a1 Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada H3A 2K6
This article deals with several related questions in harmonic analysis which are well understood for non-degenerate curves in n, but poorly understood in the degenerate case. These questions invariably involve a positive ‘reference’ measure on the curve. In the non-degenerate case the choice of measure is not particularly critical and it is usually taken to be the Euclidean arclength measure. Since the questions considered here are invariant under the group of affine motions (of determinant 1), the correct choice of reference measure is the affine arclength measure. We refer the reader to Guggenheimer  for information on affine geometry. When the curve has degeneracies, the choice of measure becomes critical and it is the affine arclength measure which yields the most powerful results. From the Euclidean point of view the affine arclength measure has correspondingly little mass near the degeneracies and thus compensates automatically for the poor behaviour there. This principle should also be valid for general submanifolds of n but alas the affine geometry of submanifolds is itself not well understood in general.
(Received October 10 1989)