Classes of singular integrals along curves and surfaces

This paper is concerned with singular convolution operators in $\mathbb{R}^{d}$, $d\ge 2$, with convolution kernels supported on radial surfaces $y_{d}=\Gamma (|y'|)$. We show that if $\Gamma (s)=\log s$, then $L^{p}$ boundedness holds if and only if $p=2$. This statement can be reduced to a similar statement about the multiplier $m(\tau ,\eta )=|\tau |^{-i\eta }$ in $\mathbb{R}^{2}$. We also construct smooth $\Gamma$ for which the corresponding operators are bounded for $p_{0}<p\le 2$ but unbounded for $p\le p_{0}$, for given $p_{0}\in [1,2)$. Finally we discuss some examples of singular integrals along convex curves in the plane, with odd extensions.