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

Ziesler, S.

doi:10.1090/S0002-9939-1994-1213872-5

$L^ p$-boundedness of the Hilbert transform and maximal function associated to flat plane curves
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Proc. Amer. Math. Soc. 122 (1994), 1035-1043 Request permission

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.

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