On the Fourier transform of measures in Besov spaces

We prove quantitative estimates for the decay of the Fourier transform of the Riesz potential of measures that are in homogeneous Besov spaces of the negative exponent: (Formula presented.) where (Formula presented.) with (Formula presented.) and (Formula presented.) is the Riesz potential of (Formula presented.) of order (Formula presented.). Our results are naturally applicable to the Morrey space (Formula presented.), including for example the Frostman measure (Formula presented.) of any compact set (Formula presented.) with (Formula presented.) for some (Formula presented.). When (Formula presented.) for (Formula presented.), (Formula presented.), and (Formula presented.), our results extend the work of Herz and Ko–Lee. We provide examples which show the sharpness of our results.