J. Aust. Math. Soc.  75 (2003), 247-261
The $L^p$-$L^q$ mapping properties of convolution operators with the affine arclength measure on space curves

Youngwoo Choi
  Department of Mathematics
  Ajou University
  Suwon 442-749
  Korea
  youngwoo@ajou.ac.kr


Abstract
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 $L^{3/2}(\mathbb{R}^3)$ into $L^2(\mathbb{R}^3)$. 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 $t\mapsto (t,t^2,t^k)$, $t > 0$ for $k\ge 4$. This was further generalized by Pan to curves $t\mapsto (t,t^k,t^l)$, $t > 0$ for $1 < k < l$, $k + l \ge 5$. 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.
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