| In this thesis we study operator of the form T&d5;fx =-infinityinfinityf x-ctc tdt, where c is a smooth Rn curve with non-zero torsion and c is some smooth cut-off function. We find that the operator T˜n becomes a pretty simple operator, namely a singular integral operator with a kernel which has integrable singularities along a codimension 1 sub-variety. Moreover, we can indeed investigate Lr class of this kernel and therefore obtain an Lp-Lq estimate for T˜ n. We also take this opportunity to discuss L p-Lq property of T˜ for various c.;The iterates of T˜n lead to a natural problem, which is the problem of describing the singularities on the unit sphere of the kernel K1*K2&vbm0;Sn-1 in terms of K1&vbm0;Sn-1 and K2&vbm0;Sn-1 . This is an interesting multi-linear formula, the singularities of which we fully analyze when n = 2. |
