Stability and Finiteness Problems in Analysis

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.