$L^ p$-boundedness of the Hilbert transform and maximal function associated to flat plane curves

Abstract:

We give a sufficient condition for the Hilbert transform and maximal function associated to a flat plane convex curve $\Gamma (t) = (t,\gamma (t))$ to be bounded on ${L^p},1 < p < \infty$. Our result includes the previously known sufficient conditions, i.e., $\gamma ’$ doubling or h, defined by $h(t) = t\gamma ’(t) - \gamma (t),t > 0$, infinitesimally doubling, as special cases.