Endpoint bounds for quasiradial Fourier multipliers

We consider quasiradial Fourier multipliers, i.e. multipliers of the form $m(a(ξ))$ 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 $(0,\infty)$, we give a similar result for associated maximal operators.

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