Abstract Oberlin This is a project in Fourier analysis. It is concerned with problems related to certain operators and to certain oscillatory integrals which are naturally associated with those operators. The operators are given by convolution with measures on curves in Euclidean spaces. In the simplest case the oscillatory integrals are one-dimensional integrals with an exponential integrand having polynomial phase function. These integrals arise naturally as the Fourier transforms of the measures defining the convolution operators. The questions of interest here have been fairly well understood in dimensions 2 and 3 since about 1985. The investigator has recently had some success with these problems in 4 dimensions. The goal of this project is to continue that work by extending the range and scope of the methods employed in dimension 4. Pure mathematics is traditionally divided into the areas of analysis, algebra, and topology. This is a project in analysis. Very roughly, the roots of analysis are to be found in calculus. (The roots of algebra are found in high school algebra, and those of topology are in geometry.) The objects of study in calculus are functions, derivatives, and integrals. The derivative of a function is an extremely important tool- when it exists. But not all functions have derivatives. The existence of a function's derivative is tied up with the idea of that function's smoothness. A smoothing operator is a device which transforms a function into a closely related but smoother function. (Applications of mathematics to the real world, e.g., problems in fluid mechanics like airplane design, almost always make the tacit assumption that the functions involved possess a certain degree of smoothness. When, as is often the case, the actual function is not that smooth, it must first be passed through a smoothing operator. Smoothing operators are also extremely useful in communications theory, where they are associated with the processes of noise removal and image enhan cement.) Most smoothing operators are of a type known as convolution operators. The motivation for this project is the desire to understand better certain of these convolution operators. The oscillatory integrals of the title are just tools which aid in this understanding.
