Singular kernels adapted to curved flags

The classical theory of Calderon-Zugmund kernels in $\mathbb{R}^n$ exploits
the study of distributions that are almost invariant under the class of
isotropic dilations.
In this talk we consider convolution operators with distributions that behave
simply under a specified class of nonisotropic dilations in the plane.
The prototypical example being the operator of Hilbert transform along the
parabola in $\mathbb{R}^2$.
Using machinery of flag kernels and multipliers,
developed by Nagel, Ricci, Stein,
we describe the form of the multiplier for such operators, and,
as a corollary, prove their $L^p$ boundedness for $1