In mathematical analysis, the importance of estimates for oscillatory integrals cannot be overestimated. Oscillatory integrals of various forms naturally arise in many areas and estimates for them play crucial roles. At the heart of the problem is to understand (quantitatively) the decay property which arises from cancelation due to oscillation. Many outstanding open problems in classical harmonic analysis can also be recast in this perspective. Fourier transforms of functions and measures are most typical examples of oscillatory integrals. In this book the authors undertake the work of presenting various decay properties of Fourier transforms (most of them are closely related to the authors’ previous works) with special emphasis on interplay of analytic and geometric aspects.
The book is divided into two parts. After a brief introduction to basic properties of the Fourier transform, the first part (Chapters 2–4) mainly concerns pointwise estimates and the second part (Chapters 5–8) deals with decay properties of Fourier transforms when they are averaged over the unit sphere (over rotation). In Chapter 2 results about Fourier transforms on convex sets are presented and these are applied to hyperbolic means and generalized Bochner-Riesz means. Chapter 3 is devoted to Fourier transforms of convex and oscillating functions. In Chapter 4 existence of Fourier transforms of radial functions \(f(x)=f_0(|x|)\) is shown when \(f_0\) is in \(MV_{\alpha}^b\). In Chapter 5 the authors prove \(L^2\) averaged decay of Fourier transforms of indicator functions of convex sets. In Chapter 6 and Chapter 7, specializing to convex sets in \(\mathbb R^2\), the authors further the study of the previous chapter by considering \(L^1\) averaged decay and a maximal estimate in radial direction. In Chapter 8, the optimal averaged decay estimates for Fourier transform of measures on the space curves are shown under a certain nondegeneracy condition.