New book published by Michael Wilson on Littlewood-Paley Theory
Release Date: 10-30-2007
Littlewood-Paley theory was created by J. E. Littlewood and R. E. A. C. Paley. It provides a handle for knowing when and how a decompose-and-recombine process can be made to work. "In analysis," Wilson explains, "one is constantly decomposing complicated functions, such as signals, into sums usually infinite series of simpler functions. One solves a problem for a given function by solving it on each of its pieces and then summing them all back together. The difficulty is that sometimes the recombined function doesn't make sense."
In addition, the connection between a function and its square function is tight but not super-tight. Sometimes one needs to understand how a function's behavior on one set affects the square function on another set, or one needs to know the reverse. A way to attack this connection is through weighted integrals, thus the "weighted" in the book's title. "Exponential-square integrability" refers to how closely the function and the square function control each other.
Applications of weighted Littlewood-Paley theory include positivity results for Schroedinger operators (where the connection with kinetic energies becomes explicit), boundedness of singular integral and Fourier multiplier operators, and the convergence of the Calderon reproducing formula (one of the foundations of wavelet theory), which is much more robust than many people seem to be aware of.
The book covers some standard material, some fairly recent theorems, and some new approaches to old results. Wilson has geared the book toward graduate students who have an understanding of measure theory and who have seen some functional analysis.
The idea of writing a book came to Wilson in 2000 after a presentation at the Universidad Autonoma de Madrid, Spain, where he was asked, "Where does one go to learn these things?" Most of the book was written during a sabbatical at the Universidad de Sevilla. Wilson gives credit for its completion to the kindness and hospitality of Carlos Perez Moreno, at the Universidad de Sevilla, and to Roger Cooke, for insightful criticisms of the earlier chapters.
Wilson received his PhD in mathematics from UCLA in 1981 and came to UVM after post-docs at the University of Chicago and the University of Wisconsin (Madison) in 1986. He has held visiting positions at Rutgers University (New Brunswick) and the Universidad de Sevilla.