Hilbert Transforms Associated With Plane Curves

Let <!-- MATH $(t,\gamma (t))$ --> be a plane curve. Set <!-- MATH ${H_\gamma }f(x,y) = \text{p.v.}\;\smallint f(x - t,y - \gamma (t))dt/t$ --> p.v. for <!-- MATH $f \in C_0^\infty ({R^2})$ --> . For a large class of curves, the authors prove <!-- MATH ${\left\| {{H_\gamma }f} \right\|_p} \leqslant {A_p}{\left\| f \right\|_p},5/3 < p < 5/2$ --> <img width="297" height="41" align="MIDDLE" border="0" src="images/img5.gif" alt="$ {\left\Vert {{H_\gamma }f} \right\Vert _p} \leqslant {A_p}{\left\Vert f \right\Vert _p},5/3 < p < 5/2$">. Various examples are given to show that some condition on the curve <!-- MATH $(t,\gamma (t))$ --> is necessary.