On the Fourier transform of measures in Besov spaces

Abstract: 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.