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Harmonic analysis. (English) Zbl 1491.42001

Courant Lecture Notes in Mathematics 31. Providence, RI: American Mathematical Society (AMS); New York, NY: Courant Institute of Mathematical Sciences (ISBN 978-1-4704-6507-0/pbk; 978-1-4704-6893-4/ebook). vii, 101 p. (2022).
This book presents some topics related to Harmonic Analysis in various chapters not always connected to each other.
Below we make some comments on the content of the book and its presentation.
For the Fourier series of a given function, the convergence in the \(L^p\) norm of the Cesaro means of the partial sums of the series is proved by using elementary arguments. The proof of the convergence of the Fourier series is based on the boundedness of the Hilbert operator, that is the fact that convolution by the kernel \(1/x\) is a bounded operator in \(L^p\). This turns out to be a special case of operators whose kernel has a bounded Fourier transform as well as bounded integral increments for the truncated operator.
Previously the Fourier transform is considered and proved to be a bijection between functions in the Schwartz space and a unitary map of \(L^2(R^d)\). The functions that are Fourier transforms of nonnegative functions are also characterized.
As a generalization of the Hilbert transform, the author deals with convolution in \(R^d\) with the kernel \(x_j/|x|^(d+1)\), which gives rise to the Riesz transform. This operator is bounded in \(L^p(R^d)\) and this fact is applied to prove the existence of solutions of the partial differential equation \(u-Lu=f\), for \(f\) in \(L^p\) and \(L\) being a differential operator of the second order with variable coefficients that satisfy some boundedness conditions. The result is that the equation above has a solution in the Sobolev space \(W_{2,p}\) and that the \(L^p\) norm of the solution is bounded by the \(L^p\) norm of \(f\).
The Sobolev spaces are introduced, and a particular case of the Sobolev embedding theorem is proved as well as the classical boundedness of the \(L^{p^\prime}\) norm of a function by the \(L^p\) norm of its gradient, where \(p^\prime=pd/(d-p)\).
Two more topics that the author deals with are Hardy spaces and BMO spaces. For the first one, the factorization of a function in \(H_p\) as a product of an inner function and an outer function is established. This fact is applied to prediction theory, which in mathematical terms consists in approximating a function in \(L^2\) with respect to some measure by linear combinations of a sequence of stationary functions, calculating the minimum of the \(L^2\) norm of the error, and finding the minimizer.
Concerning BMO spaces, a deep theorem due to Charles Fefferman is proved: the fact that BMO is the dual space of \(H_1\). However, this result is not applied in the rest of the book.
After proving that the inverse of a function with an absolutely convergent Fourier series has also an absolutely convergent Fourier series, based on Gelfand’s theory on Banach algebras, the author finishes the book with some considerations about representations of a compact group equipped with the Haar measure. The main result presented is the Peter-Weyl theorem about finite-dimensional irreducible representations, and two examples concerning the permutation group and SO(3) are given.
In the end, some references are provided but they are not mentioned in the text.
Sometimes the author uses some notation that is not previously introduced. This and some minor writing errors should be corrected in a forthcoming edition.

MSC:

42-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces
42A20 Convergence and absolute convergence of Fourier and trigonometric series
42B15 Multipliers for harmonic analysis in several variables
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J15 Second-order elliptic equations
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