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.