Dilations associated to flat curves (Q1174948)

The author has found a useful family of dilations to curves in which a curvature condition fails. The dilations are used in the proof of the following theorem in two ways: Theorem: Assume \(\gamma(t)\) is odd and convex for \(t>0\). Then if for some \(\epsilon>0\), \(h'(t)\geq\epsilon h(t)/t\), \[ \| H_ \Gamma f\|_{L^ p}\leq A(p,\Gamma)\| f\|_{L^ p}, \quad 1<p<\infty, \quad\hbox{and} \quad \| M_ \Gamma f\|_{L^ p}\leq A(p,\Gamma)\| f\|_{L^ p}, \quad 1<p<\infty. \] The first application is to obtain uniform decay estimates for measures supported on \(\gamma(t)\), and the second is to be able to develop a Caldéron-Zygmund theory.