Stability and Finiteness Problems in Analysis

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

The first question motivating the subject of our project is the following: Let f be a holomorphic function in a neighborhood of the origin of affine n-space. Can the singularities of f(z)=0 get "worse" under small perturbations of the function f? By "get worse" we mean: Can the critical exponent of f get smaller, where the critical exponent is the supremum of all exponents d for which the absolute value of f raised to the power -d is locally integrable. This question, and its effective counterpart, play an important role in proving the existence of Kahler-Einstein metrics on certain Fano manifolds (as has been recently demonstrated by Demailly and Kollar). We plan to apply our method of "algebraic estimates" to this problem. A related question which we will pursue is the following: Let R be a rational function with complex coefficients. How do we decide if R is in L^p? To what extent is the L^p norm of R a continuous function of its coefficients? The third motivating question concerns decay rates of oscillatory integral operators is one which has attracted much aattention is recent years: What is the best decay rate of such operators? We have been able to attack this problem in the multilinear one-dimensional case when the phase function is a polynomial, and have succeeded in finding the best decay rate (modulo logarithimic terms). Our method employs in an improved version of the curved trapezoid technique developed by Phong-Stein. We plan to apply this method to the higher dimensional case and in the case of damped operators. The main themes of this proposal - oscillatory and singular integrals and the method of stationary phase - are central to the field of classical analysis, with foundational results dating back to the nineteenth century: Harmonic Analysis plays a critical role in the solution of wide spectrum of problems in physics and applied mathematics - solving the heat equation, wave equation, Laplace equation, Schrodinger equation all make use of the Fourier analysis technique. The use of harmonic analysis in X-ray diffraction is indispensible to determining the structure of large molecules (such as DNA), and the Fourier transform method in signal processing is at the core of much of the modern technology involving the transfer of information by electronic means. In modern applications to a variety of questions, traditional techniques do not suffice and the need for a more general theory has arisen. In particular, the issues of bounds and stability for oscillatory integrals and operators, and the related problem of regularity of Radon transforms have been the focus of much recent work. The principal investigator, working with D.H. Phong (Columbia University) and Elias M. Stein (Princeton University), plans to continue investigating this circle of problems using the tools from geometry and analysis which were recently developed in our joint work.