Endpoint bounds for quasiradial Fourier multipliers

We consider quasiradial Fourier multipliers, i.e. multipliers of the form $m(a(\xi ))$ for a class of distance functions $a$. We give a necessary and sufficient condition for the multiplier transformations to be bounded on $L^p$ for a certain range of $p$. In addition, when $m$ is compactly supported in $(0,\mathrm{\infty })$, we give a similar result for associated maximal operators.