𝐿^{𝑝}-boundedness for the Hilbert transform and maximal operator along a class of nonconvex curves

Bez, Neal

doi:10.1090/S0002-9939-06-08603-5

$L^p$-boundedness for the Hilbert transform and maximal operator along a class of nonconvex curves
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Proc. Amer. Math. Soc. 135 (2007), 151-161 Request permission

Abstract:

Some sufficient conditions on a real polynomial $P$ and a convex function $\gamma$ are given in order for the Hilbert transform and maximal operator along $(t,P(\gamma (t)))$ to be bounded on $L^p$, for all $p$ in $(1,\infty )$, with bounds independent of the coefficients of $P$. The same conclusion is shown to hold for the corresponding hypersurface in $\mathbb {R}^{d+1}$ $(d \geq 2)$ under weaker hypotheses on $\gamma$.

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