Convolution operators with the affine arclength measure on plane curves

摘要

Let $gamma : I ightarrow mathbb{R}^{2}$ 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.