I'm reading a paper where the following inequality appears. $ \| \widehat{f} \|^2_{L^2(d\mu)} \leq \| f \ast \widehat{\mu} \|_p \| f \|_{p^\prime} $ where $f$ is a real-valued measurable function on $\mathbb{R}^n$, $\mu$ is a positive measure on $\mathbb{R}^n$, and $\frac{1}{p} + \frac{1}{p^{\prime}} = 1$. I think $\| \cdot \|_p$ and $\| \cdot \|_{p^{\prime}}$ are with respect to Lebesgue measure.
$ \widehat{\mu}(\xi) = \int e^{-2 \pi i x \xi} d\mu(x) $
I feel like this should be a consequence of Hölder's inequality and some identities relating convolution and the Fourier transform, but I can't figure it out.
Can someone please help?