Professor
Ph.D., McMaster University (1987)
Specializations:
Weighted Norm Inequalities, Harmonic Analysis, Production Scheduling
Current research interests
Hardy-type inequalities, Weighted Fourier inequalities, Weighted Hardy Spaces.
Telephone: 519-661-3638 x 86535
Fax: 519-661-3610
E-mail: sinnamon@uwo.ca
Research Profile: Integrals and Estimates.
Integration is the unifying concept of Analysis. Our various ideas of quantity; length, area, temperature, probability, electrical charge, and a great many others; are made precise using integration. Estimates are comparisons between integrals, our method of relating one to another. Where integrals represent measurable quantities, estimates give us new truths about what we have measured. Estimates also give us information about quantities we can not measure directly. Results involving integrals and estimates are central to many areas of mathematics: Harmonic Analysis, Real Analysis, Inequalities, Function Spaces, Probability Theory and others. My work is in Harmonic Analysis.
The Fourier Transform converts information about the amplitude of a wave to information about its frequency. Although estimates of the integral defining the Fourier transform have been intensively studied for well over a century I have developed new Fourier estimates, the first of their kind in thirty years. They are based on new, very sensitive techniques that take into account both the amplitude and frequency. In addition to exploring the limits of this new technique there is now a great deal of work to be done to incorporate the new estimates into the huge body of existing Fourier Transform knowledge.
I revived and reworked the level function construction from the 1950's in order to prove, in 1987, the last outstanding case of the weighted Hardy inequality. The construction improves estimates involving decreasing functions. I have shown that the method applies to functions in all rearrangement invariant spaces and that the other main method of working with decreasing functions applies to almost the same extent. This work involves not only integrals and estimates but also the theory of function spaces, further bridging the narrowing gap between the two.
The Lebesgue norms are integrals which represent different notions of the size of a function and Schur's Lemma is a standard tool to generate Lebesgue norm estimates. My sharp converse of Schur's Lemma provides the best possible Lebesgue norm estimates for a large class of transforms and sets up a common framework for the study of all positive integral transformations.