Princeton Mathematical Series. 43. Princeton, NJ: Princeton University Press. xiii, 695 pp. $ 69.50; £ 55.00 (1993).

The well-known author, specialist in Harmonic Analysis, adds an extensive volume to his preceding monographs in the Princeton University Press collection: {\it E. M. Stein} “Singular integrals and differentiability properties of functions” (1970;

Zbl 0207.13501); {\it E. M. Stein} and {\it G. Weiss}, “Introduction to Fourier analysis on Euclidean spaces” (1971;

Zbl 0232.42007). The book provides an up-to-date description of a broad subject featuring the author’s own contributions and a comprehensive, largely self-contained presentation of the last 20 years account. The text is an outcome of lectures delivered at Princeton University and was realized in collaboration with T. S. Murphy.
The book is organized along 13 chapters each of which is followed by an appendix with sketches of proofs and {\it further comments}. In view of a brief outline of the enormous material one should say that three major notions are investigated: 1) Maximal averages for which a simple instance is $${1 \over 2t} \int\sp t\sb{-t} f(x-y)\,dy\quad (t > 0).$$ 2) Singular integrals; an example is provided by the Hilbert transform ${1 \over\pi} \int\sp \infty\sb{- \infty} f(x-y) {dy \over y}$. 3) Oscillatory integrals; the Fourier transform is the most striking example. The author establishes new connexions giving rise to applications to PDEs, several complex variables theory, semisimple Lie groups, symmetric spaces.
Part I is mostly devoted to real-variable theory. Chapter I explains that the underlying space is to be endowed with a metric structure. As a model one may choose $\Bbb R\sp n$ with a family ${\cal B}$ of Euclidean balls, and a Borel measure $\mu$ on $B$. Let $$A\sb \delta f(x) = {1 \over \mu \bigl( B(x, \delta) \bigr) } \int\sb{B (x, \delta)} f(y)\,d \mu (y).$$ The problem is to find out about $A\sb \delta f \to f$ a.e. as $\delta \to 0$. Covering lemmas (often of Vitali type) are obtained. Singular integrals are defined by relations $$Tf(x) = \int\sb{\Bbb R\sp n} K(x,y) f(y)\,d \mu(y)$$ where the kernel $K$ is singular near $x = y$. Chapter II introduces the maximal vector-valued function $$Mf(x) = \sup\sb{B \in {\cal B}} {1 \over \vert B \vert} \int\sb B \bigl \vert f(x - y) \bigr \vert \,dy.$$ Chapter III deals with Hardy spaces; by definition $f \in H\sp p (\Bbb R\sp n)$ if there exists a Schwartz function $\Phi$ such that $\int \Phi = 1$ and $M\sb \Phi f \in L\sp p$ with $$(M\sb \Phi f) (x) = \sup\sb{t > 0} \bigl \vert (f* \Phi\sb t) (x) \bigr \vert,$$ $\Phi\sb t (x) = t\sp{-n} \Phi \left({x \over t}\right)$. Various other characterizations are established. The utility of the notion for Harmonic Analysis is described, for instance, in relation to Wiener’s Generalized Harmonic Analysis. Chapter IV presents a result due to C. Fefferman, the identification of the dual space of $H\sp 1$ with the space BMO (introduced by Nirenberg) of functions $f$ with bounded mean oscillations; $f$ is locally integrable and for each $B \in {\cal B}$, $${1 \over \vert B \vert} \int\sb B \bigl \vert f(x) - f\sb B \bigr \vert \,dx \le A,$$ with $f\sb B = \vert B \vert\sp{-1} \int\sb B f(x)\,dx$. Many singular integrals map $L\sp \infty$ to BMO. Among further results, links with martingales, Brownian motion, wavelets theory are shown. Chapter V deals with weighted inequalities for maximal functions and singular integrals.
Part II is largely “devoted to an exposition of concepts that belong to the $L\sp 2$ theory, when this is taken in its broader sense”. Chapter VI is about pseudodifferential operators, symbolic calculus related to Fourier transforms, Sobolev spaces [with $\Vert f \Vert = \sum\sb{\vert \alpha \vert < k} \Vert \partial\sp \alpha\sb x f \Vert\sb p]$, Lipschitz spaces [with $\sup\sb x \vert f(x - y) - f(x) \vert \le A \vert y \vert\sp \gamma$ $(0 < \gamma < 1)]$. Chapter VII describes a more general approach than Fourier transforms allowing almost orthogonality, i.e., the consideration of an operator $T = \sum T\sb j$ where the values $\Vert T\sb j \Vert$ are