This project consists of two lines of investigation. The first part of the project studies the integration theory of real constructible functions, which by definition are sums of products of real globally subanalytic functions and their logarithms. The significance of the constructible functions is that they form the smallest class of functions that extends the globally subanalytic functions and is stable under integration. Properties of Lebesgue spaces and also multivariate harmonic analysis will be studied in the context of constructible functions, including asymptotic estimates of oscillatory integrals and possibly questions concerning the integrability of Fourier transforms of constructible functions. The main tool employed to study integration of constructible functions is the subanalytic preparation theorem. The second part of the project aims to show through a purely analytic proof that the subanalytic preparation theorem holds in a more general quasianalytic setting. An important motivation for doing so is to obtain a more informative proof of the preparation theorem that could be used to study the decidability of expansions of the real field by functions from quasianalytic classes and power functions in order to combine the principal investigator's previous work on decidability, which dealt only with functions from quasianalytic classes, and the work of Jones and Servi on decidability, which dealt only with power functions.
The origins of this project stem from two very classical questions which are pervasive throughout much of mathematics and its applications to science and engineering: 1) how to solve equations and inequalities, and related to this, how to determine the truth or falsity of statements built up through the use of equations and inequalities and also logical operations; 2) how to compute and study properties of functions defined by integral formulas. The subanalytic preparation theorem shows that, at least in theory, a wide class of equations (which includes all polynomial equations) can be solved by radicals in a more liberal sense that allows the use of locally defined analytic functions in addition to arithmetic operations and radicals. One aim of this project is to obtain a new algorithmic proof of the preparation theorem which, incidentally, would also generalize the theorem to wider classes of functions. In addition to studying equations, the preparation theorem is an important tool for studying the asymptotic behavior and integrals of constructible functions, which is a class of functions that contains, in particular, all algebraic functions. The Fourier transform is an important operation used in many areas of mathematics and its applications, and it is defined by an integral formula. Another aim of the project is to the lay groundwork for a theory of integration that could be used to show that Fourier transforms of constructible functions have simple properties.
Two lines of investigation in the area of o-minimal structures were proposed for this project. The first proposed line of investigation was to continue the author’s collaboration with Raf Cluckers in developing the theory of integration of (real) constructible functions by studying questions concerning their L^p theory and multivariate harmonic analysis. (Constructible functions are members of function algebras generated from globally subanalytic functions and logarithms of positively-valued globally subanalytic functions.) Three papers were written in this regard: R. Cluckers and D. J. Miller, Locus of integrability, zero loci, and stability under integration for constructible functions on Euclidean spaces with Lebesgue measure, Int. Math. Res. Notices (2012), vol. 2012, 3182-3191 (first published online July 12, 2011 doi:10.1093/imrn/rnr133); R. Cluckers and D. J. Miller, Lebesgue Classes and Preparation of Real Constructible Functions, J. Func. Anal. 264 (2013), no. 7, 1599-1642. R. Cluckers and D. J. Miller, Uniform bounds on the decay rate of families of oscillatory integrals with a constructible amplitude function and a globally subanalytic phase function, currently under review. The work on the first paper was begun after the submission of this project proposal, but was completed prior to start of the funding period; the second and third papers were extensively worked upon during the funding period. The first paper studies L^1 classes of parameterized families of constructible, single-variable functions, and it served as a warm-up for the much more technical second paper. The second paper relates analysis (the study of L^p spaces) with analytic geometry (the study of zero loci of functions) in a general multivariate setting, and it also proves a related preparation theorem that serves as an important technical tool used by our third paper. The third paper shows that the decay rate at infinity of families of oscillatory integrals, of the type described in its title, may be bounded by a decaying power function, provided that the amplitude function is integrable and the phase function satisfies a natural condition we call the "hyperplane condition" (which is both necessary and sufficient to obtain such a decay). This theorem compares closely to a classical theorem of harmonic analysis, but unlike our theorem, the classical theorem imposes much stronger analytic assumptions upon the amplitude and phase functions which diminish its applicability to algebraic geometry. The third paper also proves that an integrable, single-variable, constructible function has an integrable Fourier transform if and only if it is continuous. The necessity of continuity is both well-known and elementary, but the sufficiency of continuity (for constructible functions) is new. The second proposed line of investigation was to show that Lion and Rolin’s preparation theorem for globally subanalytic functions generalizes to a certain quasianalytic setting. The proposed plan of attack was to obtain this from a modification of a local resolution of singularities procedure worked out by the principal investigator (PI) in his work on the decidablity of certain first-order theories. However, the PI determined through the input of other reviewers that the current form of the resolution procedure was much too technical and needs to be simplified before work on the preparation can proceed, so work on the preparation theorem was temporarily abandoned. Instead, a much improved proof of resolution of singularities was worked on. This work is far along although not complete, and is currently is currently being written up in the following incomplete preprint: 4. D.J. Miller, Resolution of singularities in a quasianalytic setting without coordinate division. This is a global, rather than a local, resolution of singularities procedure which is proven in a simple and geometrically clear manner. Also, the proof is conducted under a weaker axiomatic setting than has heretofore been considered, and this enables the resolution procedure to be applied to the class of Pfaffian functions. Because of the importance of these functions to model theory and number theory, the PI deems this to be a more significant result than the originally proposed preparation theorem, and it also greatly increases the chances of eventually obtaining a preparation theorem for functions that are first-order definable from the Pfaffian functions. In addition to the above research work, the PI also had the opportunity to serve in an informal mentoring role towards an M.S. Mathematics student at his university (in conjunction with another faculty member, who was her thesis advisor). The student did a reading course with the PI in measure theory and integration , during which the PI related his research work to certain assigned exercises about L^p spaces. As a part of the course, the student gave a small number of oral presentations, and in turn, the PI gave a seminar-style talk on his research on L^p spaces. The PI also served on the student’s thesis committee and wrote recommendation letter for her to support her successful admittance to a Ph.D. program in statistics at another university.
