Let $A\subset [0,1]$ be closed and let $\tilde{A}:=\{ x\in \mathbb{R}^n : \|x\|\in A\}$. Assume $\nu$ is a finite measure supported on $\tilde{A}$ s.t. its Fourier transform decays as $\frac{n-1+\alpha}{2}$. In symbols
$$ \hat{\nu}(\xi) = \int e^{-i\langle \xi, x \rangle}d\nu(x) $$ $$ | \hat{\nu}(\xi)| \le \|\xi\|^{-\frac{n-1+\alpha}{2}}$$
Let $\mu$ be the pushforward measure of $\nu$ on $[0,1]$ via $\|\cdot\|$, i.e. $$ \mu([a,b])= \nu(\{x\in \mathbb{R}^n : \|x\| \in [a,b] \} )$$ What can I conclude on the decay of $\mu$? I would expect that $ | \hat{\mu}(t)| \le |t|^{-\frac{\alpha}{2}}$ but I'm having a hard time in trying to prove it. If $\nu$ were radial I should be able to get to the conclusion using Bessel functions, but in the general case I don't know how to proceed. I thought I could consider the averages of $\hat{\nu}$ over the spheres, i.e. $$ f(\xi) = \int_{S^{n-1}} \hat{\nu}(\|\xi\| \theta) \, d\sigma^{n-1}(\theta) $$ but then I don't know how to conclude that $f$ is the Fourier transform of some measure supported in $\tilde{A}$.