uniformly bounded and $T\sb i T\sp*\sb j$, $T\sp*\sb j T\sb i$ tend to zero suitably as $\vert i - j \vert \to \infty$. Chapter VIII deals with oscillatory integrals of first kind $\lim\sb{\lambda \to \infty} \int e\sp{i \lambda \Phi (x)} \psi (x)\,dx$ where $\Phi$ [resp. $\psi]$ is a real-valued [resp. complex-valued] smooth function. If $S$ is submanifold of $\Bbb R\sp n$ with appropriate curvature, there exists $p\sb 0 \in ]1,2[$ such that for all $1 \le p < p\sb 0$, $f \in L\sp p (\Bbb R\sp n)$ admits a Fourier transform that restricts to $S$. The author comments: “That this phenomenon was observed so late in the development of the subject is an indication of the slowness of our progress in understanding the genuinely $n$-dimensional aspects of Fourier analysis”. Chapter IX considers oscillatory integrals of second kind where $\Phi \in {\cal C}\sp \infty\sb 0 (\Bbb R\sp{n-1}\times\Bbb R\sp n)$ satisfies a nondegeneracy condition and $\psi$ is defined on $\Bbb R\sp{n-1} \times \Bbb R\sp n$ also. Chapter X is back to maximal operators. The general problem admits the following formulation: For what collection ${\cal C}$ of sets $C$ is it true that for all $f$, $\lim\sb{\delta (C) \to 0} {1 \over \vert C \vert} \int\sb C f(x - y)\,dy = f(x)$ a.e., $\delta$ denoting the diameter? The question is linked to the boundedness in $L\sp p$ and the study of $$M\sb{\cal C} f(x) = \sup\sb{C \in {\cal C}} {1 \over \vert C \vert} \int\sb C \bigl \vert f(x - y) \bigr \vert \,dy.$$ In Chapter II the question was solved affirmatively in case ${\cal C}$ admits {\it bounded eccentricity}, i.e., the ratio (in volume) between the smallest ball containing $C$ to the largest ball contained in $C$ is uniformly bounded on ${\cal C}$. A large collection of specific results are proved. In $\Bbb R\sp 2$, for the 3-parameter collection ${\cal C}$ of all rectangles centered at the origin, $M\sb{\cal C}$ is not bounded in $L\sp p$ and the a.e. limit conclusion may fail even for characteristic functions of sets having finite measures. Nevertheless, if the centered rectangles admit major axes pointed in a fixed direction or an infinite number of suitable {\it lacunary} directions, the conclusion holds for $p > 1$, but fails for $p = 1$. The convenient item is the Besicovitch set, a union of a large number of congruent thin rectangles having a high degree of overlap; it had shown its efficiency in the study of Kakeya’s {\it needle problem}. Stein can make use of triangles instead of rectangles and may comment:
“While this historical aspect has remained something of a curiosity, the Besicovitch set has come to play an increasingly significant role in real-variable theory and Fourier analysis. Indeed, our accumulated experience allows us to regard the structure of this set as, in many ways, representative of the complexities of two-dimensional sets, in the same sense that Cantor-like sets already display some of the technical features that arise in the one-dimensional case”.
Counterexamples are produced. If $1 < p < \infty$, there exists $f \in L\sp p (\Bbb R\sp 2)$ such that with respect to all rectangles $R$ centered at the origin $$\limsup\sb{\delta (R) \to 0} {1 \over [1]} \int\sb R f(x-y)\,dy = \infty \quad\text{a.e.}$$ The same situation occurs for $p = 1$ and all such rectangles admitting their sides parallel to coordinate axes. Chapter XI examines averages over variable hyperspaces.
Finally, Part III is about Heisenberg groups. The two last chapters show their applications to several complex variables and PDEs.
In order to facilitate assimilation of all this information an index of symbols could have helped. The monumental work constitutes an indispensable reference source for the present state of affairs in the domain; it can be exploited by specialists in various areas of analysis and is stimulating also in Abstract Harmonic Analysis. As a complement, readers should refer to volume 2 in the same collection: “Essay on Fourier analysis in honour of Elias M. Stein”, Princeton Mathematical Series 42 (1995;

Zbl 0810.00019), edited by {\it C. Fefferman}, {\it R. Fefferman} and {\it S. Wainger}.