The book is related to “a discrete version of Harmonic Analysis” based on deep ideas and titanic efforts of the outstanding mathematicians Elias M. Stein and Jean Bourgain. Here I can only give a brief description of the book since it is very large and includes many conceptual and technical novelties.
The book consists of six parts and certain additional auxiliary material. Each part includes a few sections. There is a good introduction with a historical survey and a lot of exercises and an extended bibliography.
Part 1. This part includes some auxiliary material that is used in forthcoming chapters. Interpolation, the Hardy-Littlewood maximal function and consequences, linear and sublinear operators on infinite dimensional vector spaces, the Euclidean Fourier transform, and the discrete Fourier transform including Poisson summation in the part. Moreover, the author includes a lot of classical inequalities: Young’s inequality, the Cauchy-Schwarz inequality, Hölder’s inequality, Minkowski’s inequality, Young’s convolution inequality, and the Hausdorff-Young inequality. Further, this chapter presents special techniques that will be used in other chapters: stopping time arguments, square functions, and Littlewood-Paley theory.
Part 2. This part starts with the \(\ell^2(\mathbb Z)\)-boundedness of Bourgain’s maximal functions along polynomial orbits. The real-variable topics developed are entropy arguments, generic chaining, and maximal Fourier multipliers on \(L^2(\mathbb R^D)\). Further, the author considers more modern approaches related to the \(L^1\)-theory: probability theory and large deviation inequalities and the Calderón-Zygmund theory, in the context of both smooth and oscillatory operators. The tools developed in this part are applied to a problem in the discrete Ramsey theory related to number theory and combinatorics. The author introduces the density increment argument and the energy increment argument, fundamental techniques in combinatorics that are transplanted to the harmonic-analytic setting. This part closes with a \(\ell^p(\mathbb Z)\)-boundedness theorem for the above maximal function, \(p>1\).
Part 3. The author presents the Ionescu-Wainger multiplier theory. He applies it to prove \(\ell^p(\mathbb Z)\)-estimates for the variational variant of Bourgain’s maximal theory. Further development of this theory uses techniques from combinatorics, abstract Hilbert space methods, probabilistic decoupling, and elementary number theory.
Next, the author considers the Magyar-Stein- Wainger’s discrete spherical maximal function in dimension \(D\geq 3\). The following generalizations are discussed and also considered: lacunary maximal operators (in particular, the lacunary spherical maximal function) and the related variation of truncations of (rough) singular integrals. The author describes also studies related to fractional integration along polynomial orbits improving certain inequalities.
Part 4. This part deals with the \(\ell^p(\mathbb Z)\)-theory, \(1<p<\infty\), of maximally modulated Hilbert transforms. For this purpose \(TT^\ast\)-arguments and monomial generalizations of Stein’s purely quadratic Carleson-type operator are developed.
Multidimensional generalizations are given in this part. So, the author establishes \(\ell^p(\mathbb Z^D)\)-estimates, \(1<p<\infty\), for the operator \[ \operatorname{sup}\limits_{P,N}\left|\sum\limits_{1\leq|n|\leq n,\, N\in\mathbb Z^D}f(m-n)K(n)e^{2\pi iP(n)}\right|, \] where the supremum is taken over all polynomials \(P : \mathbb Z^D\rightarrow\mathbb R\) of degree \(\leq d\) which do not have any linear terms, and \(K\) is a suitably normalized Calderón-Zygmund kernel. In this place, one uses a certain technique of time-frequency analysis and sublevel estimates for polynomials and Stein-Wainger’s maximally modulated oscillatory singular integrals. The author presents certain own results here.
Part 5. This part is devoted to bilinear theory, more exactly to the pointwise convergence of bilinear ergodic averages along polynomial orbits. Arithmetic Sobolev estimates are used to prove a multi-linear discrete Ramsey theorem, extending the work done in Chapter 6.
Further, one discusses an analogous theory in two separate settings: the finite field and Euclidean models. The author describes here Hahn-Banach convexity arguments and the bilinear maximal function along the parabola.
Part 6. This part consists of a conclusion and appendices and includes select open problems and further open questions. Certain personal remarks related to J. Bourgain and E.M. Stein are also given. The appendices include certain additional material. This part contains also many exercises. They are the key to understanding the material in this book.
Finally, I would like to add that Alexandru Ionescu and Terence Tao have given a very good recommendation for the book and I can join to this opinion.