The Lp- Lp mapping properties of convolution operators with the affine arclength measure on space curves

Youngwoo Choi
2003 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
more » ... 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 [3] 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.
doi:10.1017/s144678870000375x fatcat:ku763ukccjabdhitomho6xteqq