Hilbert transforms associated with plane curves

Nagel, Alexander; Wainger, Stephen

doi:10.1090/S0002-9947-1976-0423010-8

Hilbert transforms associated with plane curves
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by Alexander Nagel and Stephen Wainger PDF

Trans. Amer. Math. Soc. 223 (1976), 235-252 Request permission

Abstract:

Let $(t,\gamma (t))$ be a plane curve. Set ${H_\gamma }f(x,y) = \text {p.v.}\;\smallint f(x - t,y - \gamma (t))dt/t$ for $f \in C_0^\infty ({R^2})$. For a large class of curves, the authors prove ${\left \| {{H_\gamma }f} \right \|_p} \leqslant {A_p}{\left \| f \right \|_p},5/3 < p < 5/2$. Various examples are given to show that some condition on the curve $(t,\gamma (t))$ is necessary.

References

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