q \| u^{1/p} \|_{A,Q} \| v^{-1/p} \|_{B,Q} < \infty, \] where \(\| \cdot \|_{A,Q}\) is the normalized Orlicz norm and \(A\) is a Young function. The presence of the Orlicz norms causes the proofs to be more technical than the proofs in the one-weight cases in Part I. However the authors say that the essential ingredients are the same as in the proof of the Rubio de Francia extrapolation theorem.
Chapters 6 and 7 explain two-weight factorization and extrapolation theorems. As the authors say, these chapters are unavoidably technical, both because of the nature of the conditions on the weights and because they want to develop their results in a fairly general setting. However, to clarify the situation they give a number of examples and special cases. The last two chapters discuss many applications to classical operators: the sharp maximal operator, the dyadic square function, singular integrals and fractional integrals.

MSC:

46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
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