Mathematical Sciences: Singular Integrals and Averages of Functions

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At the heart of harmonic analysis are two fundamental issues: how to decompose a function into elementary constituent parts and how to recover a function from those parts. The genesis for this mathematical framework is the Fourier series. In trying to understand how a function can be recovered from its Fourier series, one is led to questions of averages of functions and singular integral operators. It is the study of these concepts which forms the basis for this research. The genesis of this particular project lies in the classical real-variable result of Lebesgue which gives a precise solution of when one can differentiate an integral and recover the integrand. There is no complete analogue to this situation in higher dimensions since the method of passing to the limit allows many interpretations. This work focuses on a limiting procedure where one restricts functions to surfaces of various dimensions and either takes limits or makes maximizing estimates of approximate derivatives (averages). These tasks amount to estimating norms of singular integrals over surfaces. While there has been considerable progress made on over the past two decades, the main problems of describing the correct class of curves or surfaces along which a differentiation theory can be built remains daunting. The shape of the surface plays a critical role and this work will continue to clarify the conditions needed to produce a comprehensive theory. Although the general description of this research is phrased in the language of harmonic analysis, much of its influence derives from applications to partial differential equations.