Journal of the
Korean Mathematical Society JKMS

Let $\gamma : I \rightarrow \mathbb{R}^{2}$ be a sufficiently smooth curve and $\sigma_{\gamma}$ be the %corresponding affine arclength measure supported on $\gamma$. In this paper, we study the $L^{p}-$improving properties of the convolution operators $T_{\sigma_{\gamma}}$ associated with $\sigma_{\gamma}$ for various curves $\gamma$. Optimal results are obtained for all finite type plane curves and homogeneous curves (possibly blowing up at the origin). As an attempt to extend this result to infinitely flat curves we give an example of a family of flat curves whose affine arclength measure has the same $L^{p}$-improvement property. All of these results will be based on uniform estimates of damping oscillatory integrals.

Keywords: convolution operator, affine arclength measure

MSC numbers: Primary 42B15; Secondary 42B20