The Lp- Lp mapping properties of convolution operators with the affine arclength measure on space curves
Journal of the Australian Mathematical Society
The L p -improving properties of convolution operators with measures supported on space curves have been studied by various authors. If the underlying curve is non-degenerate, the convolution with the (Euclidean) arclength measure is a bounded operator from L 3/2 (R 3 ) into L 2 (R 3 ). Drury suggested that in case the underlying curve has degeneracies the appropriate measure to consider should be the affine arclength measure and he obtained a similar result for homogeneous curves t i-> (/, t 2
... urves t i-> (/, t 2 , t k ), / > 0 for k > 4. This was further generalized by Pan to curves t i->-(/, t k ,«'), / > 0 for 1 < Jt < /, it + / > 5. In this article, we will extend Pan's result to (smooth) compact curves of finite type whose tangents never vanish. In addition, we give an example of a flat curve with the same mapping properties. 2000 Mathematics subject classification: primary 42B15; secondary 42B10. Keywords and phrases: convolution, affine arclength measure, flat curve. Drury  suggested that in case the underlying curve has degeneracies the appropriate measure to consider should be the affine arclength measure and obtained This work is based on a research during the author's stay at Department of Mathematics, University of Wisconsin-Madison. The author would like to express his gratitude to both the university and the department for providing excellent environment for research.