On Dilations of Fourier Multipliers on Weighted Lebesgue Spaces

For a given \(p\in (1,\infty )\) , we present an example of a Muckenhoupt weight \(w\in A_p\) and a Fourier multiplier a on the weighted Lebesgue space \(L^p(w)\) such that, for any \(\tau \in \mathbb {R}\setminus \{0,1\}\) , the dilation of a given by \(a_\tau (\xi )=a(\tau \xi )\) is not a Fourier multiplier on \(L^p(w)\) .