We prove quantitative estimates for the decay of the Fourier transform of the Riesz potential of measures that are in homogeneous Besov spaces of negative exponent: \begin{ align*} \|\widehat{ I_{ \alpha}\mu}\|_{ L^{ p, \infty}} \leq C \|\mu\|_{ M_b}^{ \frac{ 1}{ 2}}\left(\sup_{ t>0} t^{ \frac{ d-\beta}{ 2}}\|p_{ t}\ast \mu\|_{ \infty}\right)^{ \frac{ 1}{ 2}}, \end{ align*} where $p=\frac{ 2d}{ 2\alpha+\beta}$ with $\beta \in (0,d)$ and $I_\alpha \mu$ is the Riesz potential of $\mu$ of order $\alpha \in ((d-\beta)/2,d-\beta/2)$. Our results are naturally applicable to the Morrey space $\mathcal{ M}^{ \beta}$, including for example the Frostman measure $\mu_K$ of any compact set $K$ with $0<\mathcal{ H}^\beta(K)<+\infty$ for some $\beta \in (0,d]$. When $\mu=D\chi_E$ for $\chi_E \in \operatorname*{ BV}(\mathbb{ R}^d)$, $\alpha =1$, and $\beta=d-1$, our results extend the work of Herz and Ko--Lee. We provide examples which show the sharpness of our results.
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